[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝² : TopologicalSpace α
inst✝¹ : T1Space α
inst✝ : Countable α
⊢ borel α = ⊤
[PROOFSTEP]
refine' top_le_iff.1 fun s _ => biUnion_of_singleton s ▸ _
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝² : TopologicalSpace α
inst✝¹ : T1Space α
inst✝ : Countable α
s : Set α
x✝ : MeasurableSet s
⊢ MeasurableSet (⋃ (x : α) (_ : x ∈ s), {x})
[PROOFSTEP]
apply MeasurableSet.biUnion s.to_countable
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝² : TopologicalSpace α
inst✝¹ : T1Space α
inst✝ : Countable α
s : Set α
x✝ : MeasurableSet s
⊢ ∀ (b : α), b ∈ s → MeasurableSet {b}
[PROOFSTEP]
intro x _
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝² : TopologicalSpace α
inst✝¹ : T1Space α
inst✝ : Countable α
s : Set α
x✝ : MeasurableSet s
x : α
a✝ : x ∈ s
⊢ MeasurableSet {x}
[PROOFSTEP]
apply MeasurableSet.of_compl
[GOAL]
case h
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝² : TopologicalSpace α
inst✝¹ : T1Space α
inst✝ : Countable α
s : Set α
x✝ : MeasurableSet s
x : α
a✝ : x ∈ s
⊢ MeasurableSet {x}ᶜ
[PROOFSTEP]
apply GenerateMeasurable.basic
[GOAL]
case h.a
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝² : TopologicalSpace α
inst✝¹ : T1Space α
inst✝ : Countable α
s : Set α
x✝ : MeasurableSet s
x : α
a✝ : x ∈ s
⊢ {x}ᶜ ∈ {s | IsOpen s}
[PROOFSTEP]
exact isClosed_singleton.isOpen_compl
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u✝ : Set α
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u : Set α
hu : TopologicalSpace.IsOpen u
⊢ MeasurableSet u
[PROOFSTEP]
rw [hs] at hu
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u✝ : Set α
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u : Set α
hu : TopologicalSpace.IsOpen u
⊢ MeasurableSet u
[PROOFSTEP]
induction hu
[GOAL]
case basic
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝¹ t✝ u✝ : Set α
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u s✝ : Set α
a✝ : s✝ ∈ s
⊢ MeasurableSet s✝
case univ
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u✝ : Set α
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u : Set α
⊢ MeasurableSet univ
case inter
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝¹ t✝¹ u✝ : Set α
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u s✝ t✝ : Set α
a✝¹ : GenerateOpen s s✝
a✝ : GenerateOpen s t✝
a_ih✝¹ : MeasurableSet s✝
a_ih✝ : MeasurableSet t✝
⊢ MeasurableSet (s✝ ∩ t✝)
case sUnion
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u✝ : Set α
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u : Set α
S✝ : Set (Set α)
a✝ : ∀ (s_1 : Set α), s_1 ∈ S✝ → GenerateOpen s s_1
a_ih✝ : ∀ (s_1 : Set α), s_1 ∈ S✝ → MeasurableSet s_1
⊢ MeasurableSet (⋃₀ S✝)
[PROOFSTEP]
case basic u hu => exact GenerateMeasurable.basic u hu
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u✝¹ : Set α
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u✝ u : Set α
hu : u ∈ s
⊢ MeasurableSet u
[PROOFSTEP]
case basic u hu => exact GenerateMeasurable.basic u hu
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u✝¹ : Set α
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u✝ u : Set α
hu : u ∈ s
⊢ MeasurableSet u
[PROOFSTEP]
exact GenerateMeasurable.basic u hu
[GOAL]
case univ
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u✝ : Set α
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u : Set α
⊢ MeasurableSet univ
case inter
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝¹ t✝¹ u✝ : Set α
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u s✝ t✝ : Set α
a✝¹ : GenerateOpen s s✝
a✝ : GenerateOpen s t✝
a_ih✝¹ : MeasurableSet s✝
a_ih✝ : MeasurableSet t✝
⊢ MeasurableSet (s✝ ∩ t✝)
case sUnion
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u✝ : Set α
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u : Set α
S✝ : Set (Set α)
a✝ : ∀ (s_1 : Set α), s_1 ∈ S✝ → GenerateOpen s s_1
a_ih✝ : ∀ (s_1 : Set α), s_1 ∈ S✝ → MeasurableSet s_1
⊢ MeasurableSet (⋃₀ S✝)
[PROOFSTEP]
case univ => exact @MeasurableSet.univ α (generateFrom s)
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u✝ : Set α
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u : Set α
⊢ MeasurableSet univ
[PROOFSTEP]
case univ => exact @MeasurableSet.univ α (generateFrom s)
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u✝ : Set α
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u : Set α
⊢ MeasurableSet univ
[PROOFSTEP]
exact @MeasurableSet.univ α (generateFrom s)
[GOAL]
case inter
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝¹ t✝¹ u✝ : Set α
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u s✝ t✝ : Set α
a✝¹ : GenerateOpen s s✝
a✝ : GenerateOpen s t✝
a_ih✝¹ : MeasurableSet s✝
a_ih✝ : MeasurableSet t✝
⊢ MeasurableSet (s✝ ∩ t✝)
case sUnion
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u✝ : Set α
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u : Set α
S✝ : Set (Set α)
a✝ : ∀ (s_1 : Set α), s_1 ∈ S✝ → GenerateOpen s s_1
a_ih✝ : ∀ (s_1 : Set α), s_1 ∈ S✝ → MeasurableSet s_1
⊢ MeasurableSet (⋃₀ S✝)
[PROOFSTEP]
case inter s₁ s₂ _ _ hs₁ hs₂ => exact @MeasurableSet.inter α (generateFrom s) _ _ hs₁ hs₂
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u✝ : Set α
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u s₁ s₂ : Set α
a✝¹ : GenerateOpen s s₁
a✝ : GenerateOpen s s₂
hs₁ : MeasurableSet s₁
hs₂ : MeasurableSet s₂
⊢ MeasurableSet (s₁ ∩ s₂)
[PROOFSTEP]
case inter s₁ s₂ _ _ hs₁ hs₂ => exact @MeasurableSet.inter α (generateFrom s) _ _ hs₁ hs₂
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u✝ : Set α
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u s₁ s₂ : Set α
a✝¹ : GenerateOpen s s₁
a✝ : GenerateOpen s s₂
hs₁ : MeasurableSet s₁
hs₂ : MeasurableSet s₂
⊢ MeasurableSet (s₁ ∩ s₂)
[PROOFSTEP]
exact @MeasurableSet.inter α (generateFrom s) _ _ hs₁ hs₂
[GOAL]
case sUnion
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u✝ : Set α
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u : Set α
S✝ : Set (Set α)
a✝ : ∀ (s_1 : Set α), s_1 ∈ S✝ → GenerateOpen s s_1
a_ih✝ : ∀ (s_1 : Set α), s_1 ∈ S✝ → MeasurableSet s_1
⊢ MeasurableSet (⋃₀ S✝)
[PROOFSTEP]
case sUnion f hf ih =>
  rcases isOpen_sUnion_countable f (by rwa [hs]) with ⟨v, hv, vf, vu⟩
  rw [← vu]
  exact @MeasurableSet.sUnion α (generateFrom s) _ hv fun x xv => ih _ (vf xv)
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u✝ : Set α
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u : Set α
f : Set (Set α)
hf : ∀ (s_1 : Set α), s_1 ∈ f → GenerateOpen s s_1
ih : ∀ (s_1 : Set α), s_1 ∈ f → MeasurableSet s_1
⊢ MeasurableSet (⋃₀ f)
[PROOFSTEP]
case sUnion f hf ih =>
  rcases isOpen_sUnion_countable f (by rwa [hs]) with ⟨v, hv, vf, vu⟩
  rw [← vu]
  exact @MeasurableSet.sUnion α (generateFrom s) _ hv fun x xv => ih _ (vf xv)
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u✝ : Set α
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u : Set α
f : Set (Set α)
hf : ∀ (s_1 : Set α), s_1 ∈ f → GenerateOpen s s_1
ih : ∀ (s_1 : Set α), s_1 ∈ f → MeasurableSet s_1
⊢ MeasurableSet (⋃₀ f)
[PROOFSTEP]
rcases isOpen_sUnion_countable f (by rwa [hs]) with ⟨v, hv, vf, vu⟩
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u✝ : Set α
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u : Set α
f : Set (Set α)
hf : ∀ (s_1 : Set α), s_1 ∈ f → GenerateOpen s s_1
ih : ∀ (s_1 : Set α), s_1 ∈ f → MeasurableSet s_1
⊢ ∀ (s : Set α), s ∈ f → IsOpen s
[PROOFSTEP]
rwa [hs]
[GOAL]
case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u✝ : Set α
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u : Set α
f : Set (Set α)
hf : ∀ (s_1 : Set α), s_1 ∈ f → GenerateOpen s s_1
ih : ∀ (s_1 : Set α), s_1 ∈ f → MeasurableSet s_1
v : Set (Set α)
hv : Set.Countable v
vf : v ⊆ f
vu : ⋃₀ v = ⋃₀ f
⊢ MeasurableSet (⋃₀ f)
[PROOFSTEP]
rw [← vu]
[GOAL]
case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u✝ : Set α
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u : Set α
f : Set (Set α)
hf : ∀ (s_1 : Set α), s_1 ∈ f → GenerateOpen s s_1
ih : ∀ (s_1 : Set α), s_1 ∈ f → MeasurableSet s_1
v : Set (Set α)
hv : Set.Countable v
vf : v ⊆ f
vu : ⋃₀ v = ⋃₀ f
⊢ MeasurableSet (⋃₀ v)
[PROOFSTEP]
exact @MeasurableSet.sUnion α (generateFrom s) _ hv fun x xv => ih _ (vf xv)
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u✝ : Set α
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u : Set α
hu : u ∈ s
⊢ TopologicalSpace.IsOpen u
[PROOFSTEP]
rw [hs]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u✝ : Set α
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u : Set α
hu : u ∈ s
⊢ TopologicalSpace.IsOpen u
[PROOFSTEP]
exact GenerateOpen.basic _ hu
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
⊢ borel α = MeasurableSpace.generateFrom (range Iio)
[PROOFSTEP]
refine' le_antisymm _ (generateFrom_le _)
[GOAL]
case refine'_1
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
⊢ borel α ≤ MeasurableSpace.generateFrom (range Iio)
[PROOFSTEP]
rw [borel_eq_generateFrom_of_subbasis (@OrderTopology.topology_eq_generate_intervals α _ _ _)]
[GOAL]
case refine'_1
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
⊢ MeasurableSpace.generateFrom {s | ∃ a, s = Ioi a ∨ s = Iio a} ≤ MeasurableSpace.generateFrom (range Iio)
[PROOFSTEP]
letI : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
[GOAL]
case refine'_1
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
⊢ MeasurableSpace.generateFrom {s | ∃ a, s = Ioi a ∨ s = Iio a} ≤ MeasurableSpace.generateFrom (range Iio)
[PROOFSTEP]
have H : ∀ a : α, MeasurableSet (Iio a) := fun a => GenerateMeasurable.basic _ ⟨_, rfl⟩
[GOAL]
case refine'_1
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
⊢ MeasurableSpace.generateFrom {s | ∃ a, s = Ioi a ∨ s = Iio a} ≤ MeasurableSpace.generateFrom (range Iio)
[PROOFSTEP]
refine' generateFrom_le _
[GOAL]
case refine'_1
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
⊢ ∀ (t : Set α), t ∈ {s | ∃ a, s = Ioi a ∨ s = Iio a} → MeasurableSet t
[PROOFSTEP]
rintro _ ⟨a, rfl | rfl⟩ <;> [skip; apply H]
[GOAL]
case refine'_1
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
⊢ ∀ (t : Set α), t ∈ {s | ∃ a, s = Ioi a ∨ s = Iio a} → MeasurableSet t
[PROOFSTEP]
rintro _ ⟨a, rfl | rfl⟩
[GOAL]
case refine'_1.intro.inl
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
a : α
⊢ MeasurableSet (Ioi a)
[PROOFSTEP]
skip
[GOAL]
case refine'_1.intro.inr
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
a : α
⊢ MeasurableSet (Iio a)
[PROOFSTEP]
apply H
[GOAL]
case refine'_1.intro.inl
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
a : α
⊢ MeasurableSet (Ioi a)
[PROOFSTEP]
by_cases h : ∃ a', ∀ b, a < b ↔ a' ≤ b
[GOAL]
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
a : α
h : ∃ a', ∀ (b : α), a < b ↔ a' ≤ b
⊢ MeasurableSet (Ioi a)
[PROOFSTEP]
rcases h with ⟨a', ha'⟩
[GOAL]
case pos.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
a a' : α
ha' : ∀ (b : α), a < b ↔ a' ≤ b
⊢ MeasurableSet (Ioi a)
[PROOFSTEP]
rw [(_ : Ioi a = (Iio a')ᶜ)]
[GOAL]
case pos.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
a a' : α
ha' : ∀ (b : α), a < b ↔ a' ≤ b
⊢ MeasurableSet (Iio a')ᶜ
[PROOFSTEP]
exact (H _).compl
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
a a' : α
ha' : ∀ (b : α), a < b ↔ a' ≤ b
⊢ Ioi a = (Iio a')ᶜ
[PROOFSTEP]
simp [Set.ext_iff, ha']
[GOAL]
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
a : α
h : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b
⊢ MeasurableSet (Ioi a)
[PROOFSTEP]
rcases isOpen_iUnion_countable (fun a' : { a' : α // a < a' } => {b | a'.1 < b}) fun a' => isOpen_lt' _ with
  ⟨v, ⟨hv⟩, vu⟩
[GOAL]
case neg.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
a : α
h : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b
v : Set { a' // a < a' }
vu : ⋃ (i : { a' // a < a' }) (_ : i ∈ v), {b | ↑i < b} = ⋃ (i : { a' // a < a' }), {b | ↑i < b}
hv : Encodable ↑v
⊢ MeasurableSet (Ioi a)
[PROOFSTEP]
simp [Set.ext_iff] at vu
[GOAL]
case neg.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
a : α
h : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b
v : Set { a' // a < a' }
hv : Encodable ↑v
vu : ∀ (x : α), (∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 < x) ↔ ∃ a_1, a < a_1 ∧ a_1 < x
⊢ MeasurableSet (Ioi a)
[PROOFSTEP]
have : Ioi a = ⋃ x : v, (Iio x.1.1)ᶜ := by
  simp [Set.ext_iff]
  refine' fun x => ⟨fun ax => _, fun ⟨a', ⟨h, _⟩, ax⟩ => lt_of_lt_of_le h ax⟩
  rcases(vu x).2
      (by
        refine' not_imp_comm.1 (fun h => _) h
        exact ⟨x, fun b => ⟨fun ab => le_of_not_lt fun h' => h ⟨b, ab, h'⟩, lt_of_lt_of_le ax⟩⟩) with
    ⟨a', h₁, h₂⟩
  · exact ⟨a', h₁, le_of_lt h₂⟩
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
a : α
h : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b
v : Set { a' // a < a' }
hv : Encodable ↑v
vu : ∀ (x : α), (∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 < x) ↔ ∃ a_1, a < a_1 ∧ a_1 < x
⊢ Ioi a = ⋃ (x : ↑v), (Iio ↑↑x)ᶜ
[PROOFSTEP]
simp [Set.ext_iff]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
a : α
h : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b
v : Set { a' // a < a' }
hv : Encodable ↑v
vu : ∀ (x : α), (∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 < x) ↔ ∃ a_1, a < a_1 ∧ a_1 < x
⊢ ∀ (x : α), a < x ↔ ∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 ≤ x
[PROOFSTEP]
refine' fun x => ⟨fun ax => _, fun ⟨a', ⟨h, _⟩, ax⟩ => lt_of_lt_of_le h ax⟩
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
a : α
h : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b
v : Set { a' // a < a' }
hv : Encodable ↑v
vu : ∀ (x : α), (∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 < x) ↔ ∃ a_1, a < a_1 ∧ a_1 < x
x : α
ax : a < x
⊢ ∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 ≤ x
[PROOFSTEP]
rcases(vu x).2
    (by
      refine' not_imp_comm.1 (fun h => _) h
      exact ⟨x, fun b => ⟨fun ab => le_of_not_lt fun h' => h ⟨b, ab, h'⟩, lt_of_lt_of_le ax⟩⟩) with
  ⟨a', h₁, h₂⟩
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
a : α
h : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b
v : Set { a' // a < a' }
hv : Encodable ↑v
vu : ∀ (x : α), (∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 < x) ↔ ∃ a_1, a < a_1 ∧ a_1 < x
x : α
ax : a < x
⊢ ∃ a_1, a < a_1 ∧ a_1 < x
[PROOFSTEP]
refine' not_imp_comm.1 (fun h => _) h
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
a : α
h✝ : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b
v : Set { a' // a < a' }
hv : Encodable ↑v
vu : ∀ (x : α), (∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 < x) ↔ ∃ a_1, a < a_1 ∧ a_1 < x
x : α
ax : a < x
h : ¬∃ a_1, a < a_1 ∧ a_1 < x
⊢ ∃ a', ∀ (b : α), a < b ↔ a' ≤ b
[PROOFSTEP]
exact ⟨x, fun b => ⟨fun ab => le_of_not_lt fun h' => h ⟨b, ab, h'⟩, lt_of_lt_of_le ax⟩⟩
[GOAL]
case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
a : α
h : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b
v : Set { a' // a < a' }
hv : Encodable ↑v
vu : ∀ (x : α), (∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 < x) ↔ ∃ a_1, a < a_1 ∧ a_1 < x
x : α
ax : a < x
a' : α
h₁ : ∃ x, { val := a', property := (_ : a < a') } ∈ v
h₂ : a' < x
⊢ ∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 ≤ x
[PROOFSTEP]
exact ⟨a', h₁, le_of_lt h₂⟩
[GOAL]
case neg.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this✝ : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
a : α
h : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b
v : Set { a' // a < a' }
hv : Encodable ↑v
vu : ∀ (x : α), (∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 < x) ↔ ∃ a_1, a < a_1 ∧ a_1 < x
this : Ioi a = ⋃ (x : ↑v), (Iio ↑↑x)ᶜ
⊢ MeasurableSet (Ioi a)
[PROOFSTEP]
rw [this]
[GOAL]
case neg.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this✝ : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
a : α
h : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b
v : Set { a' // a < a' }
hv : Encodable ↑v
vu : ∀ (x : α), (∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 < x) ↔ ∃ a_1, a < a_1 ∧ a_1 < x
this : Ioi a = ⋃ (x : ↑v), (Iio ↑↑x)ᶜ
⊢ MeasurableSet (⋃ (x : ↑v), (Iio ↑↑x)ᶜ)
[PROOFSTEP]
skip
[GOAL]
case neg.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this✝ : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
a : α
h : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b
v : Set { a' // a < a' }
hv : Encodable ↑v
vu : ∀ (x : α), (∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 < x) ↔ ∃ a_1, a < a_1 ∧ a_1 < x
this : Ioi a = ⋃ (x : ↑v), (Iio ↑↑x)ᶜ
⊢ MeasurableSet (⋃ (x : ↑v), (Iio ↑↑x)ᶜ)
[PROOFSTEP]
apply MeasurableSet.iUnion
[GOAL]
case neg.intro.intro.intro.h
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this✝ : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
a : α
h : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b
v : Set { a' // a < a' }
hv : Encodable ↑v
vu : ∀ (x : α), (∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 < x) ↔ ∃ a_1, a < a_1 ∧ a_1 < x
this : Ioi a = ⋃ (x : ↑v), (Iio ↑↑x)ᶜ
⊢ ∀ (b : ↑v), MeasurableSet (Iio ↑↑b)ᶜ
[PROOFSTEP]
exact fun _ => (H _).compl
[GOAL]
case refine'_2
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
⊢ ∀ (t : Set α), t ∈ range Iio → MeasurableSet t
[PROOFSTEP]
rw [forall_range_iff]
[GOAL]
case refine'_2
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
⊢ ∀ (i : α), MeasurableSet (Iio i)
[PROOFSTEP]
intro a
[GOAL]
case refine'_2
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a : α
⊢ MeasurableSet (Iio a)
[PROOFSTEP]
exact GenerateMeasurable.basic _ isOpen_Iio
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
⊢ SecondCountableTopology α
[PROOFSTEP]
infer_instance
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
⊢ borel α = MeasurableSpace.generateFrom (range Iic)
[PROOFSTEP]
rw [borel_eq_generateFrom_Ioi]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
⊢ MeasurableSpace.generateFrom (range Ioi) = MeasurableSpace.generateFrom (range Iic)
[PROOFSTEP]
refine' le_antisymm _ _
[GOAL]
case refine'_1
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
⊢ MeasurableSpace.generateFrom (range Ioi) ≤ MeasurableSpace.generateFrom (range Iic)
[PROOFSTEP]
refine' MeasurableSpace.generateFrom_le fun t ht => _
[GOAL]
case refine'_1
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t✝ u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
t : Set α
ht : t ∈ range Ioi
⊢ MeasurableSet t
[PROOFSTEP]
obtain ⟨u, rfl⟩ := ht
[GOAL]
case refine'_1.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u✝ : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
u : α
⊢ MeasurableSet (Ioi u)
[PROOFSTEP]
rw [← compl_Iic]
[GOAL]
case refine'_1.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u✝ : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
u : α
⊢ MeasurableSet (Iic u)ᶜ
[PROOFSTEP]
exact (MeasurableSpace.measurableSet_generateFrom (mem_range.mpr ⟨u, rfl⟩)).compl
[GOAL]
case refine'_2
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
⊢ MeasurableSpace.generateFrom (range Iic) ≤ MeasurableSpace.generateFrom (range Ioi)
[PROOFSTEP]
refine' MeasurableSpace.generateFrom_le fun t ht => _
[GOAL]
case refine'_2
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t✝ u : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
t : Set α
ht : t ∈ range Iic
⊢ MeasurableSet t
[PROOFSTEP]
obtain ⟨u, rfl⟩ := ht
[GOAL]
case refine'_2.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u✝ : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
u : α
⊢ MeasurableSet (Iic u)
[PROOFSTEP]
rw [← compl_Ioi]
[GOAL]
case refine'_2.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u✝ : Set α
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
u : α
⊢ MeasurableSet (Ioi u)ᶜ
[PROOFSTEP]
exact (MeasurableSpace.measurableSet_generateFrom (mem_range.mpr ⟨u, rfl⟩)).compl
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α✝
α : Type u_6
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
hα : BorelSpace α
s : Set α
⊢ instMeasurableSpace = borel ↑s
[PROOFSTEP]
borelize α
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α✝
α : Type u_6
inst✝ : TopologicalSpace α
s : Set α
hα : BorelSpace α
this✝ : MeasurableSpace α := borel α
⊢ instMeasurableSpace = borel ↑s
[PROOFSTEP]
symm
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α✝
α : Type u_6
inst✝ : TopologicalSpace α
s : Set α
hα : BorelSpace α
this✝ : MeasurableSpace α := borel α
⊢ borel ↑s = instMeasurableSpace
[PROOFSTEP]
apply borel_comap
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α✝
α : Type u_6
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
h : OpensMeasurableSpace α
s : Set α
⊢ borel ↑s ≤ instMeasurableSpace
[PROOFSTEP]
rw [borel_comap]
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α✝
α : Type u_6
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
h : OpensMeasurableSpace α
s : Set α
⊢ MeasurableSpace.comap val (borel α) ≤ instMeasurableSpace
[PROOFSTEP]
exact comap_mono h.1
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
α : Type u_6
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
inst✝ : SecondCountableTopology α
⊢ CountablyGenerated α
[PROOFSTEP]
obtain ⟨b, bct, -, hb⟩ := exists_countable_basis α
[GOAL]
case intro.intro.intro
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
α : Type u_6
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
inst✝ : SecondCountableTopology α
b : Set (Set α)
bct : Set.Countable b
hb : IsTopologicalBasis b
⊢ CountablyGenerated α
[PROOFSTEP]
refine' ⟨⟨b, bct, _⟩⟩
[GOAL]
case intro.intro.intro
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
α : Type u_6
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
inst✝ : SecondCountableTopology α
b : Set (Set α)
bct : Set.Countable b
hb : IsTopologicalBasis b
⊢ inst✝² = MeasurableSpace.generateFrom b
[PROOFSTEP]
borelize α
[GOAL]
case intro.intro.intro
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
α : Type u_6
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
b : Set (Set α)
bct : Set.Countable b
hb : IsTopologicalBasis b
inst✝ : BorelSpace α
this✝ : MeasurableSpace α := borel α
⊢ borel α = MeasurableSpace.generateFrom b
[PROOFSTEP]
exact hb.borel_eq_generateFrom
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
h : IsGδ s
⊢ MeasurableSet s
[PROOFSTEP]
rcases h with ⟨S, hSo, hSc, rfl⟩
[GOAL]
case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
S : Set (Set α)
hSo : ∀ (t : Set α), t ∈ S → IsOpen t
hSc : Set.Countable S
⊢ MeasurableSet (⋂₀ S)
[PROOFSTEP]
exact MeasurableSet.sInter hSc fun t ht => (hSo t ht).measurableSet
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsOpen s → MeasurableSet (f ⁻¹' s)
⊢ Measurable f
[PROOFSTEP]
rw [‹BorelSpace γ›.measurable_eq]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsOpen s → MeasurableSet (f ⁻¹' s)
⊢ Measurable f
[PROOFSTEP]
exact measurable_generateFrom hf
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsClosed s → MeasurableSet (f ⁻¹' s)
⊢ Measurable f
[PROOFSTEP]
apply measurable_of_isOpen
[GOAL]
case hf
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsClosed s → MeasurableSet (f ⁻¹' s)
⊢ ∀ (s : Set γ), IsOpen s → MeasurableSet (f ⁻¹' s)
[PROOFSTEP]
intro s hs
[GOAL]
case hf
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsClosed s → MeasurableSet (f ⁻¹' s)
s : Set γ
hs : IsOpen s
⊢ MeasurableSet (f ⁻¹' s)
[PROOFSTEP]
rw [← MeasurableSet.compl_iff, ← preimage_compl]
[GOAL]
case hf
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsClosed s → MeasurableSet (f ⁻¹' s)
s : Set γ
hs : IsOpen s
⊢ MeasurableSet (f ⁻¹' sᶜ)
[PROOFSTEP]
apply hf
[GOAL]
case hf.a
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsClosed s → MeasurableSet (f ⁻¹' s)
s : Set γ
hs : IsOpen s
⊢ IsClosed sᶜ
[PROOFSTEP]
rw [isClosed_compl_iff]
[GOAL]
case hf.a
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsClosed s → MeasurableSet (f ⁻¹' s)
s : Set γ
hs : IsOpen s
⊢ IsOpen s
[PROOFSTEP]
exact hs
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsClosed s → Set.Nonempty s → s ≠ univ → MeasurableSet (f ⁻¹' s)
⊢ Measurable f
[PROOFSTEP]
apply measurable_of_isClosed
[GOAL]
case hf
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsClosed s → Set.Nonempty s → s ≠ univ → MeasurableSet (f ⁻¹' s)
⊢ ∀ (s : Set γ), IsClosed s → MeasurableSet (f ⁻¹' s)
[PROOFSTEP]
intro s hs
[GOAL]
case hf
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsClosed s → Set.Nonempty s → s ≠ univ → MeasurableSet (f ⁻¹' s)
s : Set γ
hs : IsClosed s
⊢ MeasurableSet (f ⁻¹' s)
[PROOFSTEP]
cases' eq_empty_or_nonempty s with h1 h1
[GOAL]
case hf.inl
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsClosed s → Set.Nonempty s → s ≠ univ → MeasurableSet (f ⁻¹' s)
s : Set γ
hs : IsClosed s
h1 : s = ∅
⊢ MeasurableSet (f ⁻¹' s)
[PROOFSTEP]
simp [h1]
[GOAL]
case hf.inr
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsClosed s → Set.Nonempty s → s ≠ univ → MeasurableSet (f ⁻¹' s)
s : Set γ
hs : IsClosed s
h1 : Set.Nonempty s
⊢ MeasurableSet (f ⁻¹' s)
[PROOFSTEP]
by_cases h2 : s = univ
[GOAL]
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsClosed s → Set.Nonempty s → s ≠ univ → MeasurableSet (f ⁻¹' s)
s : Set γ
hs : IsClosed s
h1 : Set.Nonempty s
h2 : s = univ
⊢ MeasurableSet (f ⁻¹' s)
[PROOFSTEP]
simp [h2]
[GOAL]
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsClosed s → Set.Nonempty s → s ≠ univ → MeasurableSet (f ⁻¹' s)
s : Set γ
hs : IsClosed s
h1 : Set.Nonempty s
h2 : ¬s = univ
⊢ MeasurableSet (f ⁻¹' s)
[PROOFSTEP]
exact hf s hs h1 h2
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
a : α
⊢ IsMeasurablyGenerated (𝓝 a)
[PROOFSTEP]
rw [nhds, iInf_subtype']
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
a : α
⊢ IsMeasurablyGenerated (⨅ (x : { s // s ∈ {s | a ∈ s ∧ IsOpen s} }), 𝓟 ↑x)
[PROOFSTEP]
refine' @Filter.iInf_isMeasurablyGenerated α _ _ _ fun i => _
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
a : α
i : { s // s ∈ {s | a ∈ s ∧ IsOpen s} }
⊢ IsMeasurablyGenerated (𝓟 ↑i)
[PROOFSTEP]
exact i.2.2.measurableSet.principal_isMeasurablyGenerated
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹⁶ : TopologicalSpace α
inst✝¹⁵ : MeasurableSpace α
inst✝¹⁴ : OpensMeasurableSpace α
inst✝¹³ : TopologicalSpace β
inst✝¹² : MeasurableSpace β
inst✝¹¹ : OpensMeasurableSpace β
inst✝¹⁰ : TopologicalSpace γ
inst✝⁹ : MeasurableSpace γ
inst✝⁸ : BorelSpace γ
inst✝⁷ : TopologicalSpace γ₂
inst✝⁶ : MeasurableSpace γ₂
inst✝⁵ : BorelSpace γ₂
inst✝⁴ : MeasurableSpace δ
ι : Type u_6
π : ι → Type u_7
inst✝³ : Countable ι
t' : (i : ι) → TopologicalSpace (π i)
inst✝² : (i : ι) → MeasurableSpace (π i)
inst✝¹ : ∀ (i : ι), SecondCountableTopology (π i)
inst✝ : ∀ (i : ι), OpensMeasurableSpace (π i)
⊢ OpensMeasurableSpace ((i : ι) → π i)
[PROOFSTEP]
constructor
[GOAL]
case borel_le
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹⁶ : TopologicalSpace α
inst✝¹⁵ : MeasurableSpace α
inst✝¹⁴ : OpensMeasurableSpace α
inst✝¹³ : TopologicalSpace β
inst✝¹² : MeasurableSpace β
inst✝¹¹ : OpensMeasurableSpace β
inst✝¹⁰ : TopologicalSpace γ
inst✝⁹ : MeasurableSpace γ
inst✝⁸ : BorelSpace γ
inst✝⁷ : TopologicalSpace γ₂
inst✝⁶ : MeasurableSpace γ₂
inst✝⁵ : BorelSpace γ₂
inst✝⁴ : MeasurableSpace δ
ι : Type u_6
π : ι → Type u_7
inst✝³ : Countable ι
t' : (i : ι) → TopologicalSpace (π i)
inst✝² : (i : ι) → MeasurableSpace (π i)
inst✝¹ : ∀ (i : ι), SecondCountableTopology (π i)
inst✝ : ∀ (i : ι), OpensMeasurableSpace (π i)
⊢ borel ((i : ι) → π i) ≤ MeasurableSpace.pi
[PROOFSTEP]
have :
  Pi.topologicalSpace =
    .generateFrom {t | ∃ (s : ∀ a, Set (π a)) (i : Finset ι), (∀ a ∈ i, s a ∈ countableBasis (π a)) ∧ t = pi (↑i) s} :=
  by rw [funext fun a => @eq_generateFrom_countableBasis (π a) _ _, pi_generateFrom_eq]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹⁶ : TopologicalSpace α
inst✝¹⁵ : MeasurableSpace α
inst✝¹⁴ : OpensMeasurableSpace α
inst✝¹³ : TopologicalSpace β
inst✝¹² : MeasurableSpace β
inst✝¹¹ : OpensMeasurableSpace β
inst✝¹⁰ : TopologicalSpace γ
inst✝⁹ : MeasurableSpace γ
inst✝⁸ : BorelSpace γ
inst✝⁷ : TopologicalSpace γ₂
inst✝⁶ : MeasurableSpace γ₂
inst✝⁵ : BorelSpace γ₂
inst✝⁴ : MeasurableSpace δ
ι : Type u_6
π : ι → Type u_7
inst✝³ : Countable ι
t' : (i : ι) → TopologicalSpace (π i)
inst✝² : (i : ι) → MeasurableSpace (π i)
inst✝¹ : ∀ (i : ι), SecondCountableTopology (π i)
inst✝ : ∀ (i : ι), OpensMeasurableSpace (π i)
⊢ topologicalSpace =
    TopologicalSpace.generateFrom {t | ∃ s i, (∀ (a : ι), a ∈ i → s a ∈ countableBasis (π a)) ∧ t = Set.pi (↑i) s}
[PROOFSTEP]
rw [funext fun a => @eq_generateFrom_countableBasis (π a) _ _, pi_generateFrom_eq]
[GOAL]
case borel_le
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹⁶ : TopologicalSpace α
inst✝¹⁵ : MeasurableSpace α
inst✝¹⁴ : OpensMeasurableSpace α
inst✝¹³ : TopologicalSpace β
inst✝¹² : MeasurableSpace β
inst✝¹¹ : OpensMeasurableSpace β
inst✝¹⁰ : TopologicalSpace γ
inst✝⁹ : MeasurableSpace γ
inst✝⁸ : BorelSpace γ
inst✝⁷ : TopologicalSpace γ₂
inst✝⁶ : MeasurableSpace γ₂
inst✝⁵ : BorelSpace γ₂
inst✝⁴ : MeasurableSpace δ
ι : Type u_6
π : ι → Type u_7
inst✝³ : Countable ι
t' : (i : ι) → TopologicalSpace (π i)
inst✝² : (i : ι) → MeasurableSpace (π i)
inst✝¹ : ∀ (i : ι), SecondCountableTopology (π i)
inst✝ : ∀ (i : ι), OpensMeasurableSpace (π i)
this :
  topologicalSpace =
    TopologicalSpace.generateFrom {t | ∃ s i, (∀ (a : ι), a ∈ i → s a ∈ countableBasis (π a)) ∧ t = Set.pi (↑i) s}
⊢ borel ((i : ι) → π i) ≤ MeasurableSpace.pi
[PROOFSTEP]
rw [borel_eq_generateFrom_of_subbasis this]
[GOAL]
case borel_le
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹⁶ : TopologicalSpace α
inst✝¹⁵ : MeasurableSpace α
inst✝¹⁴ : OpensMeasurableSpace α
inst✝¹³ : TopologicalSpace β
inst✝¹² : MeasurableSpace β
inst✝¹¹ : OpensMeasurableSpace β
inst✝¹⁰ : TopologicalSpace γ
inst✝⁹ : MeasurableSpace γ
inst✝⁸ : BorelSpace γ
inst✝⁷ : TopologicalSpace γ₂
inst✝⁶ : MeasurableSpace γ₂
inst✝⁵ : BorelSpace γ₂
inst✝⁴ : MeasurableSpace δ
ι : Type u_6
π : ι → Type u_7
inst✝³ : Countable ι
t' : (i : ι) → TopologicalSpace (π i)
inst✝² : (i : ι) → MeasurableSpace (π i)
inst✝¹ : ∀ (i : ι), SecondCountableTopology (π i)
inst✝ : ∀ (i : ι), OpensMeasurableSpace (π i)
this :
  topologicalSpace =
    TopologicalSpace.generateFrom {t | ∃ s i, (∀ (a : ι), a ∈ i → s a ∈ countableBasis (π a)) ∧ t = Set.pi (↑i) s}
⊢ MeasurableSpace.generateFrom {t | ∃ s i, (∀ (a : ι), a ∈ i → s a ∈ countableBasis (π a)) ∧ t = Set.pi (↑i) s} ≤
    MeasurableSpace.pi
[PROOFSTEP]
apply generateFrom_le
[GOAL]
case borel_le.h
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹⁶ : TopologicalSpace α
inst✝¹⁵ : MeasurableSpace α
inst✝¹⁴ : OpensMeasurableSpace α
inst✝¹³ : TopologicalSpace β
inst✝¹² : MeasurableSpace β
inst✝¹¹ : OpensMeasurableSpace β
inst✝¹⁰ : TopologicalSpace γ
inst✝⁹ : MeasurableSpace γ
inst✝⁸ : BorelSpace γ
inst✝⁷ : TopologicalSpace γ₂
inst✝⁶ : MeasurableSpace γ₂
inst✝⁵ : BorelSpace γ₂
inst✝⁴ : MeasurableSpace δ
ι : Type u_6
π : ι → Type u_7
inst✝³ : Countable ι
t' : (i : ι) → TopologicalSpace (π i)
inst✝² : (i : ι) → MeasurableSpace (π i)
inst✝¹ : ∀ (i : ι), SecondCountableTopology (π i)
inst✝ : ∀ (i : ι), OpensMeasurableSpace (π i)
this :
  topologicalSpace =
    TopologicalSpace.generateFrom {t | ∃ s i, (∀ (a : ι), a ∈ i → s a ∈ countableBasis (π a)) ∧ t = Set.pi (↑i) s}
⊢ ∀ (t : Set ((a : ι) → π a)),
    t ∈ {t | ∃ s i, (∀ (a : ι), a ∈ i → s a ∈ countableBasis (π a)) ∧ t = Set.pi (↑i) s} → MeasurableSet t
[PROOFSTEP]
rintro _ ⟨s, i, hi, rfl⟩
[GOAL]
case borel_le.h.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s✝ t u : Set α
inst✝¹⁶ : TopologicalSpace α
inst✝¹⁵ : MeasurableSpace α
inst✝¹⁴ : OpensMeasurableSpace α
inst✝¹³ : TopologicalSpace β
inst✝¹² : MeasurableSpace β
inst✝¹¹ : OpensMeasurableSpace β
inst✝¹⁰ : TopologicalSpace γ
inst✝⁹ : MeasurableSpace γ
inst✝⁸ : BorelSpace γ
inst✝⁷ : TopologicalSpace γ₂
inst✝⁶ : MeasurableSpace γ₂
inst✝⁵ : BorelSpace γ₂
inst✝⁴ : MeasurableSpace δ
ι : Type u_6
π : ι → Type u_7
inst✝³ : Countable ι
t' : (i : ι) → TopologicalSpace (π i)
inst✝² : (i : ι) → MeasurableSpace (π i)
inst✝¹ : ∀ (i : ι), SecondCountableTopology (π i)
inst✝ : ∀ (i : ι), OpensMeasurableSpace (π i)
this :
  topologicalSpace =
    TopologicalSpace.generateFrom {t | ∃ s i, (∀ (a : ι), a ∈ i → s a ∈ countableBasis (π a)) ∧ t = Set.pi (↑i) s}
s : (a : ι) → Set (π a)
i : Finset ι
hi : ∀ (a : ι), a ∈ i → s a ∈ countableBasis (π a)
⊢ MeasurableSet (Set.pi (↑i) s)
[PROOFSTEP]
refine' MeasurableSet.pi i.countable_toSet fun a ha => IsOpen.measurableSet _
[GOAL]
case borel_le.h.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s✝ t u : Set α
inst✝¹⁶ : TopologicalSpace α
inst✝¹⁵ : MeasurableSpace α
inst✝¹⁴ : OpensMeasurableSpace α
inst✝¹³ : TopologicalSpace β
inst✝¹² : MeasurableSpace β
inst✝¹¹ : OpensMeasurableSpace β
inst✝¹⁰ : TopologicalSpace γ
inst✝⁹ : MeasurableSpace γ
inst✝⁸ : BorelSpace γ
inst✝⁷ : TopologicalSpace γ₂
inst✝⁶ : MeasurableSpace γ₂
inst✝⁵ : BorelSpace γ₂
inst✝⁴ : MeasurableSpace δ
ι : Type u_6
π : ι → Type u_7
inst✝³ : Countable ι
t' : (i : ι) → TopologicalSpace (π i)
inst✝² : (i : ι) → MeasurableSpace (π i)
inst✝¹ : ∀ (i : ι), SecondCountableTopology (π i)
inst✝ : ∀ (i : ι), OpensMeasurableSpace (π i)
this :
  topologicalSpace =
    TopologicalSpace.generateFrom {t | ∃ s i, (∀ (a : ι), a ∈ i → s a ∈ countableBasis (π a)) ∧ t = Set.pi (↑i) s}
s : (a : ι) → Set (π a)
i : Finset ι
hi : ∀ (a : ι), a ∈ i → s a ∈ countableBasis (π a)
a : ι
ha : a ∈ ↑i
⊢ IsOpen (s a)
[PROOFSTEP]
rw [eq_generateFrom_countableBasis (π a)]
[GOAL]
case borel_le.h.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s✝ t u : Set α
inst✝¹⁶ : TopologicalSpace α
inst✝¹⁵ : MeasurableSpace α
inst✝¹⁴ : OpensMeasurableSpace α
inst✝¹³ : TopologicalSpace β
inst✝¹² : MeasurableSpace β
inst✝¹¹ : OpensMeasurableSpace β
inst✝¹⁰ : TopologicalSpace γ
inst✝⁹ : MeasurableSpace γ
inst✝⁸ : BorelSpace γ
inst✝⁷ : TopologicalSpace γ₂
inst✝⁶ : MeasurableSpace γ₂
inst✝⁵ : BorelSpace γ₂
inst✝⁴ : MeasurableSpace δ
ι : Type u_6
π : ι → Type u_7
inst✝³ : Countable ι
t' : (i : ι) → TopologicalSpace (π i)
inst✝² : (i : ι) → MeasurableSpace (π i)
inst✝¹ : ∀ (i : ι), SecondCountableTopology (π i)
inst✝ : ∀ (i : ι), OpensMeasurableSpace (π i)
this :
  topologicalSpace =
    TopologicalSpace.generateFrom {t | ∃ s i, (∀ (a : ι), a ∈ i → s a ∈ countableBasis (π a)) ∧ t = Set.pi (↑i) s}
s : (a : ι) → Set (π a)
i : Finset ι
hi : ∀ (a : ι), a ∈ i → s a ∈ countableBasis (π a)
a : ι
ha : a ∈ ↑i
⊢ IsOpen (s a)
[PROOFSTEP]
exact .basic _ (hi a ha)
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹⁴ : TopologicalSpace α
inst✝¹³ : MeasurableSpace α
inst✝¹² : OpensMeasurableSpace α
inst✝¹¹ : TopologicalSpace β
inst✝¹⁰ : MeasurableSpace β
inst✝⁹ : OpensMeasurableSpace β
inst✝⁸ : TopologicalSpace γ
inst✝⁷ : MeasurableSpace γ
inst✝⁶ : BorelSpace γ
inst✝⁵ : TopologicalSpace γ₂
inst✝⁴ : MeasurableSpace γ₂
inst✝³ : BorelSpace γ₂
inst✝² : MeasurableSpace δ
inst✝¹ : SecondCountableTopology α
inst✝ : SecondCountableTopology β
⊢ OpensMeasurableSpace (α × β)
[PROOFSTEP]
constructor
[GOAL]
case borel_le
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹⁴ : TopologicalSpace α
inst✝¹³ : MeasurableSpace α
inst✝¹² : OpensMeasurableSpace α
inst✝¹¹ : TopologicalSpace β
inst✝¹⁰ : MeasurableSpace β
inst✝⁹ : OpensMeasurableSpace β
inst✝⁸ : TopologicalSpace γ
inst✝⁷ : MeasurableSpace γ
inst✝⁶ : BorelSpace γ
inst✝⁵ : TopologicalSpace γ₂
inst✝⁴ : MeasurableSpace γ₂
inst✝³ : BorelSpace γ₂
inst✝² : MeasurableSpace δ
inst✝¹ : SecondCountableTopology α
inst✝ : SecondCountableTopology β
⊢ borel (α × β) ≤ instMeasurableSpace
[PROOFSTEP]
rw [((isBasis_countableBasis α).prod (isBasis_countableBasis β)).borel_eq_generateFrom]
[GOAL]
case borel_le
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹⁴ : TopologicalSpace α
inst✝¹³ : MeasurableSpace α
inst✝¹² : OpensMeasurableSpace α
inst✝¹¹ : TopologicalSpace β
inst✝¹⁰ : MeasurableSpace β
inst✝⁹ : OpensMeasurableSpace β
inst✝⁸ : TopologicalSpace γ
inst✝⁷ : MeasurableSpace γ
inst✝⁶ : BorelSpace γ
inst✝⁵ : TopologicalSpace γ₂
inst✝⁴ : MeasurableSpace γ₂
inst✝³ : BorelSpace γ₂
inst✝² : MeasurableSpace δ
inst✝¹ : SecondCountableTopology α
inst✝ : SecondCountableTopology β
⊢ MeasurableSpace.generateFrom (image2 (fun x x_1 => x ×ˢ x_1) (countableBasis α) (countableBasis β)) ≤
    instMeasurableSpace
[PROOFSTEP]
apply generateFrom_le
[GOAL]
case borel_le.h
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹⁴ : TopologicalSpace α
inst✝¹³ : MeasurableSpace α
inst✝¹² : OpensMeasurableSpace α
inst✝¹¹ : TopologicalSpace β
inst✝¹⁰ : MeasurableSpace β
inst✝⁹ : OpensMeasurableSpace β
inst✝⁸ : TopologicalSpace γ
inst✝⁷ : MeasurableSpace γ
inst✝⁶ : BorelSpace γ
inst✝⁵ : TopologicalSpace γ₂
inst✝⁴ : MeasurableSpace γ₂
inst✝³ : BorelSpace γ₂
inst✝² : MeasurableSpace δ
inst✝¹ : SecondCountableTopology α
inst✝ : SecondCountableTopology β
⊢ ∀ (t : Set (α × β)), t ∈ image2 (fun x x_1 => x ×ˢ x_1) (countableBasis α) (countableBasis β) → MeasurableSet t
[PROOFSTEP]
rintro _ ⟨u, v, hu, hv, rfl⟩
[GOAL]
case borel_le.h.intro.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u✝ : Set α
inst✝¹⁴ : TopologicalSpace α
inst✝¹³ : MeasurableSpace α
inst✝¹² : OpensMeasurableSpace α
inst✝¹¹ : TopologicalSpace β
inst✝¹⁰ : MeasurableSpace β
inst✝⁹ : OpensMeasurableSpace β
inst✝⁸ : TopologicalSpace γ
inst✝⁷ : MeasurableSpace γ
inst✝⁶ : BorelSpace γ
inst✝⁵ : TopologicalSpace γ₂
inst✝⁴ : MeasurableSpace γ₂
inst✝³ : BorelSpace γ₂
inst✝² : MeasurableSpace δ
inst✝¹ : SecondCountableTopology α
inst✝ : SecondCountableTopology β
u : Set α
v : Set β
hu : u ∈ countableBasis α
hv : v ∈ countableBasis β
⊢ MeasurableSet ((fun x x_1 => x ×ˢ x_1) u v)
[PROOFSTEP]
exact (isOpen_of_mem_countableBasis hu).measurableSet.prod (isOpen_of_mem_countableBasis hv).measurableSet
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹⁶ : TopologicalSpace α
inst✝¹⁵ : MeasurableSpace α
inst✝¹⁴ : OpensMeasurableSpace α
inst✝¹³ : TopologicalSpace β
inst✝¹² : MeasurableSpace β
inst✝¹¹ : OpensMeasurableSpace β
inst✝¹⁰ : TopologicalSpace γ
inst✝⁹ : MeasurableSpace γ
inst✝⁸ : BorelSpace γ
inst✝⁷ : TopologicalSpace γ₂
inst✝⁶ : MeasurableSpace γ₂
inst✝⁵ : BorelSpace γ₂
inst✝⁴ : MeasurableSpace δ
α' : Type u_6
inst✝³ : TopologicalSpace α'
inst✝² : MeasurableSpace α'
inst✝¹ : Preorder α
inst✝ : OrderClosedTopology α
a b x : α
⊢ IsMeasurablyGenerated (𝓝[Icc a b] x)
[PROOFSTEP]
rw [← Ici_inter_Iic, nhdsWithin_inter]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹⁶ : TopologicalSpace α
inst✝¹⁵ : MeasurableSpace α
inst✝¹⁴ : OpensMeasurableSpace α
inst✝¹³ : TopologicalSpace β
inst✝¹² : MeasurableSpace β
inst✝¹¹ : OpensMeasurableSpace β
inst✝¹⁰ : TopologicalSpace γ
inst✝⁹ : MeasurableSpace γ
inst✝⁸ : BorelSpace γ
inst✝⁷ : TopologicalSpace γ₂
inst✝⁶ : MeasurableSpace γ₂
inst✝⁵ : BorelSpace γ₂
inst✝⁴ : MeasurableSpace δ
α' : Type u_6
inst✝³ : TopologicalSpace α'
inst✝² : MeasurableSpace α'
inst✝¹ : Preorder α
inst✝ : OrderClosedTopology α
a b x : α
⊢ IsMeasurablyGenerated (𝓝[Ici a] x ⊓ 𝓝[Iic b] x)
[PROOFSTEP]
infer_instance
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹⁶ : TopologicalSpace α
inst✝¹⁵ : MeasurableSpace α
inst✝¹⁴ : OpensMeasurableSpace α
inst✝¹³ : TopologicalSpace β
inst✝¹² : MeasurableSpace β
inst✝¹¹ : OpensMeasurableSpace β
inst✝¹⁰ : TopologicalSpace γ
inst✝⁹ : MeasurableSpace γ
inst✝⁸ : BorelSpace γ
inst✝⁷ : TopologicalSpace γ₂
inst✝⁶ : MeasurableSpace γ₂
inst✝⁵ : BorelSpace γ₂
inst✝⁴ : MeasurableSpace δ
α' : Type u_6
inst✝³ : TopologicalSpace α'
inst✝² : MeasurableSpace α'
inst✝¹ : LinearOrder α
inst✝ : OrderClosedTopology α
a b x : α
h : OrdConnected s
⊢ MeasurableSet s
[PROOFSTEP]
let u := ⋃ (x ∈ s) (y ∈ s), Ioo x y
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u✝ : Set α
inst✝¹⁶ : TopologicalSpace α
inst✝¹⁵ : MeasurableSpace α
inst✝¹⁴ : OpensMeasurableSpace α
inst✝¹³ : TopologicalSpace β
inst✝¹² : MeasurableSpace β
inst✝¹¹ : OpensMeasurableSpace β
inst✝¹⁰ : TopologicalSpace γ
inst✝⁹ : MeasurableSpace γ
inst✝⁸ : BorelSpace γ
inst✝⁷ : TopologicalSpace γ₂
inst✝⁶ : MeasurableSpace γ₂
inst✝⁵ : BorelSpace γ₂
inst✝⁴ : MeasurableSpace δ
α' : Type u_6
inst✝³ : TopologicalSpace α'
inst✝² : MeasurableSpace α'
inst✝¹ : LinearOrder α
inst✝ : OrderClosedTopology α
a b x : α
h : OrdConnected s
u : Set α := ⋃ (x : α) (_ : x ∈ s) (y : α) (_ : y ∈ s), Ioo x y
⊢ MeasurableSet s
[PROOFSTEP]
have huopen : IsOpen u := isOpen_biUnion fun _ _ => isOpen_biUnion fun _ _ => isOpen_Ioo
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u✝ : Set α
inst✝¹⁶ : TopologicalSpace α
inst✝¹⁵ : MeasurableSpace α
inst✝¹⁴ : OpensMeasurableSpace α
inst✝¹³ : TopologicalSpace β
inst✝¹² : MeasurableSpace β
inst✝¹¹ : OpensMeasurableSpace β
inst✝¹⁰ : TopologicalSpace γ
inst✝⁹ : MeasurableSpace γ
inst✝⁸ : BorelSpace γ
inst✝⁷ : TopologicalSpace γ₂
inst✝⁶ : MeasurableSpace γ₂
inst✝⁵ : BorelSpace γ₂
inst✝⁴ : MeasurableSpace δ
α' : Type u_6
inst✝³ : TopologicalSpace α'
inst✝² : MeasurableSpace α'
inst✝¹ : LinearOrder α
inst✝ : OrderClosedTopology α
a b x : α
h : OrdConnected s
u : Set α := ⋃ (x : α) (_ : x ∈ s) (y : α) (_ : y ∈ s), Ioo x y
huopen : IsOpen u
⊢ MeasurableSet s
[PROOFSTEP]
have humeas : MeasurableSet u := huopen.measurableSet
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u✝ : Set α
inst✝¹⁶ : TopologicalSpace α
inst✝¹⁵ : MeasurableSpace α
inst✝¹⁴ : OpensMeasurableSpace α
inst✝¹³ : TopologicalSpace β
inst✝¹² : MeasurableSpace β
inst✝¹¹ : OpensMeasurableSpace β
inst✝¹⁰ : TopologicalSpace γ
inst✝⁹ : MeasurableSpace γ
inst✝⁸ : BorelSpace γ
inst✝⁷ : TopologicalSpace γ₂
inst✝⁶ : MeasurableSpace γ₂
inst✝⁵ : BorelSpace γ₂
inst✝⁴ : MeasurableSpace δ
α' : Type u_6
inst✝³ : TopologicalSpace α'
inst✝² : MeasurableSpace α'
inst✝¹ : LinearOrder α
inst✝ : OrderClosedTopology α
a b x : α
h : OrdConnected s
u : Set α := ⋃ (x : α) (_ : x ∈ s) (y : α) (_ : y ∈ s), Ioo x y
huopen : IsOpen u
humeas : MeasurableSet u
⊢ MeasurableSet s
[PROOFSTEP]
have hfinite : (s \ u).Finite := s.finite_diff_iUnion_Ioo
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u✝ : Set α
inst✝¹⁶ : TopologicalSpace α
inst✝¹⁵ : MeasurableSpace α
inst✝¹⁴ : OpensMeasurableSpace α
inst✝¹³ : TopologicalSpace β
inst✝¹² : MeasurableSpace β
inst✝¹¹ : OpensMeasurableSpace β
inst✝¹⁰ : TopologicalSpace γ
inst✝⁹ : MeasurableSpace γ
inst✝⁸ : BorelSpace γ
inst✝⁷ : TopologicalSpace γ₂
inst✝⁶ : MeasurableSpace γ₂
inst✝⁵ : BorelSpace γ₂
inst✝⁴ : MeasurableSpace δ
α' : Type u_6
inst✝³ : TopologicalSpace α'
inst✝² : MeasurableSpace α'
inst✝¹ : LinearOrder α
inst✝ : OrderClosedTopology α
a b x : α
h : OrdConnected s
u : Set α := ⋃ (x : α) (_ : x ∈ s) (y : α) (_ : y ∈ s), Ioo x y
huopen : IsOpen u
humeas : MeasurableSet u
hfinite : Set.Finite (s \ u)
⊢ MeasurableSet s
[PROOFSTEP]
have : u ⊆ s := iUnion₂_subset fun x hx => iUnion₂_subset fun y hy => Ioo_subset_Icc_self.trans (h.out hx hy)
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u✝ : Set α
inst✝¹⁶ : TopologicalSpace α
inst✝¹⁵ : MeasurableSpace α
inst✝¹⁴ : OpensMeasurableSpace α
inst✝¹³ : TopologicalSpace β
inst✝¹² : MeasurableSpace β
inst✝¹¹ : OpensMeasurableSpace β
inst✝¹⁰ : TopologicalSpace γ
inst✝⁹ : MeasurableSpace γ
inst✝⁸ : BorelSpace γ
inst✝⁷ : TopologicalSpace γ₂
inst✝⁶ : MeasurableSpace γ₂
inst✝⁵ : BorelSpace γ₂
inst✝⁴ : MeasurableSpace δ
α' : Type u_6
inst✝³ : TopologicalSpace α'
inst✝² : MeasurableSpace α'
inst✝¹ : LinearOrder α
inst✝ : OrderClosedTopology α
a b x : α
h : OrdConnected s
u : Set α := ⋃ (x : α) (_ : x ∈ s) (y : α) (_ : y ∈ s), Ioo x y
huopen : IsOpen u
humeas : MeasurableSet u
hfinite : Set.Finite (s \ u)
this : u ⊆ s
⊢ MeasurableSet s
[PROOFSTEP]
rw [← union_diff_cancel this]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u✝ : Set α
inst✝¹⁶ : TopologicalSpace α
inst✝¹⁵ : MeasurableSpace α
inst✝¹⁴ : OpensMeasurableSpace α
inst✝¹³ : TopologicalSpace β
inst✝¹² : MeasurableSpace β
inst✝¹¹ : OpensMeasurableSpace β
inst✝¹⁰ : TopologicalSpace γ
inst✝⁹ : MeasurableSpace γ
inst✝⁸ : BorelSpace γ
inst✝⁷ : TopologicalSpace γ₂
inst✝⁶ : MeasurableSpace γ₂
inst✝⁵ : BorelSpace γ₂
inst✝⁴ : MeasurableSpace δ
α' : Type u_6
inst✝³ : TopologicalSpace α'
inst✝² : MeasurableSpace α'
inst✝¹ : LinearOrder α
inst✝ : OrderClosedTopology α
a b x : α
h : OrdConnected s
u : Set α := ⋃ (x : α) (_ : x ∈ s) (y : α) (_ : y ∈ s), Ioo x y
huopen : IsOpen u
humeas : MeasurableSet u
hfinite : Set.Finite (s \ u)
this : u ⊆ s
⊢ MeasurableSet (u ∪ s \ u)
[PROOFSTEP]
exact humeas.union hfinite.measurableSet
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α✝
inst✝¹⁹ : TopologicalSpace α✝
inst✝¹⁸ : MeasurableSpace α✝
inst✝¹⁷ : OpensMeasurableSpace α✝
inst✝¹⁶ : TopologicalSpace β
inst✝¹⁵ : MeasurableSpace β
inst✝¹⁴ : OpensMeasurableSpace β
inst✝¹³ : TopologicalSpace γ
inst✝¹² : MeasurableSpace γ
inst✝¹¹ : BorelSpace γ
inst✝¹⁰ : TopologicalSpace γ₂
inst✝⁹ : MeasurableSpace γ₂
inst✝⁸ : BorelSpace γ₂
inst✝⁷ : MeasurableSpace δ
α' : Type u_6
inst✝⁶ : TopologicalSpace α'
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : LinearOrder α✝
inst✝³ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderClosedTopology α
s t : Set α
⊢ MeasurableSpace.generateFrom {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ t ∧ l < u ∧ Ico l u = S} ≤ borel α
[PROOFSTEP]
apply generateFrom_le
[GOAL]
case h
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α✝
inst✝¹⁹ : TopologicalSpace α✝
inst✝¹⁸ : MeasurableSpace α✝
inst✝¹⁷ : OpensMeasurableSpace α✝
inst✝¹⁶ : TopologicalSpace β
inst✝¹⁵ : MeasurableSpace β
inst✝¹⁴ : OpensMeasurableSpace β
inst✝¹³ : TopologicalSpace γ
inst✝¹² : MeasurableSpace γ
inst✝¹¹ : BorelSpace γ
inst✝¹⁰ : TopologicalSpace γ₂
inst✝⁹ : MeasurableSpace γ₂
inst✝⁸ : BorelSpace γ₂
inst✝⁷ : MeasurableSpace δ
α' : Type u_6
inst✝⁶ : TopologicalSpace α'
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : LinearOrder α✝
inst✝³ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderClosedTopology α
s t : Set α
⊢ ∀ (t_1 : Set α), t_1 ∈ {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ t ∧ l < u ∧ Ico l u = S} → MeasurableSet t_1
[PROOFSTEP]
borelize α
[GOAL]
case h
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α✝
inst✝¹⁹ : TopologicalSpace α✝
inst✝¹⁸ : MeasurableSpace α✝
inst✝¹⁷ : OpensMeasurableSpace α✝
inst✝¹⁶ : TopologicalSpace β
inst✝¹⁵ : MeasurableSpace β
inst✝¹⁴ : OpensMeasurableSpace β
inst✝¹³ : TopologicalSpace γ
inst✝¹² : MeasurableSpace γ
inst✝¹¹ : BorelSpace γ
inst✝¹⁰ : TopologicalSpace γ₂
inst✝⁹ : MeasurableSpace γ₂
inst✝⁸ : BorelSpace γ₂
inst✝⁷ : MeasurableSpace δ
α' : Type u_6
inst✝⁶ : TopologicalSpace α'
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : LinearOrder α✝
inst✝³ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderClosedTopology α
s t : Set α
this✝¹ : MeasurableSpace α := borel α
this✝ : BorelSpace α
⊢ ∀ (t_1 : Set α), t_1 ∈ {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ t ∧ l < u ∧ Ico l u = S} → MeasurableSet t_1
[PROOFSTEP]
rintro _ ⟨a, -, b, -, -, rfl⟩
[GOAL]
case h.intro.intro.intro.intro.intro
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α✝
inst✝¹⁹ : TopologicalSpace α✝
inst✝¹⁸ : MeasurableSpace α✝
inst✝¹⁷ : OpensMeasurableSpace α✝
inst✝¹⁶ : TopologicalSpace β
inst✝¹⁵ : MeasurableSpace β
inst✝¹⁴ : OpensMeasurableSpace β
inst✝¹³ : TopologicalSpace γ
inst✝¹² : MeasurableSpace γ
inst✝¹¹ : BorelSpace γ
inst✝¹⁰ : TopologicalSpace γ₂
inst✝⁹ : MeasurableSpace γ₂
inst✝⁸ : BorelSpace γ₂
inst✝⁷ : MeasurableSpace δ
α' : Type u_6
inst✝⁶ : TopologicalSpace α'
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : LinearOrder α✝
inst✝³ : OrderClosedTopology α✝
a✝ b✝ x : α✝
α : Type u_7
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderClosedTopology α
s t : Set α
this✝¹ : MeasurableSpace α := borel α
this✝ : BorelSpace α
a b : α
⊢ MeasurableSet (Ico a b)
[PROOFSTEP]
exact measurableSet_Ico
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
⊢ borel α = MeasurableSpace.generateFrom {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}
[PROOFSTEP]
set S : Set (Set α) := {S | ∃ l ∈ s, ∃ u ∈ s, l < u ∧ Ico l u = S}
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}
⊢ borel α = MeasurableSpace.generateFrom S
[PROOFSTEP]
refine' le_antisymm _ (generateFrom_Ico_mem_le_borel _ _)
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}
⊢ borel α ≤ MeasurableSpace.generateFrom S
[PROOFSTEP]
letI : MeasurableSpace α := generateFrom S
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
⊢ borel α ≤ MeasurableSpace.generateFrom S
[PROOFSTEP]
rw [borel_eq_generateFrom_Iio]
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
⊢ MeasurableSpace.generateFrom (range Iio) ≤ MeasurableSpace.generateFrom S
[PROOFSTEP]
refine' generateFrom_le (forall_range_iff.2 fun a => _)
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a✝ b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
⊢ MeasurableSet (Iio a)
[PROOFSTEP]
rcases hd.exists_countable_dense_subset_bot_top with ⟨t, hts, hc, htd, htb, -⟩
[GOAL]
case intro.intro.intro.intro.intro
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a✝ b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : Set.Countable t
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
⊢ MeasurableSet (Iio a)
[PROOFSTEP]
by_cases ha : ∀ b < a, (Ioo b a).Nonempty
[GOAL]
case pos
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a✝ b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : Set.Countable t
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : ∀ (b : α), b < a → Set.Nonempty (Ioo b a)
⊢ MeasurableSet (Iio a)
[PROOFSTEP]
convert_to MeasurableSet (⋃ (l ∈ t) (u ∈ t) (_ : l < u) (_ : u ≤ a), Ico l u)
[GOAL]
case h.e'_3
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a✝ b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : Set.Countable t
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : ∀ (b : α), b < a → Set.Nonempty (Ioo b a)
⊢ Iio a = ⋃ (l : α) (_ : l ∈ t) (u : α) (_ : u ∈ t) (_ : l < u) (_ : u ≤ a), Ico l u
[PROOFSTEP]
ext y
[GOAL]
case h.e'_3.h
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a✝ b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : Set.Countable t
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : ∀ (b : α), b < a → Set.Nonempty (Ioo b a)
y : α
⊢ y ∈ Iio a ↔ y ∈ ⋃ (l : α) (_ : l ∈ t) (u : α) (_ : u ∈ t) (_ : l < u) (_ : u ≤ a), Ico l u
[PROOFSTEP]
simp only [mem_iUnion, mem_Iio, mem_Ico]
[GOAL]
case h.e'_3.h
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a✝ b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : Set.Countable t
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : ∀ (b : α), b < a → Set.Nonempty (Ioo b a)
y : α
⊢ y < a ↔ ∃ i h i_1 h h h, i ≤ y ∧ y < i_1
[PROOFSTEP]
constructor
[GOAL]
case h.e'_3.h.mp
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a✝ b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : Set.Countable t
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : ∀ (b : α), b < a → Set.Nonempty (Ioo b a)
y : α
⊢ y < a → ∃ i h i_1 h h h, i ≤ y ∧ y < i_1
[PROOFSTEP]
intro hy
[GOAL]
case h.e'_3.h.mp
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a✝ b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : Set.Countable t
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : ∀ (b : α), b < a → Set.Nonempty (Ioo b a)
y : α
hy : y < a
⊢ ∃ i h i_1 h h h, i ≤ y ∧ y < i_1
[PROOFSTEP]
rcases htd.exists_le' (fun b hb => htb _ hb (hbot b hb)) y with ⟨l, hlt, hly⟩
[GOAL]
case h.e'_3.h.mp.intro.intro
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a✝ b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : Set.Countable t
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : ∀ (b : α), b < a → Set.Nonempty (Ioo b a)
y : α
hy : y < a
l : α
hlt : l ∈ t
hly : l ≤ y
⊢ ∃ i h i_1 h h h, i ≤ y ∧ y < i_1
[PROOFSTEP]
rcases htd.exists_mem_open isOpen_Ioo (ha y hy) with ⟨u, hut, hyu, hua⟩
[GOAL]
case h.e'_3.h.mp.intro.intro.intro.intro.intro
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u✝ : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a✝ b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : Set.Countable t
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : ∀ (b : α), b < a → Set.Nonempty (Ioo b a)
y : α
hy : y < a
l : α
hlt : l ∈ t
hly : l ≤ y
u : α
hut : u ∈ t
hyu : y < u
hua : u < a
⊢ ∃ i h i_1 h h h, i ≤ y ∧ y < i_1
[PROOFSTEP]
exact ⟨l, hlt, u, hut, hly.trans_lt hyu, hua.le, hly, hyu⟩
[GOAL]
case h.e'_3.h.mpr
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a✝ b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : Set.Countable t
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : ∀ (b : α), b < a → Set.Nonempty (Ioo b a)
y : α
⊢ (∃ i h i_1 h h h, i ≤ y ∧ y < i_1) → y < a
[PROOFSTEP]
rintro ⟨l, -, u, -, -, hua, -, hyu⟩
[GOAL]
case h.e'_3.h.mpr.intro.intro.intro.intro.intro.intro.intro
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u✝ : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a✝ b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : Set.Countable t
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : ∀ (b : α), b < a → Set.Nonempty (Ioo b a)
y l u : α
hua : u ≤ a
hyu : y < u
⊢ y < a
[PROOFSTEP]
exact hyu.trans_le hua
[GOAL]
case pos
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a✝ b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : Set.Countable t
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : ∀ (b : α), b < a → Set.Nonempty (Ioo b a)
⊢ MeasurableSet (⋃ (l : α) (_ : l ∈ t) (u : α) (_ : u ∈ t) (_ : l < u) (_ : u ≤ a), Ico l u)
[PROOFSTEP]
refine' MeasurableSet.biUnion hc fun a ha => MeasurableSet.biUnion hc fun b hb => _
[GOAL]
case pos
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a✝¹ b✝ x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a✝ : α
t : Set α
hts : t ⊆ s
hc : Set.Countable t
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha✝ : ∀ (b : α), b < a✝ → Set.Nonempty (Ioo b a✝)
a : α
ha : a ∈ t
b : α
hb : b ∈ t
⊢ MeasurableSet (⋃ (_ : a < b) (_ : b ≤ a✝), Ico a b)
[PROOFSTEP]
refine' MeasurableSet.iUnion fun hab => MeasurableSet.iUnion fun _ => _
[GOAL]
case pos
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a✝¹ b✝ x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a✝ : α
t : Set α
hts : t ⊆ s
hc : Set.Countable t
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha✝ : ∀ (b : α), b < a✝ → Set.Nonempty (Ioo b a✝)
a : α
ha : a ∈ t
b : α
hb : b ∈ t
hab : a < b
x✝ : b ≤ a✝
⊢ MeasurableSet (Ico a b)
[PROOFSTEP]
exact .basic _ ⟨a, hts ha, b, hts hb, hab, mem_singleton _⟩
[GOAL]
case neg
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a✝ b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : Set.Countable t
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : ¬∀ (b : α), b < a → Set.Nonempty (Ioo b a)
⊢ MeasurableSet (Iio a)
[PROOFSTEP]
simp only [not_forall, not_nonempty_iff_eq_empty] at ha
[GOAL]
case neg
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a✝ b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : Set.Countable t
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : ∃ x h, Ioo x a = ∅
⊢ MeasurableSet (Iio a)
[PROOFSTEP]
replace ha : a ∈ s := hIoo ha.choose a ha.choose_spec.fst ha.choose_spec.snd
[GOAL]
case neg
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a✝ b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : Set.Countable t
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : a ∈ s
⊢ MeasurableSet (Iio a)
[PROOFSTEP]
convert_to MeasurableSet (⋃ (l ∈ t) (_ : l < a), Ico l a)
[GOAL]
case h.e'_3
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a✝ b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : Set.Countable t
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : a ∈ s
⊢ Iio a = ⋃ (l : α) (_ : l ∈ t) (_ : l < a), Ico l a
[PROOFSTEP]
symm
[GOAL]
case h.e'_3
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a✝ b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : Set.Countable t
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : a ∈ s
⊢ ⋃ (l : α) (_ : l ∈ t) (_ : l < a), Ico l a = Iio a
[PROOFSTEP]
simp only [← Ici_inter_Iio, ← iUnion_inter, inter_eq_right_iff_subset, subset_def, mem_iUnion, mem_Ici, mem_Iio]
[GOAL]
case h.e'_3
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a✝ b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : Set.Countable t
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : a ∈ s
⊢ ∀ (x : α), x < a → ∃ i h h, i ≤ x
[PROOFSTEP]
intro x hx
[GOAL]
case h.e'_3
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a✝ b x✝ : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : Set.Countable t
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : a ∈ s
x : α
hx : x < a
⊢ ∃ i h h, i ≤ x
[PROOFSTEP]
rcases htd.exists_le' (fun b hb => htb _ hb (hbot b hb)) x with ⟨z, hzt, hzx⟩
[GOAL]
case h.e'_3.intro.intro
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a✝ b x✝ : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : Set.Countable t
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : a ∈ s
x : α
hx : x < a
z : α
hzt : z ∈ t
hzx : z ≤ x
⊢ ∃ i h h, i ≤ x
[PROOFSTEP]
exact ⟨z, hzt, hzx.trans_lt hx, hzx⟩
[GOAL]
case neg
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a✝ b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : Set.Countable t
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : a ∈ s
⊢ MeasurableSet (⋃ (l : α) (_ : l ∈ t) (_ : l < a), Ico l a)
[PROOFSTEP]
refine' .biUnion hc fun x hx => MeasurableSet.iUnion fun hlt => _
[GOAL]
case neg
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a✝ b x✝ : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : Set.Countable t
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : a ∈ s
x : α
hx : x ∈ t
hlt : x < a
⊢ MeasurableSet (Ico x a)
[PROOFSTEP]
exact .basic _ ⟨x, hts hx, a, ha, hlt, mem_singleton _⟩
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : OrderTopology α
inst✝² : SecondCountableTopology α
inst✝¹ : DenselyOrdered α
inst✝ : NoMinOrder α
s : Set α
hd : Dense s
⊢ ∀ (x : α), IsBot x → x ∈ s
[PROOFSTEP]
simp
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
⊢ borel α = MeasurableSpace.generateFrom {S | ∃ l u, l < u ∧ Ico l u = S}
[PROOFSTEP]
simpa only [exists_prop, mem_univ, true_and_iff] using
  (@dense_univ α _).borel_eq_generateFrom_Ico_mem_aux (fun _ _ => mem_univ _) fun _ _ _ _ => mem_univ _
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsTop x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → x ∈ s
⊢ borel α = MeasurableSpace.generateFrom {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ioc l u = S}
[PROOFSTEP]
convert hd.orderDual.borel_eq_generateFrom_Ico_mem_aux hbot fun x y hlt he => hIoo y x hlt _ using 2
[GOAL]
case h.e'_3.h.h.e'_2.h
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsTop x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → x ∈ s
e_1✝¹ : MeasurableSpace α = MeasurableSpace αᵒᵈ
e_1✝ : α = αᵒᵈ
⊢ {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ioc l u = S} =
    {S | ∃ l, l ∈ ↑OrderDual.ofDual ⁻¹' s ∧ ∃ u, u ∈ ↑OrderDual.ofDual ⁻¹' s ∧ l < u ∧ Ico l u = S}
[PROOFSTEP]
ext s
[GOAL]
case h.e'_3.h.h.e'_2.h.h
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝¹ t u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s✝ : Set α
hd : Dense s✝
hbot : ∀ (x : α), IsTop x → x ∈ s✝
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → x ∈ s✝
e_1✝¹ : MeasurableSpace α = MeasurableSpace αᵒᵈ
e_1✝ : α = αᵒᵈ
s : Set α
⊢ s ∈ {S | ∃ l, l ∈ s✝ ∧ ∃ u, u ∈ s✝ ∧ l < u ∧ Ioc l u = S} ↔
    s ∈ {S | ∃ l, l ∈ ↑OrderDual.ofDual ⁻¹' s✝ ∧ ∃ u, u ∈ ↑OrderDual.ofDual ⁻¹' s✝ ∧ l < u ∧ Ico l u = S}
[PROOFSTEP]
constructor
[GOAL]
case h.e'_3.h.h.e'_2.h.h.mp
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝¹ t u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s✝ : Set α
hd : Dense s✝
hbot : ∀ (x : α), IsTop x → x ∈ s✝
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → x ∈ s✝
e_1✝¹ : MeasurableSpace α = MeasurableSpace αᵒᵈ
e_1✝ : α = αᵒᵈ
s : Set α
⊢ s ∈ {S | ∃ l, l ∈ s✝ ∧ ∃ u, u ∈ s✝ ∧ l < u ∧ Ioc l u = S} →
    s ∈ {S | ∃ l, l ∈ ↑OrderDual.ofDual ⁻¹' s✝ ∧ ∃ u, u ∈ ↑OrderDual.ofDual ⁻¹' s✝ ∧ l < u ∧ Ico l u = S}
[PROOFSTEP]
rintro ⟨l, hl, u, hu, hlt, rfl⟩
[GOAL]
case h.e'_3.h.h.e'_2.h.h.mpr
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝¹ t u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s✝ : Set α
hd : Dense s✝
hbot : ∀ (x : α), IsTop x → x ∈ s✝
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → x ∈ s✝
e_1✝¹ : MeasurableSpace α = MeasurableSpace αᵒᵈ
e_1✝ : α = αᵒᵈ
s : Set α
⊢ s ∈ {S | ∃ l, l ∈ ↑OrderDual.ofDual ⁻¹' s✝ ∧ ∃ u, u ∈ ↑OrderDual.ofDual ⁻¹' s✝ ∧ l < u ∧ Ico l u = S} →
    s ∈ {S | ∃ l, l ∈ s✝ ∧ ∃ u, u ∈ s✝ ∧ l < u ∧ Ioc l u = S}
[PROOFSTEP]
rintro ⟨l, hl, u, hu, hlt, rfl⟩
[GOAL]
case h.e'_3.h.h.e'_2.h.h.mp.intro.intro.intro.intro.intro
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u✝ : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsTop x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → x ∈ s
e_1✝¹ : MeasurableSpace α = MeasurableSpace αᵒᵈ
e_1✝ : α = αᵒᵈ
l : α
hl : l ∈ s
u : α
hu : u ∈ s
hlt : l < u
⊢ Ioc l u ∈ {S | ∃ l, l ∈ ↑OrderDual.ofDual ⁻¹' s ∧ ∃ u, u ∈ ↑OrderDual.ofDual ⁻¹' s ∧ l < u ∧ Ico l u = S}
case h.e'_3.h.h.e'_2.h.h.mpr.intro.intro.intro.intro.intro
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u✝ : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsTop x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → x ∈ s
e_1✝¹ : MeasurableSpace α = MeasurableSpace αᵒᵈ
e_1✝ : α = αᵒᵈ
l : αᵒᵈ
hl : l ∈ ↑OrderDual.ofDual ⁻¹' s
u : αᵒᵈ
hu : u ∈ ↑OrderDual.ofDual ⁻¹' s
hlt : l < u
⊢ Ico l u ∈ {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ioc l u = S}
[PROOFSTEP]
exacts [⟨u, hu, l, hl, hlt, dual_Ico⟩, ⟨u, hu, l, hl, hlt, dual_Ioc⟩]
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a b x✝ : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsTop x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → x ∈ s
x y : αᵒᵈ
hlt : x < y
he : Ioo x y = ∅
⊢ Ioo y x = ∅
[PROOFSTEP]
erw [dual_Ioo]
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a b x✝ : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsTop x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → x ∈ s
x y : αᵒᵈ
hlt : x < y
he : Ioo x y = ∅
⊢ ↑OrderDual.ofDual ⁻¹' Ioo x y = ∅
[PROOFSTEP]
exact he
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : OrderTopology α
inst✝² : SecondCountableTopology α
inst✝¹ : DenselyOrdered α
inst✝ : NoMaxOrder α
s : Set α
hd : Dense s
⊢ ∀ (x : α), IsTop x → x ∈ s
[PROOFSTEP]
simp
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝²⁰ : TopologicalSpace α✝
inst✝¹⁹ : MeasurableSpace α✝
inst✝¹⁸ : OpensMeasurableSpace α✝
inst✝¹⁷ : TopologicalSpace β
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : OpensMeasurableSpace β
inst✝¹⁴ : TopologicalSpace γ
inst✝¹³ : MeasurableSpace γ
inst✝¹² : BorelSpace γ
inst✝¹¹ : TopologicalSpace γ₂
inst✝¹⁰ : MeasurableSpace γ₂
inst✝⁹ : BorelSpace γ₂
inst✝⁸ : MeasurableSpace δ
α' : Type u_6
inst✝⁷ : TopologicalSpace α'
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : LinearOrder α✝
inst✝⁴ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
⊢ borel α = MeasurableSpace.generateFrom {S | ∃ l u, l < u ∧ Ioc l u = S}
[PROOFSTEP]
simpa only [exists_prop, mem_univ, true_and_iff] using
  (@dense_univ α _).borel_eq_generateFrom_Ioc_mem_aux (fun _ _ => mem_univ _) fun _ _ _ _ => mem_univ _
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
hμν : ↑↑μ univ = ↑↑ν univ
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
⊢ μ = ν
[PROOFSTEP]
refine'
  ext_of_generate_finite _ (BorelSpace.measurable_eq.trans (borel_eq_generateFrom_Ico α))
    (isPiSystem_Ico (id : α → α) id) _ hμν
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
hμν : ↑↑μ univ = ↑↑ν univ
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
⊢ ∀ (s : Set α), s ∈ {S | ∃ l u, l < u ∧ Ico l u = S} → ↑↑μ s = ↑↑ν s
[PROOFSTEP]
rintro - ⟨a, b, hlt, rfl⟩
[GOAL]
case intro.intro.intro
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a✝ b✝ x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
hμν : ↑↑μ univ = ↑↑ν univ
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
a b : α
hlt : a < b
⊢ ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
[PROOFSTEP]
exact h hlt
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
hμν : ↑↑μ univ = ↑↑ν univ
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ioc a b) = ↑↑ν (Ioc a b)
⊢ μ = ν
[PROOFSTEP]
refine' @ext_of_Ico_finite αᵒᵈ _ _ _ _ _ ‹_› μ ν _ hμν fun a b hab => _
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a✝ b✝ x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
hμν : ↑↑μ univ = ↑↑ν univ
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ioc a b) = ↑↑ν (Ioc a b)
a b : αᵒᵈ
hab : a < b
⊢ ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
[PROOFSTEP]
erw [dual_Ico (α := α)]
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a✝ b✝ x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
hμν : ↑↑μ univ = ↑↑ν univ
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ioc a b) = ↑↑ν (Ioc a b)
a b : αᵒᵈ
hab : a < b
⊢ ↑↑μ (↑OrderDual.ofDual ⁻¹' Ioc b a) = ↑↑ν (↑OrderDual.ofDual ⁻¹' Ioc b a)
[PROOFSTEP]
exact h hab
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
⊢ μ = ν
[PROOFSTEP]
rcases exists_countable_dense_bot_top α with ⟨s, hsc, hsd, hsb, _⟩
[GOAL]
case intro.intro.intro.intro
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
s : Set α
hsc : Set.Countable s
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
⊢ μ = ν
[PROOFSTEP]
have : (⋃ (l ∈ s) (u ∈ s) (_ : l < u), {Ico l u} : Set (Set α)).Countable :=
  hsc.biUnion fun l _ => hsc.biUnion fun u _ => countable_iUnion fun _ => countable_singleton _
[GOAL]
case intro.intro.intro.intro
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
s : Set α
hsc : Set.Countable s
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
this : Set.Countable (⋃ (l : α) (_ : l ∈ s) (u : α) (_ : u ∈ s) (_ : l < u), {Ico l u})
⊢ μ = ν
[PROOFSTEP]
simp only [← setOf_eq_eq_singleton, ← setOf_exists] at this
[GOAL]
case intro.intro.intro.intro
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
s : Set α
hsc : Set.Countable s
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
this : Set.Countable {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2}
⊢ μ = ν
[PROOFSTEP]
refine'
  Measure.ext_of_generateFrom_of_cover_subset (BorelSpace.measurable_eq.trans (borel_eq_generateFrom_Ico α))
    (isPiSystem_Ico id id) _ this _ _ _
[GOAL]
case intro.intro.intro.intro.refine'_1
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
s : Set α
hsc : Set.Countable s
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
this : Set.Countable {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2}
⊢ {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2} ⊆ {S | ∃ l u, l < u ∧ Ico l u = S}
[PROOFSTEP]
rintro _ ⟨l, -, u, -, h, rfl⟩
[GOAL]
case intro.intro.intro.intro.refine'_1.intro.intro.intro.intro.intro
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u✝ : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤
h✝ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
s : Set α
hsc : Set.Countable s
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
this : Set.Countable {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2}
l u : α
h : l < u
⊢ Ico l u ∈ {S | ∃ l u, l < u ∧ Ico l u = S}
[PROOFSTEP]
exact ⟨l, u, h, rfl⟩
[GOAL]
case intro.intro.intro.intro.refine'_2
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
s : Set α
hsc : Set.Countable s
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
this : Set.Countable {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2}
⊢ ⋃₀ {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2} = univ
[PROOFSTEP]
refine' sUnion_eq_univ_iff.2 fun x => _
[GOAL]
case intro.intro.intro.intro.refine'_2
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x✝ : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
s : Set α
hsc : Set.Countable s
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
this : Set.Countable {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2}
x : α
⊢ ∃ b, b ∈ {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2} ∧ x ∈ b
[PROOFSTEP]
rcases hsd.exists_le' hsb x with ⟨l, hls, hlx⟩
[GOAL]
case intro.intro.intro.intro.refine'_2.intro.intro
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x✝ : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
s : Set α
hsc : Set.Countable s
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
this : Set.Countable {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2}
x l : α
hls : l ∈ s
hlx : l ≤ x
⊢ ∃ b, b ∈ {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2} ∧ x ∈ b
[PROOFSTEP]
rcases hsd.exists_gt x with ⟨u, hus, hxu⟩
[GOAL]
case intro.intro.intro.intro.refine'_2.intro.intro.intro.intro
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u✝ : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x✝ : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
s : Set α
hsc : Set.Countable s
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
this : Set.Countable {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2}
x l : α
hls : l ∈ s
hlx : l ≤ x
u : α
hus : u ∈ s
hxu : x < u
⊢ ∃ b, b ∈ {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2} ∧ x ∈ b
[PROOFSTEP]
exact ⟨_, ⟨l, hls, u, hus, hlx.trans_lt hxu, rfl⟩, hlx, hxu⟩
[GOAL]
case intro.intro.intro.intro.refine'_3
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
s : Set α
hsc : Set.Countable s
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
this : Set.Countable {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2}
⊢ ∀ (s_1 : Set α), s_1 ∈ {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2} → ↑↑μ s_1 ≠ ⊤
[PROOFSTEP]
rintro _ ⟨l, -, u, -, hlt, rfl⟩
[GOAL]
case intro.intro.intro.intro.refine'_3.intro.intro.intro.intro.intro
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u✝ : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
s : Set α
hsc : Set.Countable s
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
this : Set.Countable {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2}
l u : α
hlt : l < u
⊢ ↑↑μ (Ico l u) ≠ ⊤
[PROOFSTEP]
exact hμ hlt
[GOAL]
case intro.intro.intro.intro.refine'_4
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
s : Set α
hsc : Set.Countable s
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
this : Set.Countable {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2}
⊢ ∀ (s : Set α), s ∈ {S | ∃ l u, l < u ∧ Ico l u = S} → ↑↑μ s = ↑↑ν s
[PROOFSTEP]
rintro _ ⟨l, u, hlt, rfl⟩
[GOAL]
case intro.intro.intro.intro.refine'_4.intro.intro.intro
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u✝ : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
s : Set α
hsc : Set.Countable s
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
this : Set.Countable {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2}
l u : α
hlt : l < u
⊢ ↑↑μ (Ico l u) = ↑↑ν (Ico l u)
[PROOFSTEP]
exact h hlt
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMinOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ioc a b) ≠ ⊤
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ioc a b) = ↑↑ν (Ioc a b)
⊢ μ = ν
[PROOFSTEP]
refine' @ext_of_Ico' αᵒᵈ _ _ _ _ _ ‹_› _ μ ν _ _
[GOAL]
case refine'_1
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMinOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ioc a b) ≠ ⊤
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ioc a b) = ↑↑ν (Ioc a b)
⊢ ∀ ⦃a b : αᵒᵈ⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤
[PROOFSTEP]
intro a b hab
[GOAL]
case refine'_2
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMinOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ioc a b) ≠ ⊤
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ioc a b) = ↑↑ν (Ioc a b)
⊢ ∀ ⦃a b : αᵒᵈ⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
[PROOFSTEP]
intro a b hab
[GOAL]
case refine'_1
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a✝ b✝ x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMinOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ioc a b) ≠ ⊤
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ioc a b) = ↑↑ν (Ioc a b)
a b : αᵒᵈ
hab : a < b
⊢ ↑↑μ (Ico a b) ≠ ⊤
[PROOFSTEP]
erw [dual_Ico (α := α)]
[GOAL]
case refine'_2
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a✝ b✝ x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMinOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ioc a b) ≠ ⊤
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ioc a b) = ↑↑ν (Ioc a b)
a b : αᵒᵈ
hab : a < b
⊢ ↑↑μ (Ico a b) = ↑↑ν (Ico a b)
[PROOFSTEP]
erw [dual_Ico (α := α)]
[GOAL]
case refine'_1
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a✝ b✝ x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMinOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ioc a b) ≠ ⊤
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ioc a b) = ↑↑ν (Ioc a b)
a b : αᵒᵈ
hab : a < b
⊢ ↑↑μ (↑OrderDual.ofDual ⁻¹' Ioc b a) ≠ ⊤
case refine'_2
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a✝ b✝ x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMinOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ioc a b) ≠ ⊤
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ioc a b) = ↑↑ν (Ioc a b)
a b : αᵒᵈ
hab : a < b
⊢ ↑↑μ (↑OrderDual.ofDual ⁻¹' Ioc b a) = ↑↑ν (↑OrderDual.ofDual ⁻¹' Ioc b a)
[PROOFSTEP]
exacts [hμ hab, h hab]
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
h : ∀ (a : α), ↑↑μ (Iic a) = ↑↑ν (Iic a)
⊢ μ = ν
[PROOFSTEP]
refine' ext_of_Ioc_finite μ ν _ fun a b hlt => _
[GOAL]
case refine'_1
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
h : ∀ (a : α), ↑↑μ (Iic a) = ↑↑ν (Iic a)
⊢ ↑↑μ univ = ↑↑ν univ
[PROOFSTEP]
rcases exists_countable_dense_bot_top α with ⟨s, hsc, hsd, -, hst⟩
[GOAL]
case refine'_1.intro.intro.intro.intro
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
h : ∀ (a : α), ↑↑μ (Iic a) = ↑↑ν (Iic a)
s : Set α
hsc : Set.Countable s
hsd : Dense s
hst : ∀ (x : α), IsTop x → x ∈ s
⊢ ↑↑μ univ = ↑↑ν univ
[PROOFSTEP]
have : DirectedOn (· ≤ ·) s := directedOn_iff_directed.2 (directed_of_sup fun _ _ => id)
[GOAL]
case refine'_1.intro.intro.intro.intro
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
h : ∀ (a : α), ↑↑μ (Iic a) = ↑↑ν (Iic a)
s : Set α
hsc : Set.Countable s
hsd : Dense s
hst : ∀ (x : α), IsTop x → x ∈ s
this : DirectedOn (fun x x_1 => x ≤ x_1) s
⊢ ↑↑μ univ = ↑↑ν univ
[PROOFSTEP]
simp only [← biSup_measure_Iic hsc (hsd.exists_ge' hst) this, h]
[GOAL]
case refine'_2
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a✝ b✝ x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
h : ∀ (a : α), ↑↑μ (Iic a) = ↑↑ν (Iic a)
a b : α
hlt : a < b
⊢ ↑↑μ (Ioc a b) = ↑↑ν (Ioc a b)
[PROOFSTEP]
rw [← Iic_diff_Iic, measure_diff (Iic_subset_Iic.2 hlt.le) measurableSet_Iic,
  measure_diff (Iic_subset_Iic.2 hlt.le) measurableSet_Iic, h a, h b]
[GOAL]
case refine'_2
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a✝ b✝ x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
h : ∀ (a : α), ↑↑μ (Iic a) = ↑↑ν (Iic a)
a b : α
hlt : a < b
⊢ ↑↑ν (Iic a) ≠ ⊤
[PROOFSTEP]
rw [← h a]
[GOAL]
case refine'_2
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a✝ b✝ x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
h : ∀ (a : α), ↑↑μ (Iic a) = ↑↑ν (Iic a)
a b : α
hlt : a < b
⊢ ↑↑μ (Iic a) ≠ ⊤
[PROOFSTEP]
exact (measure_lt_top μ _).ne
[GOAL]
case refine'_2
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type u_6
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a✝ b✝ x : α✝
α : Type u_7
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
h : ∀ (a : α), ↑↑μ (Iic a) = ↑↑ν (Iic a)
a b : α
hlt : a < b
⊢ ↑↑μ (Iic a) ≠ ⊤
[PROOFSTEP]
exact (measure_lt_top μ _).ne
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹⁷ : TopologicalSpace α
inst✝¹⁶ : MeasurableSpace α
inst✝¹⁵ : OpensMeasurableSpace α
inst✝¹⁴ : TopologicalSpace β
inst✝¹³ : MeasurableSpace β
inst✝¹² : OpensMeasurableSpace β
inst✝¹¹ : TopologicalSpace γ
inst✝¹⁰ : MeasurableSpace γ
inst✝⁹ : BorelSpace γ
inst✝⁸ : TopologicalSpace γ₂
inst✝⁷ : MeasurableSpace γ₂
inst✝⁶ : BorelSpace γ₂
inst✝⁵ : MeasurableSpace δ
α' : Type u_6
inst✝⁴ : TopologicalSpace α'
inst✝³ : MeasurableSpace α'
inst✝² : LinearOrder α
inst✝¹ : OrderClosedTopology α
a b : α
inst✝ : SecondCountableTopology α
f g : δ → α
hf : Measurable f
hg : Measurable g
⊢ Measurable fun a => Max.max (f a) (g a)
[PROOFSTEP]
simpa only [max_def'] using hf.piecewise (measurableSet_le hg hf) hg
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹⁷ : TopologicalSpace α
inst✝¹⁶ : MeasurableSpace α
inst✝¹⁵ : OpensMeasurableSpace α
inst✝¹⁴ : TopologicalSpace β
inst✝¹³ : MeasurableSpace β
inst✝¹² : OpensMeasurableSpace β
inst✝¹¹ : TopologicalSpace γ
inst✝¹⁰ : MeasurableSpace γ
inst✝⁹ : BorelSpace γ
inst✝⁸ : TopologicalSpace γ₂
inst✝⁷ : MeasurableSpace γ₂
inst✝⁶ : BorelSpace γ₂
inst✝⁵ : MeasurableSpace δ
α' : Type u_6
inst✝⁴ : TopologicalSpace α'
inst✝³ : MeasurableSpace α'
inst✝² : LinearOrder α
inst✝¹ : OrderClosedTopology α
a b : α
inst✝ : SecondCountableTopology α
f g : δ → α
hf : Measurable f
hg : Measurable g
⊢ Measurable fun a => Min.min (f a) (g a)
[PROOFSTEP]
simpa only [min_def] using hf.piecewise (measurableSet_le hf hg) hg
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹⁵ : TopologicalSpace α
inst✝¹⁴ : MeasurableSpace α
inst✝¹³ : OpensMeasurableSpace α
inst✝¹² : TopologicalSpace β
inst✝¹¹ : MeasurableSpace β
inst✝¹⁰ : OpensMeasurableSpace β
inst✝⁹ : TopologicalSpace γ
inst✝⁸ : MeasurableSpace γ
inst✝⁷ : BorelSpace γ
inst✝⁶ : TopologicalSpace γ₂
inst✝⁵ : MeasurableSpace γ₂
inst✝⁴ : BorelSpace γ₂
inst✝³ : MeasurableSpace δ
α' : Type u_6
inst✝² : TopologicalSpace α'
inst✝¹ : MeasurableSpace α'
f g : α → γ
s : Set α
inst✝ : (j : α) → Decidable (j ∈ s)
hf : ContinuousOn f s
hg : ContinuousOn g sᶜ
hs : MeasurableSet s
⊢ Measurable (Set.piecewise s f g)
[PROOFSTEP]
refine' measurable_of_isOpen fun t ht => _
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝¹⁵ : TopologicalSpace α
inst✝¹⁴ : MeasurableSpace α
inst✝¹³ : OpensMeasurableSpace α
inst✝¹² : TopologicalSpace β
inst✝¹¹ : MeasurableSpace β
inst✝¹⁰ : OpensMeasurableSpace β
inst✝⁹ : TopologicalSpace γ
inst✝⁸ : MeasurableSpace γ
inst✝⁷ : BorelSpace γ
inst✝⁶ : TopologicalSpace γ₂
inst✝⁵ : MeasurableSpace γ₂
inst✝⁴ : BorelSpace γ₂
inst✝³ : MeasurableSpace δ
α' : Type u_6
inst✝² : TopologicalSpace α'
inst✝¹ : MeasurableSpace α'
f g : α → γ
s : Set α
inst✝ : (j : α) → Decidable (j ∈ s)
hf : ContinuousOn f s
hg : ContinuousOn g sᶜ
hs : MeasurableSet s
t : Set γ
ht : IsOpen t
⊢ MeasurableSet (Set.piecewise s f g ⁻¹' t)
[PROOFSTEP]
rw [piecewise_preimage, Set.ite]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝¹⁵ : TopologicalSpace α
inst✝¹⁴ : MeasurableSpace α
inst✝¹³ : OpensMeasurableSpace α
inst✝¹² : TopologicalSpace β
inst✝¹¹ : MeasurableSpace β
inst✝¹⁰ : OpensMeasurableSpace β
inst✝⁹ : TopologicalSpace γ
inst✝⁸ : MeasurableSpace γ
inst✝⁷ : BorelSpace γ
inst✝⁶ : TopologicalSpace γ₂
inst✝⁵ : MeasurableSpace γ₂
inst✝⁴ : BorelSpace γ₂
inst✝³ : MeasurableSpace δ
α' : Type u_6
inst✝² : TopologicalSpace α'
inst✝¹ : MeasurableSpace α'
f g : α → γ
s : Set α
inst✝ : (j : α) → Decidable (j ∈ s)
hf : ContinuousOn f s
hg : ContinuousOn g sᶜ
hs : MeasurableSet s
t : Set γ
ht : IsOpen t
⊢ MeasurableSet (f ⁻¹' t ∩ s ∪ g ⁻¹' t \ s)
[PROOFSTEP]
apply MeasurableSet.union
[GOAL]
case h₁
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝¹⁵ : TopologicalSpace α
inst✝¹⁴ : MeasurableSpace α
inst✝¹³ : OpensMeasurableSpace α
inst✝¹² : TopologicalSpace β
inst✝¹¹ : MeasurableSpace β
inst✝¹⁰ : OpensMeasurableSpace β
inst✝⁹ : TopologicalSpace γ
inst✝⁸ : MeasurableSpace γ
inst✝⁷ : BorelSpace γ
inst✝⁶ : TopologicalSpace γ₂
inst✝⁵ : MeasurableSpace γ₂
inst✝⁴ : BorelSpace γ₂
inst✝³ : MeasurableSpace δ
α' : Type u_6
inst✝² : TopologicalSpace α'
inst✝¹ : MeasurableSpace α'
f g : α → γ
s : Set α
inst✝ : (j : α) → Decidable (j ∈ s)
hf : ContinuousOn f s
hg : ContinuousOn g sᶜ
hs : MeasurableSet s
t : Set γ
ht : IsOpen t
⊢ MeasurableSet (f ⁻¹' t ∩ s)
[PROOFSTEP]
rcases _root_.continuousOn_iff'.1 hf t ht with ⟨u, u_open, hu⟩
[GOAL]
case h₁.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u✝ : Set α
inst✝¹⁵ : TopologicalSpace α
inst✝¹⁴ : MeasurableSpace α
inst✝¹³ : OpensMeasurableSpace α
inst✝¹² : TopologicalSpace β
inst✝¹¹ : MeasurableSpace β
inst✝¹⁰ : OpensMeasurableSpace β
inst✝⁹ : TopologicalSpace γ
inst✝⁸ : MeasurableSpace γ
inst✝⁷ : BorelSpace γ
inst✝⁶ : TopologicalSpace γ₂
inst✝⁵ : MeasurableSpace γ₂
inst✝⁴ : BorelSpace γ₂
inst✝³ : MeasurableSpace δ
α' : Type u_6
inst✝² : TopologicalSpace α'
inst✝¹ : MeasurableSpace α'
f g : α → γ
s : Set α
inst✝ : (j : α) → Decidable (j ∈ s)
hf : ContinuousOn f s
hg : ContinuousOn g sᶜ
hs : MeasurableSet s
t : Set γ
ht : IsOpen t
u : Set α
u_open : IsOpen u
hu : f ⁻¹' t ∩ s = u ∩ s
⊢ MeasurableSet (f ⁻¹' t ∩ s)
[PROOFSTEP]
rw [hu]
[GOAL]
case h₁.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u✝ : Set α
inst✝¹⁵ : TopologicalSpace α
inst✝¹⁴ : MeasurableSpace α
inst✝¹³ : OpensMeasurableSpace α
inst✝¹² : TopologicalSpace β
inst✝¹¹ : MeasurableSpace β
inst✝¹⁰ : OpensMeasurableSpace β
inst✝⁹ : TopologicalSpace γ
inst✝⁸ : MeasurableSpace γ
inst✝⁷ : BorelSpace γ
inst✝⁶ : TopologicalSpace γ₂
inst✝⁵ : MeasurableSpace γ₂
inst✝⁴ : BorelSpace γ₂
inst✝³ : MeasurableSpace δ
α' : Type u_6
inst✝² : TopologicalSpace α'
inst✝¹ : MeasurableSpace α'
f g : α → γ
s : Set α
inst✝ : (j : α) → Decidable (j ∈ s)
hf : ContinuousOn f s
hg : ContinuousOn g sᶜ
hs : MeasurableSet s
t : Set γ
ht : IsOpen t
u : Set α
u_open : IsOpen u
hu : f ⁻¹' t ∩ s = u ∩ s
⊢ MeasurableSet (u ∩ s)
[PROOFSTEP]
exact u_open.measurableSet.inter hs
[GOAL]
case h₂
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝¹⁵ : TopologicalSpace α
inst✝¹⁴ : MeasurableSpace α
inst✝¹³ : OpensMeasurableSpace α
inst✝¹² : TopologicalSpace β
inst✝¹¹ : MeasurableSpace β
inst✝¹⁰ : OpensMeasurableSpace β
inst✝⁹ : TopologicalSpace γ
inst✝⁸ : MeasurableSpace γ
inst✝⁷ : BorelSpace γ
inst✝⁶ : TopologicalSpace γ₂
inst✝⁵ : MeasurableSpace γ₂
inst✝⁴ : BorelSpace γ₂
inst✝³ : MeasurableSpace δ
α' : Type u_6
inst✝² : TopologicalSpace α'
inst✝¹ : MeasurableSpace α'
f g : α → γ
s : Set α
inst✝ : (j : α) → Decidable (j ∈ s)
hf : ContinuousOn f s
hg : ContinuousOn g sᶜ
hs : MeasurableSet s
t : Set γ
ht : IsOpen t
⊢ MeasurableSet (g ⁻¹' t \ s)
[PROOFSTEP]
rcases _root_.continuousOn_iff'.1 hg t ht with ⟨u, u_open, hu⟩
[GOAL]
case h₂.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u✝ : Set α
inst✝¹⁵ : TopologicalSpace α
inst✝¹⁴ : MeasurableSpace α
inst✝¹³ : OpensMeasurableSpace α
inst✝¹² : TopologicalSpace β
inst✝¹¹ : MeasurableSpace β
inst✝¹⁰ : OpensMeasurableSpace β
inst✝⁹ : TopologicalSpace γ
inst✝⁸ : MeasurableSpace γ
inst✝⁷ : BorelSpace γ
inst✝⁶ : TopologicalSpace γ₂
inst✝⁵ : MeasurableSpace γ₂
inst✝⁴ : BorelSpace γ₂
inst✝³ : MeasurableSpace δ
α' : Type u_6
inst✝² : TopologicalSpace α'
inst✝¹ : MeasurableSpace α'
f g : α → γ
s : Set α
inst✝ : (j : α) → Decidable (j ∈ s)
hf : ContinuousOn f s
hg : ContinuousOn g sᶜ
hs : MeasurableSet s
t : Set γ
ht : IsOpen t
u : Set α
u_open : IsOpen u
hu : g ⁻¹' t ∩ sᶜ = u ∩ sᶜ
⊢ MeasurableSet (g ⁻¹' t \ s)
[PROOFSTEP]
rw [diff_eq_compl_inter, inter_comm, hu]
[GOAL]
case h₂.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u✝ : Set α
inst✝¹⁵ : TopologicalSpace α
inst✝¹⁴ : MeasurableSpace α
inst✝¹³ : OpensMeasurableSpace α
inst✝¹² : TopologicalSpace β
inst✝¹¹ : MeasurableSpace β
inst✝¹⁰ : OpensMeasurableSpace β
inst✝⁹ : TopologicalSpace γ
inst✝⁸ : MeasurableSpace γ
inst✝⁷ : BorelSpace γ
inst✝⁶ : TopologicalSpace γ₂
inst✝⁵ : MeasurableSpace γ₂
inst✝⁴ : BorelSpace γ₂
inst✝³ : MeasurableSpace δ
α' : Type u_6
inst✝² : TopologicalSpace α'
inst✝¹ : MeasurableSpace α'
f g : α → γ
s : Set α
inst✝ : (j : α) → Decidable (j ∈ s)
hf : ContinuousOn f s
hg : ContinuousOn g sᶜ
hs : MeasurableSet s
t : Set γ
ht : IsOpen t
u : Set α
u_open : IsOpen u
hu : g ⁻¹' t ∩ sᶜ = u ∩ sᶜ
⊢ MeasurableSet (u ∩ sᶜ)
[PROOFSTEP]
exact u_open.measurableSet.inter hs.compl
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
ι : Type u_6
π : ι → Type u_7
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : (i : ι) → MeasurableSpace (π i)
inst✝ : ∀ (i : ι), BorelSpace (π i)
⊢ MeasurableSpace.pi ≤ borel ((i : ι) → π i)
[PROOFSTEP]
have : ‹∀ i, MeasurableSpace (π i)› = fun i => borel (π i) := funext fun i => BorelSpace.measurable_eq
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
ι : Type u_6
π : ι → Type u_7
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : (i : ι) → MeasurableSpace (π i)
inst✝ : ∀ (i : ι), BorelSpace (π i)
this : inst✝¹ = fun i => borel (π i)
⊢ MeasurableSpace.pi ≤ borel ((i : ι) → π i)
[PROOFSTEP]
rw [this]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
ι : Type u_6
π : ι → Type u_7
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : (i : ι) → MeasurableSpace (π i)
inst✝ : ∀ (i : ι), BorelSpace (π i)
this : inst✝¹ = fun i => borel (π i)
⊢ MeasurableSpace.pi ≤ borel ((i : ι) → π i)
[PROOFSTEP]
exact iSup_le fun i => comap_le_iff_le_map.2 <| (continuous_apply i).borel_measurable
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : MeasurableSpace α
inst✝⁷ : BorelSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : MeasurableSpace β
inst✝⁴ : BorelSpace β
inst✝³ : TopologicalSpace γ
inst✝² : MeasurableSpace γ
inst✝¹ : BorelSpace γ
inst✝ : MeasurableSpace δ
⊢ Prod.instMeasurableSpace ≤ borel (α × β)
[PROOFSTEP]
rw [‹BorelSpace α›.measurable_eq, ‹BorelSpace β›.measurable_eq]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : MeasurableSpace α
inst✝⁷ : BorelSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : MeasurableSpace β
inst✝⁴ : BorelSpace β
inst✝³ : TopologicalSpace γ
inst✝² : MeasurableSpace γ
inst✝¹ : BorelSpace γ
inst✝ : MeasurableSpace δ
⊢ Prod.instMeasurableSpace ≤ borel (α × β)
[PROOFSTEP]
refine' sup_le _ _
[GOAL]
case refine'_1
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : MeasurableSpace α
inst✝⁷ : BorelSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : MeasurableSpace β
inst✝⁴ : BorelSpace β
inst✝³ : TopologicalSpace γ
inst✝² : MeasurableSpace γ
inst✝¹ : BorelSpace γ
inst✝ : MeasurableSpace δ
⊢ MeasurableSpace.comap Prod.fst (borel α) ≤ borel (α × β)
[PROOFSTEP]
exact comap_le_iff_le_map.mpr continuous_fst.borel_measurable
[GOAL]
case refine'_2
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : MeasurableSpace α
inst✝⁷ : BorelSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : MeasurableSpace β
inst✝⁴ : BorelSpace β
inst✝³ : TopologicalSpace γ
inst✝² : MeasurableSpace γ
inst✝¹ : BorelSpace γ
inst✝ : MeasurableSpace δ
⊢ MeasurableSpace.comap Prod.snd (borel β) ≤ borel (α × β)
[PROOFSTEP]
exact comap_le_iff_le_map.mpr continuous_snd.borel_measurable
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
f : δ → α
hf : ∀ (x : α), MeasurableSet (f ⁻¹' Iio x)
⊢ Measurable f
[PROOFSTEP]
convert measurable_generateFrom (α := δ) _
[GOAL]
case h.e'_4
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
f : δ → α
hf : ∀ (x : α), MeasurableSet (f ⁻¹' Iio x)
⊢ inst✝¹¹ = MeasurableSpace.generateFrom ?convert_2
case convert_2
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
f : δ → α
hf : ∀ (x : α), MeasurableSet (f ⁻¹' Iio x)
⊢ Set (Set α)
case convert_4
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
f : δ → α
hf : ∀ (x : α), MeasurableSet (f ⁻¹' Iio x)
⊢ ∀ (t : Set α), t ∈ ?convert_2 → MeasurableSet (f ⁻¹' t)
[PROOFSTEP]
exact BorelSpace.measurable_eq.trans (borel_eq_generateFrom_Iio _)
[GOAL]
case convert_4
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
f : δ → α
hf : ∀ (x : α), MeasurableSet (f ⁻¹' Iio x)
⊢ ∀ (t : Set α), t ∈ range Iio → MeasurableSet (f ⁻¹' t)
[PROOFSTEP]
rintro _ ⟨x, rfl⟩
[GOAL]
case convert_4.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
f : δ → α
hf : ∀ (x : α), MeasurableSet (f ⁻¹' Iio x)
x : α
⊢ MeasurableSet (f ⁻¹' Iio x)
[PROOFSTEP]
exact hf x
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
f : δ → α
hf : ∀ (x : α), MeasurableSet (f ⁻¹' Ioi x)
⊢ Measurable f
[PROOFSTEP]
convert measurable_generateFrom (α := δ) _
[GOAL]
case h.e'_4
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
f : δ → α
hf : ∀ (x : α), MeasurableSet (f ⁻¹' Ioi x)
⊢ inst✝¹¹ = MeasurableSpace.generateFrom ?convert_2
case convert_2
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
f : δ → α
hf : ∀ (x : α), MeasurableSet (f ⁻¹' Ioi x)
⊢ Set (Set α)
case convert_4
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
f : δ → α
hf : ∀ (x : α), MeasurableSet (f ⁻¹' Ioi x)
⊢ ∀ (t : Set α), t ∈ ?convert_2 → MeasurableSet (f ⁻¹' t)
[PROOFSTEP]
exact BorelSpace.measurable_eq.trans (borel_eq_generateFrom_Ioi _)
[GOAL]
case convert_4
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
f : δ → α
hf : ∀ (x : α), MeasurableSet (f ⁻¹' Ioi x)
⊢ ∀ (t : Set α), t ∈ range Ioi → MeasurableSet (f ⁻¹' t)
[PROOFSTEP]
rintro _ ⟨x, rfl⟩
[GOAL]
case convert_4.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
f : δ → α
hf : ∀ (x : α), MeasurableSet (f ⁻¹' Ioi x)
x : α
⊢ MeasurableSet (f ⁻¹' Ioi x)
[PROOFSTEP]
exact hf x
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
f : δ → α
hf : ∀ (x : α), MeasurableSet (f ⁻¹' Iic x)
⊢ Measurable f
[PROOFSTEP]
apply measurable_of_Ioi
[GOAL]
case hf
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
f : δ → α
hf : ∀ (x : α), MeasurableSet (f ⁻¹' Iic x)
⊢ ∀ (x : α), MeasurableSet (f ⁻¹' Ioi x)
[PROOFSTEP]
simp_rw [← compl_Iic, preimage_compl, MeasurableSet.compl_iff]
[GOAL]
case hf
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
f : δ → α
hf : ∀ (x : α), MeasurableSet (f ⁻¹' Iic x)
⊢ ∀ (x : α), MeasurableSet (f ⁻¹' Iic x)
[PROOFSTEP]
assumption
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
f : δ → α
hf : ∀ (x : α), MeasurableSet (f ⁻¹' Ici x)
⊢ Measurable f
[PROOFSTEP]
apply measurable_of_Iio
[GOAL]
case hf
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
f : δ → α
hf : ∀ (x : α), MeasurableSet (f ⁻¹' Ici x)
⊢ ∀ (x : α), MeasurableSet (f ⁻¹' Iio x)
[PROOFSTEP]
simp_rw [← compl_Ici, preimage_compl, MeasurableSet.compl_iff]
[GOAL]
case hf
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
f : δ → α
hf : ∀ (x : α), MeasurableSet (f ⁻¹' Ici x)
⊢ ∀ (x : α), MeasurableSet (f ⁻¹' Ici x)
[PROOFSTEP]
assumption
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), Measurable (f i)
hg : ∀ (b : δ), IsLUB {a | ∃ i, f i b = a} (g b)
⊢ Measurable g
[PROOFSTEP]
change ∀ b, IsLUB (range fun i => f i b) (g b) at hg
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), Measurable (f i)
hg : ∀ (b : δ), IsLUB (range fun i => f i b) (g b)
⊢ Measurable g
[PROOFSTEP]
rw [‹BorelSpace α›.measurable_eq, borel_eq_generateFrom_Ioi α]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), Measurable (f i)
hg : ∀ (b : δ), IsLUB (range fun i => f i b) (g b)
⊢ Measurable g
[PROOFSTEP]
apply measurable_generateFrom
[GOAL]
case h
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), Measurable (f i)
hg : ∀ (b : δ), IsLUB (range fun i => f i b) (g b)
⊢ ∀ (t : Set α), t ∈ range Ioi → MeasurableSet (g ⁻¹' t)
[PROOFSTEP]
rintro _ ⟨a, rfl⟩
[GOAL]
case h.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), Measurable (f i)
hg : ∀ (b : δ), IsLUB (range fun i => f i b) (g b)
a : α
⊢ MeasurableSet (g ⁻¹' Ioi a)
[PROOFSTEP]
simp_rw [Set.preimage, mem_Ioi, lt_isLUB_iff (hg _), exists_range_iff, setOf_exists]
[GOAL]
case h.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), Measurable (f i)
hg : ∀ (b : δ), IsLUB (range fun i => f i b) (g b)
a : α
⊢ MeasurableSet (⋃ (i : ι), {x | a < f i x})
[PROOFSTEP]
exact MeasurableSet.iUnion fun i => hf i (isOpen_lt' _).measurableSet
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
hι : Nonempty ι
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
⊢ AEMeasurable g
[PROOFSTEP]
let p : δ → (ι → α) → Prop := fun x f' => IsLUB {a | ∃ i, f' i = a} (g x)
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
hι : Nonempty ι
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
p : δ → (ι → α) → Prop := fun x f' => IsLUB {a | ∃ i, f' i = a} (g x)
⊢ AEMeasurable g
[PROOFSTEP]
let g_seq x := ite (x ∈ aeSeqSet hf p) (g x) (⟨g x⟩ : Nonempty α).some
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
hι : Nonempty ι
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
p : δ → (ι → α) → Prop := fun x f' => IsLUB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := fun x => if x ∈ aeSeqSet hf p then g x else Nonempty.some (_ : Nonempty α)
⊢ AEMeasurable g
[PROOFSTEP]
have hg_seq : ∀ b, IsLUB {a | ∃ i, aeSeq hf p i b = a} (g_seq b) :=
  by
  intro b
  haveI hα : Nonempty α := Nonempty.map g ⟨b⟩
  simp only [aeSeq]
  split_ifs with h
  · have h_set_eq : {a : α | ∃ i : ι, (hf i).mk (f i) b = a} = {a : α | ∃ i : ι, f i b = a} :=
      by
      ext x
      simp_rw [Set.mem_setOf_eq, aeSeq.mk_eq_fun_of_mem_aeSeqSet hf h]
    rw [h_set_eq]
    exact aeSeq.fun_prop_of_mem_aeSeqSet hf h
  · have h_singleton : {a : α | ∃ _ : ι, hα.some = a} = { hα.some } :=
      by
      ext1 x
      exact ⟨fun hx => hx.choose_spec.symm, fun hx => ⟨hι.some, hx.symm⟩⟩
    rw [h_singleton]
    exact isLUB_singleton
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
hι : Nonempty ι
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
p : δ → (ι → α) → Prop := fun x f' => IsLUB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := fun x => if x ∈ aeSeqSet hf p then g x else Nonempty.some (_ : Nonempty α)
⊢ ∀ (b : δ), IsLUB {a | ∃ i, aeSeq hf p i b = a} (g_seq b)
[PROOFSTEP]
intro b
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
hι : Nonempty ι
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
p : δ → (ι → α) → Prop := fun x f' => IsLUB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := fun x => if x ∈ aeSeqSet hf p then g x else Nonempty.some (_ : Nonempty α)
b : δ
⊢ IsLUB {a | ∃ i, aeSeq hf p i b = a} (g_seq b)
[PROOFSTEP]
haveI hα : Nonempty α := Nonempty.map g ⟨b⟩
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
hι : Nonempty ι
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
p : δ → (ι → α) → Prop := fun x f' => IsLUB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := fun x => if x ∈ aeSeqSet hf p then g x else Nonempty.some (_ : Nonempty α)
b : δ
hα : Nonempty α
⊢ IsLUB {a | ∃ i, aeSeq hf p i b = a} (g_seq b)
[PROOFSTEP]
simp only [aeSeq]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
hι : Nonempty ι
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
p : δ → (ι → α) → Prop := fun x f' => IsLUB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := fun x => if x ∈ aeSeqSet hf p then g x else Nonempty.some (_ : Nonempty α)
b : δ
hα : Nonempty α
⊢ IsLUB
    {a |
      ∃ i,
        (if b ∈ aeSeqSet hf fun x f' => IsLUB {a | ∃ i, f' i = a} (g x) then mk (f i) (_ : AEMeasurable (f i)) b
          else Nonempty.some (_ : Nonempty α)) =
          a}
    (if b ∈ aeSeqSet hf fun x f' => IsLUB {a | ∃ i, f' i = a} (g x) then g b else Nonempty.some (_ : Nonempty α))
[PROOFSTEP]
split_ifs with h
[GOAL]
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
hι : Nonempty ι
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
p : δ → (ι → α) → Prop := fun x f' => IsLUB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := fun x => if x ∈ aeSeqSet hf p then g x else Nonempty.some (_ : Nonempty α)
b : δ
hα : Nonempty α
h : b ∈ aeSeqSet hf fun x f' => IsLUB {a | ∃ i, f' i = a} (g x)
⊢ IsLUB {a | ∃ i, mk (f i) (_ : AEMeasurable (f i)) b = a} (g b)
[PROOFSTEP]
have h_set_eq : {a : α | ∃ i : ι, (hf i).mk (f i) b = a} = {a : α | ∃ i : ι, f i b = a} :=
  by
  ext x
  simp_rw [Set.mem_setOf_eq, aeSeq.mk_eq_fun_of_mem_aeSeqSet hf h]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
hι : Nonempty ι
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
p : δ → (ι → α) → Prop := fun x f' => IsLUB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := fun x => if x ∈ aeSeqSet hf p then g x else Nonempty.some (_ : Nonempty α)
b : δ
hα : Nonempty α
h : b ∈ aeSeqSet hf fun x f' => IsLUB {a | ∃ i, f' i = a} (g x)
⊢ {a | ∃ i, mk (f i) (_ : AEMeasurable (f i)) b = a} = {a | ∃ i, f i b = a}
[PROOFSTEP]
ext x
[GOAL]
case h
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
hι : Nonempty ι
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
p : δ → (ι → α) → Prop := fun x f' => IsLUB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := fun x => if x ∈ aeSeqSet hf p then g x else Nonempty.some (_ : Nonempty α)
b : δ
hα : Nonempty α
h : b ∈ aeSeqSet hf fun x f' => IsLUB {a | ∃ i, f' i = a} (g x)
x : α
⊢ x ∈ {a | ∃ i, mk (f i) (_ : AEMeasurable (f i)) b = a} ↔ x ∈ {a | ∃ i, f i b = a}
[PROOFSTEP]
simp_rw [Set.mem_setOf_eq, aeSeq.mk_eq_fun_of_mem_aeSeqSet hf h]
[GOAL]
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
hι : Nonempty ι
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
p : δ → (ι → α) → Prop := fun x f' => IsLUB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := fun x => if x ∈ aeSeqSet hf p then g x else Nonempty.some (_ : Nonempty α)
b : δ
hα : Nonempty α
h : b ∈ aeSeqSet hf fun x f' => IsLUB {a | ∃ i, f' i = a} (g x)
h_set_eq : {a | ∃ i, mk (f i) (_ : AEMeasurable (f i)) b = a} = {a | ∃ i, f i b = a}
⊢ IsLUB {a | ∃ i, mk (f i) (_ : AEMeasurable (f i)) b = a} (g b)
[PROOFSTEP]
rw [h_set_eq]
[GOAL]
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
hι : Nonempty ι
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
p : δ → (ι → α) → Prop := fun x f' => IsLUB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := fun x => if x ∈ aeSeqSet hf p then g x else Nonempty.some (_ : Nonempty α)
b : δ
hα : Nonempty α
h : b ∈ aeSeqSet hf fun x f' => IsLUB {a | ∃ i, f' i = a} (g x)
h_set_eq : {a | ∃ i, mk (f i) (_ : AEMeasurable (f i)) b = a} = {a | ∃ i, f i b = a}
⊢ IsLUB {a | ∃ i, f i b = a} (g b)
[PROOFSTEP]
exact aeSeq.fun_prop_of_mem_aeSeqSet hf h
[GOAL]
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
hι : Nonempty ι
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
p : δ → (ι → α) → Prop := fun x f' => IsLUB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := fun x => if x ∈ aeSeqSet hf p then g x else Nonempty.some (_ : Nonempty α)
b : δ
hα : Nonempty α
h : ¬b ∈ aeSeqSet hf fun x f' => IsLUB {a | ∃ i, f' i = a} (g x)
⊢ IsLUB {a | ∃ i, Nonempty.some (_ : Nonempty α) = a} (Nonempty.some (_ : Nonempty α))
[PROOFSTEP]
have h_singleton : {a : α | ∃ _ : ι, hα.some = a} = { hα.some } :=
  by
  ext1 x
  exact ⟨fun hx => hx.choose_spec.symm, fun hx => ⟨hι.some, hx.symm⟩⟩
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
hι : Nonempty ι
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
p : δ → (ι → α) → Prop := fun x f' => IsLUB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := fun x => if x ∈ aeSeqSet hf p then g x else Nonempty.some (_ : Nonempty α)
b : δ
hα : Nonempty α
h : ¬b ∈ aeSeqSet hf fun x f' => IsLUB {a | ∃ i, f' i = a} (g x)
⊢ {a | ∃ x, Nonempty.some hα = a} = {Nonempty.some hα}
[PROOFSTEP]
ext1 x
[GOAL]
case h
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
hι : Nonempty ι
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
p : δ → (ι → α) → Prop := fun x f' => IsLUB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := fun x => if x ∈ aeSeqSet hf p then g x else Nonempty.some (_ : Nonempty α)
b : δ
hα : Nonempty α
h : ¬b ∈ aeSeqSet hf fun x f' => IsLUB {a | ∃ i, f' i = a} (g x)
x : α
⊢ x ∈ {a | ∃ x, Nonempty.some hα = a} ↔ x ∈ {Nonempty.some hα}
[PROOFSTEP]
exact ⟨fun hx => hx.choose_spec.symm, fun hx => ⟨hι.some, hx.symm⟩⟩
[GOAL]
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
hι : Nonempty ι
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
p : δ → (ι → α) → Prop := fun x f' => IsLUB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := fun x => if x ∈ aeSeqSet hf p then g x else Nonempty.some (_ : Nonempty α)
b : δ
hα : Nonempty α
h : ¬b ∈ aeSeqSet hf fun x f' => IsLUB {a | ∃ i, f' i = a} (g x)
h_singleton : {a | ∃ x, Nonempty.some hα = a} = {Nonempty.some hα}
⊢ IsLUB {a | ∃ i, Nonempty.some (_ : Nonempty α) = a} (Nonempty.some (_ : Nonempty α))
[PROOFSTEP]
rw [h_singleton]
[GOAL]
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
hι : Nonempty ι
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
p : δ → (ι → α) → Prop := fun x f' => IsLUB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := fun x => if x ∈ aeSeqSet hf p then g x else Nonempty.some (_ : Nonempty α)
b : δ
hα : Nonempty α
h : ¬b ∈ aeSeqSet hf fun x f' => IsLUB {a | ∃ i, f' i = a} (g x)
h_singleton : {a | ∃ x, Nonempty.some hα = a} = {Nonempty.some hα}
⊢ IsLUB {Nonempty.some hα} (Nonempty.some (_ : Nonempty α))
[PROOFSTEP]
exact isLUB_singleton
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
hι : Nonempty ι
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
p : δ → (ι → α) → Prop := fun x f' => IsLUB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := fun x => if x ∈ aeSeqSet hf p then g x else Nonempty.some (_ : Nonempty α)
hg_seq : ∀ (b : δ), IsLUB {a | ∃ i, aeSeq hf p i b = a} (g_seq b)
⊢ AEMeasurable g
[PROOFSTEP]
refine' ⟨g_seq, Measurable.isLUB (aeSeq.measurable hf p) hg_seq, _⟩
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
hι : Nonempty ι
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
p : δ → (ι → α) → Prop := fun x f' => IsLUB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := fun x => if x ∈ aeSeqSet hf p then g x else Nonempty.some (_ : Nonempty α)
hg_seq : ∀ (b : δ), IsLUB {a | ∃ i, aeSeq hf p i b = a} (g_seq b)
⊢ g =ᵐ[μ] g_seq
[PROOFSTEP]
exact
  (ite_ae_eq_of_measure_compl_zero g (fun x => (⟨g x⟩ : Nonempty α).some) (aeSeqSet hf p)
      (aeSeq.measure_compl_aeSeqSet_eq_zero hf hg)).symm
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
⊢ AEMeasurable g
[PROOFSTEP]
rcases eq_zero_or_neZero μ with rfl | hμ
[GOAL]
case inl
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂0, IsLUB {a | ∃ i, f i b = a} (g b)
⊢ AEMeasurable g
[PROOFSTEP]
exact aemeasurable_zero_measure
[GOAL]
case inr
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
hμ : NeZero μ
⊢ AEMeasurable g
[PROOFSTEP]
by_cases hι : Nonempty ι
[GOAL]
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
hμ : NeZero μ
hι : Nonempty ι
⊢ AEMeasurable g
[PROOFSTEP]
exact AEMeasurable.is_lub_of_nonempty hι hf hg
[GOAL]
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
hμ : NeZero μ
hι : ¬Nonempty ι
⊢ AEMeasurable g
[PROOFSTEP]
suffices ∃ x, g =ᵐ[μ] fun _ => g x by exact ⟨fun _ => g this.choose, measurable_const, this.choose_spec⟩
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
hμ : NeZero μ
hι : ¬Nonempty ι
this : ∃ x, g =ᵐ[μ] fun x_1 => g x
⊢ AEMeasurable g
[PROOFSTEP]
exact ⟨fun _ => g this.choose, measurable_const, this.choose_spec⟩
[GOAL]
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
hμ : NeZero μ
hι : ¬Nonempty ι
⊢ ∃ x, g =ᵐ[μ] fun x_1 => g x
[PROOFSTEP]
have h_empty : ∀ x, {a : α | ∃ i : ι, f i x = a} = ∅ := by
  intro x
  ext1 y
  rw [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false_iff]
  exact fun hi => hι (nonempty_of_exists hi)
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
hμ : NeZero μ
hι : ¬Nonempty ι
⊢ ∀ (x : δ), {a | ∃ i, f i x = a} = ∅
[PROOFSTEP]
intro x
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
hμ : NeZero μ
hι : ¬Nonempty ι
x : δ
⊢ {a | ∃ i, f i x = a} = ∅
[PROOFSTEP]
ext1 y
[GOAL]
case h
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
hμ : NeZero μ
hι : ¬Nonempty ι
x : δ
y : α
⊢ y ∈ {a | ∃ i, f i x = a} ↔ y ∈ ∅
[PROOFSTEP]
rw [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false_iff]
[GOAL]
case h
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
hμ : NeZero μ
hι : ¬Nonempty ι
x : δ
y : α
⊢ ¬∃ i, f i x = y
[PROOFSTEP]
exact fun hi => hι (nonempty_of_exists hi)
[GOAL]
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
hμ : NeZero μ
hι : ¬Nonempty ι
h_empty : ∀ (x : δ), {a | ∃ i, f i x = a} = ∅
⊢ ∃ x, g =ᵐ[μ] fun x_1 => g x
[PROOFSTEP]
simp_rw [h_empty] at hg
[GOAL]
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hμ : NeZero μ
hι : ¬Nonempty ι
h_empty : ∀ (x : δ), {a | ∃ i, f i x = a} = ∅
hg : ∀ᵐ (b : δ) ∂μ, IsLUB ∅ (g b)
⊢ ∃ x, g =ᵐ[μ] fun x_1 => g x
[PROOFSTEP]
exact ⟨hg.exists.choose, hg.mono fun y hy => IsLUB.unique hy hg.exists.choose_spec⟩
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), Measurable (f i)
hg : ∀ (b : δ), IsGLB {a | ∃ i, f i b = a} (g b)
⊢ Measurable g
[PROOFSTEP]
change ∀ b, IsGLB (range fun i => f i b) (g b) at hg
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), Measurable (f i)
hg : ∀ (b : δ), IsGLB (range fun i => f i b) (g b)
⊢ Measurable g
[PROOFSTEP]
rw [‹BorelSpace α›.measurable_eq, borel_eq_generateFrom_Iio α]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), Measurable (f i)
hg : ∀ (b : δ), IsGLB (range fun i => f i b) (g b)
⊢ Measurable g
[PROOFSTEP]
apply measurable_generateFrom
[GOAL]
case h
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), Measurable (f i)
hg : ∀ (b : δ), IsGLB (range fun i => f i b) (g b)
⊢ ∀ (t : Set α), t ∈ range Iio → MeasurableSet (g ⁻¹' t)
[PROOFSTEP]
rintro _ ⟨a, rfl⟩
[GOAL]
case h.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), Measurable (f i)
hg : ∀ (b : δ), IsGLB (range fun i => f i b) (g b)
a : α
⊢ MeasurableSet (g ⁻¹' Iio a)
[PROOFSTEP]
simp_rw [Set.preimage, mem_Iio, isGLB_lt_iff (hg _), exists_range_iff, setOf_exists]
[GOAL]
case h.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), Measurable (f i)
hg : ∀ (b : δ), IsGLB (range fun i => f i b) (g b)
a : α
⊢ MeasurableSet (⋃ (i : ι), {x | f i x < a})
[PROOFSTEP]
exact MeasurableSet.iUnion fun i => hf i (isOpen_gt' _).measurableSet
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsGLB {a | ∃ i, f i b = a} (g b)
⊢ AEMeasurable g
[PROOFSTEP]
nontriviality α
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsGLB {a | ∃ i, f i b = a} (g b)
✝ : Nontrivial α
⊢ AEMeasurable g
[PROOFSTEP]
haveI hα : Nonempty α := inferInstance
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsGLB {a | ∃ i, f i b = a} (g b)
✝ : Nontrivial α
hα : Nonempty α
⊢ AEMeasurable g
[PROOFSTEP]
cases' isEmpty_or_nonempty ι with hι hι
[GOAL]
case inl
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsGLB {a | ∃ i, f i b = a} (g b)
✝ : Nontrivial α
hα : Nonempty α
hι : IsEmpty ι
⊢ AEMeasurable g
[PROOFSTEP]
simp only [IsEmpty.exists_iff, setOf_false, isGLB_empty_iff] at hg
[GOAL]
case inl
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
✝ : Nontrivial α
hα : Nonempty α
hι : IsEmpty ι
hg : ∀ᵐ (b : δ) ∂μ, IsTop (g b)
⊢ AEMeasurable g
[PROOFSTEP]
exact aemeasurable_const' (hg.mono fun a ha => hg.mono fun b hb => (hb _).antisymm (ha _))
[GOAL]
case inr
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsGLB {a | ∃ i, f i b = a} (g b)
✝ : Nontrivial α
hα : Nonempty α
hι : Nonempty ι
⊢ AEMeasurable g
[PROOFSTEP]
let p : δ → (ι → α) → Prop := fun x f' => IsGLB {a | ∃ i, f' i = a} (g x)
[GOAL]
case inr
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsGLB {a | ∃ i, f i b = a} (g b)
✝ : Nontrivial α
hα : Nonempty α
hι : Nonempty ι
p : δ → (ι → α) → Prop := fun x f' => IsGLB {a | ∃ i, f' i = a} (g x)
⊢ AEMeasurable g
[PROOFSTEP]
let g_seq := (aeSeqSet hf p).piecewise g fun _ => hα.some
[GOAL]
case inr
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsGLB {a | ∃ i, f i b = a} (g b)
✝ : Nontrivial α
hα : Nonempty α
hι : Nonempty ι
p : δ → (ι → α) → Prop := fun x f' => IsGLB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := piecewise (aeSeqSet hf p) g fun x => Nonempty.some hα
⊢ AEMeasurable g
[PROOFSTEP]
have hg_seq : ∀ b, IsGLB {a | ∃ i, aeSeq hf p i b = a} (g_seq b) :=
  by
  intro b
  simp only [aeSeq, Set.piecewise]
  split_ifs with h
  · have h_set_eq : {a : α | ∃ i : ι, (hf i).mk (f i) b = a} = {a : α | ∃ i : ι, f i b = a} :=
      by
      ext x
      simp_rw [Set.mem_setOf_eq, aeSeq.mk_eq_fun_of_mem_aeSeqSet hf h]
    rw [h_set_eq]
    exact aeSeq.fun_prop_of_mem_aeSeqSet hf h
  · exact IsLeast.isGLB ⟨(@exists_const (hα.some = hα.some) ι _).2 rfl, fun x ⟨i, hi⟩ => hi.le⟩
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsGLB {a | ∃ i, f i b = a} (g b)
✝ : Nontrivial α
hα : Nonempty α
hι : Nonempty ι
p : δ → (ι → α) → Prop := fun x f' => IsGLB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := piecewise (aeSeqSet hf p) g fun x => Nonempty.some hα
⊢ ∀ (b : δ), IsGLB {a | ∃ i, aeSeq hf p i b = a} (g_seq b)
[PROOFSTEP]
intro b
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsGLB {a | ∃ i, f i b = a} (g b)
✝ : Nontrivial α
hα : Nonempty α
hι : Nonempty ι
p : δ → (ι → α) → Prop := fun x f' => IsGLB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := piecewise (aeSeqSet hf p) g fun x => Nonempty.some hα
b : δ
⊢ IsGLB {a | ∃ i, aeSeq hf p i b = a} (g_seq b)
[PROOFSTEP]
simp only [aeSeq, Set.piecewise]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsGLB {a | ∃ i, f i b = a} (g b)
✝ : Nontrivial α
hα : Nonempty α
hι : Nonempty ι
p : δ → (ι → α) → Prop := fun x f' => IsGLB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := piecewise (aeSeqSet hf p) g fun x => Nonempty.some hα
b : δ
⊢ IsGLB
    {a |
      ∃ i,
        (if b ∈ aeSeqSet hf fun x f' => IsGLB {a | ∃ i, f' i = a} (g x) then mk (f i) (_ : AEMeasurable (f i)) b
          else Nonempty.some (_ : Nonempty α)) =
          a}
    (if b ∈ aeSeqSet hf fun x f' => IsGLB {a | ∃ i, f' i = a} (g x) then g b else Nonempty.some hα)
[PROOFSTEP]
split_ifs with h
[GOAL]
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsGLB {a | ∃ i, f i b = a} (g b)
✝ : Nontrivial α
hα : Nonempty α
hι : Nonempty ι
p : δ → (ι → α) → Prop := fun x f' => IsGLB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := piecewise (aeSeqSet hf p) g fun x => Nonempty.some hα
b : δ
h : b ∈ aeSeqSet hf fun x f' => IsGLB {a | ∃ i, f' i = a} (g x)
⊢ IsGLB {a | ∃ i, mk (f i) (_ : AEMeasurable (f i)) b = a} (g b)
[PROOFSTEP]
have h_set_eq : {a : α | ∃ i : ι, (hf i).mk (f i) b = a} = {a : α | ∃ i : ι, f i b = a} :=
  by
  ext x
  simp_rw [Set.mem_setOf_eq, aeSeq.mk_eq_fun_of_mem_aeSeqSet hf h]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsGLB {a | ∃ i, f i b = a} (g b)
✝ : Nontrivial α
hα : Nonempty α
hι : Nonempty ι
p : δ → (ι → α) → Prop := fun x f' => IsGLB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := piecewise (aeSeqSet hf p) g fun x => Nonempty.some hα
b : δ
h : b ∈ aeSeqSet hf fun x f' => IsGLB {a | ∃ i, f' i = a} (g x)
⊢ {a | ∃ i, mk (f i) (_ : AEMeasurable (f i)) b = a} = {a | ∃ i, f i b = a}
[PROOFSTEP]
ext x
[GOAL]
case h
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsGLB {a | ∃ i, f i b = a} (g b)
✝ : Nontrivial α
hα : Nonempty α
hι : Nonempty ι
p : δ → (ι → α) → Prop := fun x f' => IsGLB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := piecewise (aeSeqSet hf p) g fun x => Nonempty.some hα
b : δ
h : b ∈ aeSeqSet hf fun x f' => IsGLB {a | ∃ i, f' i = a} (g x)
x : α
⊢ x ∈ {a | ∃ i, mk (f i) (_ : AEMeasurable (f i)) b = a} ↔ x ∈ {a | ∃ i, f i b = a}
[PROOFSTEP]
simp_rw [Set.mem_setOf_eq, aeSeq.mk_eq_fun_of_mem_aeSeqSet hf h]
[GOAL]
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsGLB {a | ∃ i, f i b = a} (g b)
✝ : Nontrivial α
hα : Nonempty α
hι : Nonempty ι
p : δ → (ι → α) → Prop := fun x f' => IsGLB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := piecewise (aeSeqSet hf p) g fun x => Nonempty.some hα
b : δ
h : b ∈ aeSeqSet hf fun x f' => IsGLB {a | ∃ i, f' i = a} (g x)
h_set_eq : {a | ∃ i, mk (f i) (_ : AEMeasurable (f i)) b = a} = {a | ∃ i, f i b = a}
⊢ IsGLB {a | ∃ i, mk (f i) (_ : AEMeasurable (f i)) b = a} (g b)
[PROOFSTEP]
rw [h_set_eq]
[GOAL]
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsGLB {a | ∃ i, f i b = a} (g b)
✝ : Nontrivial α
hα : Nonempty α
hι : Nonempty ι
p : δ → (ι → α) → Prop := fun x f' => IsGLB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := piecewise (aeSeqSet hf p) g fun x => Nonempty.some hα
b : δ
h : b ∈ aeSeqSet hf fun x f' => IsGLB {a | ∃ i, f' i = a} (g x)
h_set_eq : {a | ∃ i, mk (f i) (_ : AEMeasurable (f i)) b = a} = {a | ∃ i, f i b = a}
⊢ IsGLB {a | ∃ i, f i b = a} (g b)
[PROOFSTEP]
exact aeSeq.fun_prop_of_mem_aeSeqSet hf h
[GOAL]
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsGLB {a | ∃ i, f i b = a} (g b)
✝ : Nontrivial α
hα : Nonempty α
hι : Nonempty ι
p : δ → (ι → α) → Prop := fun x f' => IsGLB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := piecewise (aeSeqSet hf p) g fun x => Nonempty.some hα
b : δ
h : ¬b ∈ aeSeqSet hf fun x f' => IsGLB {a | ∃ i, f' i = a} (g x)
⊢ IsGLB {a | ∃ i, Nonempty.some (_ : Nonempty α) = a} (Nonempty.some hα)
[PROOFSTEP]
exact IsLeast.isGLB ⟨(@exists_const (hα.some = hα.some) ι _).2 rfl, fun x ⟨i, hi⟩ => hi.le⟩
[GOAL]
case inr
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsGLB {a | ∃ i, f i b = a} (g b)
✝ : Nontrivial α
hα : Nonempty α
hι : Nonempty ι
p : δ → (ι → α) → Prop := fun x f' => IsGLB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := piecewise (aeSeqSet hf p) g fun x => Nonempty.some hα
hg_seq : ∀ (b : δ), IsGLB {a | ∃ i, aeSeq hf p i b = a} (g_seq b)
⊢ AEMeasurable g
[PROOFSTEP]
refine' ⟨g_seq, Measurable.isGLB (aeSeq.measurable hf p) hg_seq, _⟩
[GOAL]
case inr
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_6
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i)
hg : ∀ᵐ (b : δ) ∂μ, IsGLB {a | ∃ i, f i b = a} (g b)
✝ : Nontrivial α
hα : Nonempty α
hι : Nonempty ι
p : δ → (ι → α) → Prop := fun x f' => IsGLB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := piecewise (aeSeqSet hf p) g fun x => Nonempty.some hα
hg_seq : ∀ (b : δ), IsGLB {a | ∃ i, aeSeq hf p i b = a} (g_seq b)
⊢ g =ᵐ[μ] g_seq
[PROOFSTEP]
exact
  (ite_ae_eq_of_measure_compl_zero g (fun _ => hα.some) (aeSeqSet hf p)
      (aeSeq.measure_compl_aeSeqSet_eq_zero hf hg)).symm
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
h' : ∀ (x : α), x ∈ s → ∃ y, x < y
⊢ MeasurableSet s
[PROOFSTEP]
choose! M hM using h'
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
M : α → α
hM : ∀ (x : α), x ∈ s → x < M x
⊢ MeasurableSet s
[PROOFSTEP]
suffices H : (s \ interior s).Countable
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
M : α → α
hM : ∀ (x : α), x ∈ s → x < M x
H : Set.Countable (s \ interior s)
⊢ MeasurableSet s
[PROOFSTEP]
have : s = interior s ∪ s \ interior s := by rw [union_diff_cancel interior_subset]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
M : α → α
hM : ∀ (x : α), x ∈ s → x < M x
H : Set.Countable (s \ interior s)
⊢ s = interior s ∪ s \ interior s
[PROOFSTEP]
rw [union_diff_cancel interior_subset]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
M : α → α
hM : ∀ (x : α), x ∈ s → x < M x
H : Set.Countable (s \ interior s)
this : s = interior s ∪ s \ interior s
⊢ MeasurableSet s
[PROOFSTEP]
rw [this]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
M : α → α
hM : ∀ (x : α), x ∈ s → x < M x
H : Set.Countable (s \ interior s)
this : s = interior s ∪ s \ interior s
⊢ MeasurableSet (interior s ∪ s \ interior s)
[PROOFSTEP]
exact isOpen_interior.measurableSet.union H.measurableSet
[GOAL]
case H
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
M : α → α
hM : ∀ (x : α), x ∈ s → x < M x
⊢ Set.Countable (s \ interior s)
[PROOFSTEP]
have A : ∀ x ∈ s, ∃ y ∈ Ioi x, Ioo x y ⊆ s := fun x hx =>
  (mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' (hM x hx)).1 (h x hx)
[GOAL]
case H
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
M : α → α
hM : ∀ (x : α), x ∈ s → x < M x
A : ∀ (x : α), x ∈ s → ∃ y, y ∈ Ioi x ∧ Ioo x y ⊆ s
⊢ Set.Countable (s \ interior s)
[PROOFSTEP]
choose! y hy h'y using A
[GOAL]
case H
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
M : α → α
hM : ∀ (x : α), x ∈ s → x < M x
y : α → α
hy : ∀ (x : α), x ∈ s → y x ∈ Ioi x
h'y : ∀ (x : α), x ∈ s → Ioo x (y x) ⊆ s
⊢ Set.Countable (s \ interior s)
[PROOFSTEP]
have B : Set.PairwiseDisjoint (s \ interior s) fun x => Ioo x (y x) :=
  by
  intro x hx x' hx' hxx'
  rcases lt_or_gt_of_ne hxx' with (h' | h')
  · refine disjoint_left.2 fun z hz h'z => ?_
    have : x' ∈ interior s := mem_interior.2 ⟨Ioo x (y x), h'y _ hx.1, isOpen_Ioo, ⟨h', h'z.1.trans hz.2⟩⟩
    exact False.elim (hx'.2 this)
  · refine disjoint_left.2 fun z hz h'z => ?_
    have : x ∈ interior s := mem_interior.2 ⟨Ioo x' (y x'), h'y _ hx'.1, isOpen_Ioo, ⟨h', hz.1.trans h'z.2⟩⟩
    exact False.elim (hx.2 this)
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
M : α → α
hM : ∀ (x : α), x ∈ s → x < M x
y : α → α
hy : ∀ (x : α), x ∈ s → y x ∈ Ioi x
h'y : ∀ (x : α), x ∈ s → Ioo x (y x) ⊆ s
⊢ PairwiseDisjoint (s \ interior s) fun x => Ioo x (y x)
[PROOFSTEP]
intro x hx x' hx' hxx'
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
M : α → α
hM : ∀ (x : α), x ∈ s → x < M x
y : α → α
hy : ∀ (x : α), x ∈ s → y x ∈ Ioi x
h'y : ∀ (x : α), x ∈ s → Ioo x (y x) ⊆ s
x : α
hx : x ∈ s \ interior s
x' : α
hx' : x' ∈ s \ interior s
hxx' : x ≠ x'
⊢ (Disjoint on fun x => Ioo x (y x)) x x'
[PROOFSTEP]
rcases lt_or_gt_of_ne hxx' with (h' | h')
[GOAL]
case inl
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
M : α → α
hM : ∀ (x : α), x ∈ s → x < M x
y : α → α
hy : ∀ (x : α), x ∈ s → y x ∈ Ioi x
h'y : ∀ (x : α), x ∈ s → Ioo x (y x) ⊆ s
x : α
hx : x ∈ s \ interior s
x' : α
hx' : x' ∈ s \ interior s
hxx' : x ≠ x'
h' : x < x'
⊢ (Disjoint on fun x => Ioo x (y x)) x x'
[PROOFSTEP]
refine disjoint_left.2 fun z hz h'z => ?_
[GOAL]
case inl
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
M : α → α
hM : ∀ (x : α), x ∈ s → x < M x
y : α → α
hy : ∀ (x : α), x ∈ s → y x ∈ Ioi x
h'y : ∀ (x : α), x ∈ s → Ioo x (y x) ⊆ s
x : α
hx : x ∈ s \ interior s
x' : α
hx' : x' ∈ s \ interior s
hxx' : x ≠ x'
h' : x < x'
z : α
hz : z ∈ (fun x => Ioo x (y x)) x
h'z : z ∈ (fun x => Ioo x (y x)) x'
⊢ False
[PROOFSTEP]
have : x' ∈ interior s := mem_interior.2 ⟨Ioo x (y x), h'y _ hx.1, isOpen_Ioo, ⟨h', h'z.1.trans hz.2⟩⟩
[GOAL]
case inl
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
M : α → α
hM : ∀ (x : α), x ∈ s → x < M x
y : α → α
hy : ∀ (x : α), x ∈ s → y x ∈ Ioi x
h'y : ∀ (x : α), x ∈ s → Ioo x (y x) ⊆ s
x : α
hx : x ∈ s \ interior s
x' : α
hx' : x' ∈ s \ interior s
hxx' : x ≠ x'
h' : x < x'
z : α
hz : z ∈ (fun x => Ioo x (y x)) x
h'z : z ∈ (fun x => Ioo x (y x)) x'
this : x' ∈ interior s
⊢ False
[PROOFSTEP]
exact False.elim (hx'.2 this)
[GOAL]
case inr
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
M : α → α
hM : ∀ (x : α), x ∈ s → x < M x
y : α → α
hy : ∀ (x : α), x ∈ s → y x ∈ Ioi x
h'y : ∀ (x : α), x ∈ s → Ioo x (y x) ⊆ s
x : α
hx : x ∈ s \ interior s
x' : α
hx' : x' ∈ s \ interior s
hxx' : x ≠ x'
h' : x > x'
⊢ (Disjoint on fun x => Ioo x (y x)) x x'
[PROOFSTEP]
refine disjoint_left.2 fun z hz h'z => ?_
[GOAL]
case inr
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
M : α → α
hM : ∀ (x : α), x ∈ s → x < M x
y : α → α
hy : ∀ (x : α), x ∈ s → y x ∈ Ioi x
h'y : ∀ (x : α), x ∈ s → Ioo x (y x) ⊆ s
x : α
hx : x ∈ s \ interior s
x' : α
hx' : x' ∈ s \ interior s
hxx' : x ≠ x'
h' : x > x'
z : α
hz : z ∈ (fun x => Ioo x (y x)) x
h'z : z ∈ (fun x => Ioo x (y x)) x'
⊢ False
[PROOFSTEP]
have : x ∈ interior s := mem_interior.2 ⟨Ioo x' (y x'), h'y _ hx'.1, isOpen_Ioo, ⟨h', hz.1.trans h'z.2⟩⟩
[GOAL]
case inr
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
M : α → α
hM : ∀ (x : α), x ∈ s → x < M x
y : α → α
hy : ∀ (x : α), x ∈ s → y x ∈ Ioi x
h'y : ∀ (x : α), x ∈ s → Ioo x (y x) ⊆ s
x : α
hx : x ∈ s \ interior s
x' : α
hx' : x' ∈ s \ interior s
hxx' : x ≠ x'
h' : x > x'
z : α
hz : z ∈ (fun x => Ioo x (y x)) x
h'z : z ∈ (fun x => Ioo x (y x)) x'
this : x ∈ interior s
⊢ False
[PROOFSTEP]
exact False.elim (hx.2 this)
[GOAL]
case H
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
M : α → α
hM : ∀ (x : α), x ∈ s → x < M x
y : α → α
hy : ∀ (x : α), x ∈ s → y x ∈ Ioi x
h'y : ∀ (x : α), x ∈ s → Ioo x (y x) ⊆ s
B : PairwiseDisjoint (s \ interior s) fun x => Ioo x (y x)
⊢ Set.Countable (s \ interior s)
[PROOFSTEP]
exact B.countable_of_Ioo fun x hx => hy x hx.1
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
⊢ MeasurableSet s
[PROOFSTEP]
by_cases H : ∃ x ∈ s, IsTop x
[GOAL]
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
H : ∃ x, x ∈ s ∧ IsTop x
⊢ MeasurableSet s
[PROOFSTEP]
rcases H with ⟨x₀, x₀s, h₀⟩
[GOAL]
case pos.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
x₀ : α
x₀s : x₀ ∈ s
h₀ : IsTop x₀
⊢ MeasurableSet s
[PROOFSTEP]
have : s = { x₀ } ∪ s \ { x₀ } := by rw [union_diff_cancel (singleton_subset_iff.2 x₀s)]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
x₀ : α
x₀s : x₀ ∈ s
h₀ : IsTop x₀
⊢ s = {x₀} ∪ s \ {x₀}
[PROOFSTEP]
rw [union_diff_cancel (singleton_subset_iff.2 x₀s)]
[GOAL]
case pos.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
x₀ : α
x₀s : x₀ ∈ s
h₀ : IsTop x₀
this : s = {x₀} ∪ s \ {x₀}
⊢ MeasurableSet s
[PROOFSTEP]
rw [this]
[GOAL]
case pos.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
x₀ : α
x₀s : x₀ ∈ s
h₀ : IsTop x₀
this : s = {x₀} ∪ s \ {x₀}
⊢ MeasurableSet ({x₀} ∪ s \ {x₀})
[PROOFSTEP]
refine' (measurableSet_singleton _).union _
[GOAL]
case pos.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
x₀ : α
x₀s : x₀ ∈ s
h₀ : IsTop x₀
this : s = {x₀} ∪ s \ {x₀}
⊢ MeasurableSet (s \ {x₀})
[PROOFSTEP]
have A : ∀ x ∈ s \ { x₀ }, x < x₀ := fun x hx => lt_of_le_of_ne (h₀ _) (by simpa using hx.2)
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
x₀ : α
x₀s : x₀ ∈ s
h₀ : IsTop x₀
this : s = {x₀} ∪ s \ {x₀}
x : α
hx : x ∈ s \ {x₀}
⊢ x ≠ x₀
[PROOFSTEP]
simpa using hx.2
[GOAL]
case pos.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
x₀ : α
x₀s : x₀ ∈ s
h₀ : IsTop x₀
this : s = {x₀} ∪ s \ {x₀}
A : ∀ (x : α), x ∈ s \ {x₀} → x < x₀
⊢ MeasurableSet (s \ {x₀})
[PROOFSTEP]
refine' measurableSet_of_mem_nhdsWithin_Ioi_aux (fun x hx => _) fun x hx => ⟨x₀, A x hx⟩
[GOAL]
case pos.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
x₀ : α
x₀s : x₀ ∈ s
h₀ : IsTop x₀
this : s = {x₀} ∪ s \ {x₀}
A : ∀ (x : α), x ∈ s \ {x₀} → x < x₀
x : α
hx : x ∈ s \ {x₀}
⊢ s \ {x₀} ∈ 𝓝[Ioi x] x
[PROOFSTEP]
obtain ⟨u, hu, us⟩ : ∃ (u : α), u ∈ Ioi x ∧ Ioo x u ⊆ s :=
  (mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' (A x hx)).1 (h x hx.1)
[GOAL]
case pos.intro.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u✝ : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
x₀ : α
x₀s : x₀ ∈ s
h₀ : IsTop x₀
this : s = {x₀} ∪ s \ {x₀}
A : ∀ (x : α), x ∈ s \ {x₀} → x < x₀
x : α
hx : x ∈ s \ {x₀}
u : α
hu : u ∈ Ioi x
us : Ioo x u ⊆ s
⊢ s \ {x₀} ∈ 𝓝[Ioi x] x
[PROOFSTEP]
refine' (mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' (A x hx)).2 ⟨u, hu, fun y hy => ⟨us hy, _⟩⟩
[GOAL]
case pos.intro.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u✝ : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
x₀ : α
x₀s : x₀ ∈ s
h₀ : IsTop x₀
this : s = {x₀} ∪ s \ {x₀}
A : ∀ (x : α), x ∈ s \ {x₀} → x < x₀
x : α
hx : x ∈ s \ {x₀}
u : α
hu : u ∈ Ioi x
us : Ioo x u ⊆ s
y : α
hy : y ∈ Ioo x u
⊢ ¬y ∈ {x₀}
[PROOFSTEP]
exact ne_of_lt (hy.2.trans_le (h₀ _))
[GOAL]
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
H : ¬∃ x, x ∈ s ∧ IsTop x
⊢ MeasurableSet s
[PROOFSTEP]
apply measurableSet_of_mem_nhdsWithin_Ioi_aux h
[GOAL]
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
H : ¬∃ x, x ∈ s ∧ IsTop x
⊢ ∀ (x : α), x ∈ s → ∃ y, x < y
[PROOFSTEP]
simp only [IsTop] at H
[GOAL]
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
H : ¬∃ x, x ∈ s ∧ ∀ (b : α), b ≤ x
⊢ ∀ (x : α), x ∈ s → ∃ y, x < y
[PROOFSTEP]
push_neg at H
[GOAL]
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x
H : ∀ (x : α), x ∈ s → ∃ b, x < b
⊢ ∀ (x : α), x ∈ s → ∃ y, x < y
[PROOFSTEP]
exact H
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝¹¹ : TopologicalSpace α✝
inst✝¹⁰ : MeasurableSpace α✝
inst✝⁹ : BorelSpace α✝
inst✝⁸ : TopologicalSpace β
inst✝⁷ : MeasurableSpace β
inst✝⁶ : BorelSpace β
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : MeasurableSpace γ
inst✝³ : BorelSpace γ
inst✝² : MeasurableSpace δ
α : Type u_6
inst✝¹ : MeasurableSpace α
inst✝ : CompleteLattice α
p : Prop
f : δ → α
hf : Measurable f
h : p
⊢ Measurable fun b => ⨆ (_ : p), f b
[PROOFSTEP]
convert hf
[GOAL]
case h.e'_5.h
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝¹¹ : TopologicalSpace α✝
inst✝¹⁰ : MeasurableSpace α✝
inst✝⁹ : BorelSpace α✝
inst✝⁸ : TopologicalSpace β
inst✝⁷ : MeasurableSpace β
inst✝⁶ : BorelSpace β
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : MeasurableSpace γ
inst✝³ : BorelSpace γ
inst✝² : MeasurableSpace δ
α : Type u_6
inst✝¹ : MeasurableSpace α
inst✝ : CompleteLattice α
p : Prop
f : δ → α
hf : Measurable f
h : p
x✝ : δ
⊢ ⨆ (_ : p), f x✝ = f x✝
[PROOFSTEP]
funext
[GOAL]
case h.e'_5.h
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝¹¹ : TopologicalSpace α✝
inst✝¹⁰ : MeasurableSpace α✝
inst✝⁹ : BorelSpace α✝
inst✝⁸ : TopologicalSpace β
inst✝⁷ : MeasurableSpace β
inst✝⁶ : BorelSpace β
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : MeasurableSpace γ
inst✝³ : BorelSpace γ
inst✝² : MeasurableSpace δ
α : Type u_6
inst✝¹ : MeasurableSpace α
inst✝ : CompleteLattice α
p : Prop
f : δ → α
hf : Measurable f
h : p
x✝ : δ
⊢ ⨆ (_ : p), f x✝ = f x✝
[PROOFSTEP]
exact iSup_pos h
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝¹¹ : TopologicalSpace α✝
inst✝¹⁰ : MeasurableSpace α✝
inst✝⁹ : BorelSpace α✝
inst✝⁸ : TopologicalSpace β
inst✝⁷ : MeasurableSpace β
inst✝⁶ : BorelSpace β
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : MeasurableSpace γ
inst✝³ : BorelSpace γ
inst✝² : MeasurableSpace δ
α : Type u_6
inst✝¹ : MeasurableSpace α
inst✝ : CompleteLattice α
p : Prop
f : δ → α
hf : Measurable f
h : ¬p
⊢ Measurable fun b => ⨆ (_ : p), f b
[PROOFSTEP]
convert measurable_const using 1
[GOAL]
case h.e'_5
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝¹¹ : TopologicalSpace α✝
inst✝¹⁰ : MeasurableSpace α✝
inst✝⁹ : BorelSpace α✝
inst✝⁸ : TopologicalSpace β
inst✝⁷ : MeasurableSpace β
inst✝⁶ : BorelSpace β
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : MeasurableSpace γ
inst✝³ : BorelSpace γ
inst✝² : MeasurableSpace δ
α : Type u_6
inst✝¹ : MeasurableSpace α
inst✝ : CompleteLattice α
p : Prop
f : δ → α
hf : Measurable f
h : ¬p
⊢ (fun b => ⨆ (_ : p), f b) = fun x => ?convert_5
case convert_5
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝¹¹ : TopologicalSpace α✝
inst✝¹⁰ : MeasurableSpace α✝
inst✝⁹ : BorelSpace α✝
inst✝⁸ : TopologicalSpace β
inst✝⁷ : MeasurableSpace β
inst✝⁶ : BorelSpace β
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : MeasurableSpace γ
inst✝³ : BorelSpace γ
inst✝² : MeasurableSpace δ
α : Type u_6
inst✝¹ : MeasurableSpace α
inst✝ : CompleteLattice α
p : Prop
f : δ → α
hf : Measurable f
h : ¬p
⊢ α
[PROOFSTEP]
funext
[GOAL]
case h.e'_5.h
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝¹¹ : TopologicalSpace α✝
inst✝¹⁰ : MeasurableSpace α✝
inst✝⁹ : BorelSpace α✝
inst✝⁸ : TopologicalSpace β
inst✝⁷ : MeasurableSpace β
inst✝⁶ : BorelSpace β
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : MeasurableSpace γ
inst✝³ : BorelSpace γ
inst✝² : MeasurableSpace δ
α : Type u_6
inst✝¹ : MeasurableSpace α
inst✝ : CompleteLattice α
p : Prop
f : δ → α
hf : Measurable f
h : ¬p
x✝ : δ
⊢ ⨆ (_ : p), f x✝ = ?convert_5
case convert_5
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝¹¹ : TopologicalSpace α✝
inst✝¹⁰ : MeasurableSpace α✝
inst✝⁹ : BorelSpace α✝
inst✝⁸ : TopologicalSpace β
inst✝⁷ : MeasurableSpace β
inst✝⁶ : BorelSpace β
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : MeasurableSpace γ
inst✝³ : BorelSpace γ
inst✝² : MeasurableSpace δ
α : Type u_6
inst✝¹ : MeasurableSpace α
inst✝ : CompleteLattice α
p : Prop
f : δ → α
hf : Measurable f
h : ¬p
⊢ α
[PROOFSTEP]
exact iSup_neg h
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝¹¹ : TopologicalSpace α✝
inst✝¹⁰ : MeasurableSpace α✝
inst✝⁹ : BorelSpace α✝
inst✝⁸ : TopologicalSpace β
inst✝⁷ : MeasurableSpace β
inst✝⁶ : BorelSpace β
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : MeasurableSpace γ
inst✝³ : BorelSpace γ
inst✝² : MeasurableSpace δ
α : Type u_6
inst✝¹ : MeasurableSpace α
inst✝ : CompleteLattice α
p : Prop
f : δ → α
hf : Measurable f
h : p
⊢ Measurable fun b => ⨅ (_ : p), f b
[PROOFSTEP]
convert hf
[GOAL]
case h.e'_5.h
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝¹¹ : TopologicalSpace α✝
inst✝¹⁰ : MeasurableSpace α✝
inst✝⁹ : BorelSpace α✝
inst✝⁸ : TopologicalSpace β
inst✝⁷ : MeasurableSpace β
inst✝⁶ : BorelSpace β
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : MeasurableSpace γ
inst✝³ : BorelSpace γ
inst✝² : MeasurableSpace δ
α : Type u_6
inst✝¹ : MeasurableSpace α
inst✝ : CompleteLattice α
p : Prop
f : δ → α
hf : Measurable f
h : p
x✝ : δ
⊢ ⨅ (_ : p), f x✝ = f x✝
[PROOFSTEP]
funext
[GOAL]
case h.e'_5.h
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝¹¹ : TopologicalSpace α✝
inst✝¹⁰ : MeasurableSpace α✝
inst✝⁹ : BorelSpace α✝
inst✝⁸ : TopologicalSpace β
inst✝⁷ : MeasurableSpace β
inst✝⁶ : BorelSpace β
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : MeasurableSpace γ
inst✝³ : BorelSpace γ
inst✝² : MeasurableSpace δ
α : Type u_6
inst✝¹ : MeasurableSpace α
inst✝ : CompleteLattice α
p : Prop
f : δ → α
hf : Measurable f
h : p
x✝ : δ
⊢ ⨅ (_ : p), f x✝ = f x✝
[PROOFSTEP]
exact iInf_pos h
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝¹¹ : TopologicalSpace α✝
inst✝¹⁰ : MeasurableSpace α✝
inst✝⁹ : BorelSpace α✝
inst✝⁸ : TopologicalSpace β
inst✝⁷ : MeasurableSpace β
inst✝⁶ : BorelSpace β
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : MeasurableSpace γ
inst✝³ : BorelSpace γ
inst✝² : MeasurableSpace δ
α : Type u_6
inst✝¹ : MeasurableSpace α
inst✝ : CompleteLattice α
p : Prop
f : δ → α
hf : Measurable f
h : ¬p
⊢ Measurable fun b => ⨅ (_ : p), f b
[PROOFSTEP]
convert measurable_const using 1
[GOAL]
case h.e'_5
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝¹¹ : TopologicalSpace α✝
inst✝¹⁰ : MeasurableSpace α✝
inst✝⁹ : BorelSpace α✝
inst✝⁸ : TopologicalSpace β
inst✝⁷ : MeasurableSpace β
inst✝⁶ : BorelSpace β
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : MeasurableSpace γ
inst✝³ : BorelSpace γ
inst✝² : MeasurableSpace δ
α : Type u_6
inst✝¹ : MeasurableSpace α
inst✝ : CompleteLattice α
p : Prop
f : δ → α
hf : Measurable f
h : ¬p
⊢ (fun b => ⨅ (_ : p), f b) = fun x => ?convert_5
case convert_5
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝¹¹ : TopologicalSpace α✝
inst✝¹⁰ : MeasurableSpace α✝
inst✝⁹ : BorelSpace α✝
inst✝⁸ : TopologicalSpace β
inst✝⁷ : MeasurableSpace β
inst✝⁶ : BorelSpace β
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : MeasurableSpace γ
inst✝³ : BorelSpace γ
inst✝² : MeasurableSpace δ
α : Type u_6
inst✝¹ : MeasurableSpace α
inst✝ : CompleteLattice α
p : Prop
f : δ → α
hf : Measurable f
h : ¬p
⊢ α
[PROOFSTEP]
funext
[GOAL]
case h.e'_5.h
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝¹¹ : TopologicalSpace α✝
inst✝¹⁰ : MeasurableSpace α✝
inst✝⁹ : BorelSpace α✝
inst✝⁸ : TopologicalSpace β
inst✝⁷ : MeasurableSpace β
inst✝⁶ : BorelSpace β
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : MeasurableSpace γ
inst✝³ : BorelSpace γ
inst✝² : MeasurableSpace δ
α : Type u_6
inst✝¹ : MeasurableSpace α
inst✝ : CompleteLattice α
p : Prop
f : δ → α
hf : Measurable f
h : ¬p
x✝ : δ
⊢ ⨅ (_ : p), f x✝ = ?convert_5
case convert_5
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝¹¹ : TopologicalSpace α✝
inst✝¹⁰ : MeasurableSpace α✝
inst✝⁹ : BorelSpace α✝
inst✝⁸ : TopologicalSpace β
inst✝⁷ : MeasurableSpace β
inst✝⁶ : BorelSpace β
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : MeasurableSpace γ
inst✝³ : BorelSpace γ
inst✝² : MeasurableSpace δ
α : Type u_6
inst✝¹ : MeasurableSpace α
inst✝ : CompleteLattice α
p : Prop
f : δ → α
hf : Measurable f
h : ¬p
⊢ α
[PROOFSTEP]
exact iInf_neg h
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : CompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_6
s : Set ι
f : ι → δ → α
hs : Set.Countable s
hf : ∀ (i : ι), Measurable (f i)
⊢ Measurable fun b => ⨆ (i : ι) (_ : i ∈ s), f i b
[PROOFSTEP]
haveI : Encodable s := hs.toEncodable
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : CompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_6
s : Set ι
f : ι → δ → α
hs : Set.Countable s
hf : ∀ (i : ι), Measurable (f i)
this : Encodable ↑s
⊢ Measurable fun b => ⨆ (i : ι) (_ : i ∈ s), f i b
[PROOFSTEP]
simp only [iSup_subtype']
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : CompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_6
s : Set ι
f : ι → δ → α
hs : Set.Countable s
hf : ∀ (i : ι), Measurable (f i)
this : Encodable ↑s
⊢ Measurable fun b => ⨆ (x : { i // i ∈ s }), f (↑x) b
[PROOFSTEP]
exact measurable_iSup fun i => hf i
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : CompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_6
μ : Measure δ
s : Set ι
f : ι → δ → α
hs : Set.Countable s
hf : ∀ (i : ι), AEMeasurable (f i)
⊢ AEMeasurable fun b => ⨆ (i : ι) (_ : i ∈ s), f i b
[PROOFSTEP]
haveI : Encodable s := hs.toEncodable
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : CompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_6
μ : Measure δ
s : Set ι
f : ι → δ → α
hs : Set.Countable s
hf : ∀ (i : ι), AEMeasurable (f i)
this : Encodable ↑s
⊢ AEMeasurable fun b => ⨆ (i : ι) (_ : i ∈ s), f i b
[PROOFSTEP]
simp only [iSup_subtype']
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : CompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_6
μ : Measure δ
s : Set ι
f : ι → δ → α
hs : Set.Countable s
hf : ∀ (i : ι), AEMeasurable (f i)
this : Encodable ↑s
⊢ AEMeasurable fun b => ⨆ (x : { i // i ∈ s }), f (↑x) b
[PROOFSTEP]
exact aemeasurable_iSup fun i => hf i
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : CompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_6
s : Set ι
f : ι → δ → α
hs : Set.Countable s
hf : ∀ (i : ι), Measurable (f i)
⊢ Measurable fun b => ⨅ (i : ι) (_ : i ∈ s), f i b
[PROOFSTEP]
haveI : Encodable s := hs.toEncodable
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : CompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_6
s : Set ι
f : ι → δ → α
hs : Set.Countable s
hf : ∀ (i : ι), Measurable (f i)
this : Encodable ↑s
⊢ Measurable fun b => ⨅ (i : ι) (_ : i ∈ s), f i b
[PROOFSTEP]
simp only [iInf_subtype']
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : CompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_6
s : Set ι
f : ι → δ → α
hs : Set.Countable s
hf : ∀ (i : ι), Measurable (f i)
this : Encodable ↑s
⊢ Measurable fun b => ⨅ (x : { i // i ∈ s }), f (↑x) b
[PROOFSTEP]
exact measurable_iInf fun i => hf i
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : CompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_6
μ : Measure δ
s : Set ι
f : ι → δ → α
hs : Set.Countable s
hf : ∀ (i : ι), AEMeasurable (f i)
⊢ AEMeasurable fun b => ⨅ (i : ι) (_ : i ∈ s), f i b
[PROOFSTEP]
haveI : Encodable s := hs.toEncodable
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : CompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_6
μ : Measure δ
s : Set ι
f : ι → δ → α
hs : Set.Countable s
hf : ∀ (i : ι), AEMeasurable (f i)
this : Encodable ↑s
⊢ AEMeasurable fun b => ⨅ (i : ι) (_ : i ∈ s), f i b
[PROOFSTEP]
simp only [iInf_subtype']
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : CompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_6
μ : Measure δ
s : Set ι
f : ι → δ → α
hs : Set.Countable s
hf : ∀ (i : ι), AEMeasurable (f i)
this : Encodable ↑s
⊢ AEMeasurable fun b => ⨅ (x : { i // i ∈ s }), f (↑x) b
[PROOFSTEP]
exact aemeasurable_iInf fun i => hf i
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s✝ t u✝ : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : CompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_6
ι' : Type u_7
f : ι → δ → α
u : Filter ι
hf : ∀ (i : ι), Measurable (f i)
p : ι' → Prop
s : ι' → Set ι
hu : HasCountableBasis u p s
hs : ∀ (i : ι'), Set.Countable (s i)
⊢ Measurable fun x => liminf (fun i => f i x) u
[PROOFSTEP]
simp_rw [hu.toHasBasis.liminf_eq_iSup_iInf]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s✝ t u✝ : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : CompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_6
ι' : Type u_7
f : ι → δ → α
u : Filter ι
hf : ∀ (i : ι), Measurable (f i)
p : ι' → Prop
s : ι' → Set ι
hu : HasCountableBasis u p s
hs : ∀ (i : ι'), Set.Countable (s i)
⊢ Measurable fun x => ⨆ (i : ι') (_ : p i), ⨅ (a : ι) (_ : a ∈ s i), f a x
[PROOFSTEP]
refine' measurable_biSup _ hu.countable _
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s✝ t u✝ : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : CompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_6
ι' : Type u_7
f : ι → δ → α
u : Filter ι
hf : ∀ (i : ι), Measurable (f i)
p : ι' → Prop
s : ι' → Set ι
hu : HasCountableBasis u p s
hs : ∀ (i : ι'), Set.Countable (s i)
⊢ ∀ (i : ι'), Measurable fun x => ⨅ (a : ι) (_ : a ∈ s i), f a x
[PROOFSTEP]
exact fun i => measurable_biInf _ (hs i) hf
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s✝ t u✝ : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : CompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_6
ι' : Type u_7
f : ι → δ → α
u : Filter ι
hf : ∀ (i : ι), Measurable (f i)
p : ι' → Prop
s : ι' → Set ι
hu : HasCountableBasis u p s
hs : ∀ (i : ι'), Set.Countable (s i)
⊢ Measurable fun x => limsup (fun i => f i x) u
[PROOFSTEP]
simp_rw [hu.toHasBasis.limsup_eq_iInf_iSup]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s✝ t u✝ : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : CompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_6
ι' : Type u_7
f : ι → δ → α
u : Filter ι
hf : ∀ (i : ι), Measurable (f i)
p : ι' → Prop
s : ι' → Set ι
hu : HasCountableBasis u p s
hs : ∀ (i : ι'), Set.Countable (s i)
⊢ Measurable fun x => ⨅ (i : ι') (_ : p i), ⨆ (a : ι) (_ : a ∈ s i), f a x
[PROOFSTEP]
refine' measurable_biInf _ hu.countable _
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s✝ t u✝ : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : CompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_6
ι' : Type u_7
f : ι → δ → α
u : Filter ι
hf : ∀ (i : ι), Measurable (f i)
p : ι' → Prop
s : ι' → Set ι
hu : HasCountableBasis u p s
hs : ∀ (i : ι'), Set.Countable (s i)
⊢ ∀ (i : ι'), Measurable fun x => ⨆ (a : ι) (_ : a ∈ s i), f a x
[PROOFSTEP]
exact fun i => measurable_biSup _ (hs i) hf
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_6
f : ι → δ → α
s : Set ι
hs : Set.Countable s
hf : ∀ (i : ι), Measurable (f i)
bdd : ∀ (x : δ), BddAbove ((fun i => f i x) '' s)
⊢ Measurable fun x => sSup ((fun i => f i x) '' s)
[PROOFSTEP]
cases' eq_empty_or_nonempty s with h2s h2s
[GOAL]
case inl
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_6
f : ι → δ → α
s : Set ι
hs : Set.Countable s
hf : ∀ (i : ι), Measurable (f i)
bdd : ∀ (x : δ), BddAbove ((fun i => f i x) '' s)
h2s : s = ∅
⊢ Measurable fun x => sSup ((fun i => f i x) '' s)
[PROOFSTEP]
simp [h2s, measurable_const]
[GOAL]
case inr
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_6
f : ι → δ → α
s : Set ι
hs : Set.Countable s
hf : ∀ (i : ι), Measurable (f i)
bdd : ∀ (x : δ), BddAbove ((fun i => f i x) '' s)
h2s : Set.Nonempty s
⊢ Measurable fun x => sSup ((fun i => f i x) '' s)
[PROOFSTEP]
apply measurable_of_Iic
[GOAL]
case inr.hf
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_6
f : ι → δ → α
s : Set ι
hs : Set.Countable s
hf : ∀ (i : ι), Measurable (f i)
bdd : ∀ (x : δ), BddAbove ((fun i => f i x) '' s)
h2s : Set.Nonempty s
⊢ ∀ (x : α), MeasurableSet ((fun x => sSup ((fun i => f i x) '' s)) ⁻¹' Iic x)
[PROOFSTEP]
intro y
[GOAL]
case inr.hf
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_6
f : ι → δ → α
s : Set ι
hs : Set.Countable s
hf : ∀ (i : ι), Measurable (f i)
bdd : ∀ (x : δ), BddAbove ((fun i => f i x) '' s)
h2s : Set.Nonempty s
y : α
⊢ MeasurableSet ((fun x => sSup ((fun i => f i x) '' s)) ⁻¹' Iic y)
[PROOFSTEP]
simp_rw [preimage, mem_Iic, csSup_le_iff (bdd _) (h2s.image _), ball_image_iff, setOf_forall]
[GOAL]
case inr.hf
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_6
f : ι → δ → α
s : Set ι
hs : Set.Countable s
hf : ∀ (i : ι), Measurable (f i)
bdd : ∀ (x : δ), BddAbove ((fun i => f i x) '' s)
h2s : Set.Nonempty s
y : α
⊢ MeasurableSet (⋂ (i : ι) (_ : i ∈ s), {x | f i x ≤ y})
[PROOFSTEP]
exact MeasurableSet.biInter hs fun i _ => measurableSet_le (hf i) measurable_const
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Type u_6
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
bdd : ∀ (x : δ), BddAbove (range fun i => f i x)
⊢ Measurable fun x => ⨆ (i : ι), f i x
[PROOFSTEP]
change Measurable fun x => sSup (range fun i : ι => f i x)
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Type u_6
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
bdd : ∀ (x : δ), BddAbove (range fun i => f i x)
⊢ Measurable fun x => sSup (range fun i => f i x)
[PROOFSTEP]
simp_rw [← image_univ] at bdd ⊢
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Type u_6
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
bdd : ∀ (x : δ), BddAbove ((fun i => f i x) '' univ)
⊢ Measurable fun x => sSup ((fun i => f i x) '' univ)
[PROOFSTEP]
refine' measurable_cSup countable_univ hf bdd
[GOAL]
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝¹⁶ : TopologicalSpace α✝
inst✝¹⁵ : MeasurableSpace α✝
inst✝¹⁴ : BorelSpace α✝
inst✝¹³ : TopologicalSpace β✝
inst✝¹² : MeasurableSpace β✝
inst✝¹¹ : BorelSpace β✝
inst✝¹⁰ : TopologicalSpace γ
inst✝⁹ : MeasurableSpace γ
inst✝⁸ : BorelSpace γ
inst✝⁷ : MeasurableSpace δ
α : Type u_6
m0 : MeasurableSpace α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OpensMeasurableSpace α
β : Type u_7
inst✝⁴ : MeasurableSpace β
inst✝³ : TopologicalSpace β
inst✝² : BorelSpace β
inst✝¹ : T2Space β
μ : Measure α
inst✝ : IsFiniteMeasureOnCompacts μ
f : α ≃ₜ β
⊢ ∀ ⦃K : Set β⦄, IsCompact K → ↑↑(Measure.map (↑f) μ) K < ⊤
[PROOFSTEP]
intro K hK
[GOAL]
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝¹⁶ : TopologicalSpace α✝
inst✝¹⁵ : MeasurableSpace α✝
inst✝¹⁴ : BorelSpace α✝
inst✝¹³ : TopologicalSpace β✝
inst✝¹² : MeasurableSpace β✝
inst✝¹¹ : BorelSpace β✝
inst✝¹⁰ : TopologicalSpace γ
inst✝⁹ : MeasurableSpace γ
inst✝⁸ : BorelSpace γ
inst✝⁷ : MeasurableSpace δ
α : Type u_6
m0 : MeasurableSpace α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OpensMeasurableSpace α
β : Type u_7
inst✝⁴ : MeasurableSpace β
inst✝³ : TopologicalSpace β
inst✝² : BorelSpace β
inst✝¹ : T2Space β
μ : Measure α
inst✝ : IsFiniteMeasureOnCompacts μ
f : α ≃ₜ β
K : Set β
hK : IsCompact K
⊢ ↑↑(Measure.map (↑f) μ) K < ⊤
[PROOFSTEP]
rw [Measure.map_apply f.measurable hK.measurableSet]
[GOAL]
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝¹⁶ : TopologicalSpace α✝
inst✝¹⁵ : MeasurableSpace α✝
inst✝¹⁴ : BorelSpace α✝
inst✝¹³ : TopologicalSpace β✝
inst✝¹² : MeasurableSpace β✝
inst✝¹¹ : BorelSpace β✝
inst✝¹⁰ : TopologicalSpace γ
inst✝⁹ : MeasurableSpace γ
inst✝⁸ : BorelSpace γ
inst✝⁷ : MeasurableSpace δ
α : Type u_6
m0 : MeasurableSpace α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OpensMeasurableSpace α
β : Type u_7
inst✝⁴ : MeasurableSpace β
inst✝³ : TopologicalSpace β
inst✝² : BorelSpace β
inst✝¹ : T2Space β
μ : Measure α
inst✝ : IsFiniteMeasureOnCompacts μ
f : α ≃ₜ β
K : Set β
hK : IsCompact K
⊢ ↑↑μ (↑f ⁻¹' K) < ⊤
[PROOFSTEP]
apply IsCompact.measure_lt_top
[GOAL]
case hK
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝¹⁶ : TopologicalSpace α✝
inst✝¹⁵ : MeasurableSpace α✝
inst✝¹⁴ : BorelSpace α✝
inst✝¹³ : TopologicalSpace β✝
inst✝¹² : MeasurableSpace β✝
inst✝¹¹ : BorelSpace β✝
inst✝¹⁰ : TopologicalSpace γ
inst✝⁹ : MeasurableSpace γ
inst✝⁸ : BorelSpace γ
inst✝⁷ : MeasurableSpace δ
α : Type u_6
m0 : MeasurableSpace α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OpensMeasurableSpace α
β : Type u_7
inst✝⁴ : MeasurableSpace β
inst✝³ : TopologicalSpace β
inst✝² : BorelSpace β
inst✝¹ : T2Space β
μ : Measure α
inst✝ : IsFiniteMeasureOnCompacts μ
f : α ≃ₜ β
K : Set β
hK : IsCompact K
⊢ IsCompact (↑f ⁻¹' K)
[PROOFSTEP]
rwa [f.isCompact_preimage]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
⊢ ↑↑μ s = ↑↑μ (s ∩ f ⁻¹' {0}) + ↑↑μ (s ∩ f ⁻¹' {⊤}) + ∑' (n : ℤ), ↑↑μ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))
[PROOFSTEP]
have A : μ s = μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' Ioi 0) :=
  by
  rw [← measure_union]
  · congr 1
    ext x
    have : 0 = f x ∨ 0 < f x := eq_or_lt_of_le bot_le
    rw [eq_comm] at this
    simp only [← and_or_left, this, mem_singleton_iff, mem_inter_iff, and_true_iff, mem_union, mem_Ioi, mem_preimage]
  · refine disjoint_left.2 fun x hx h'x => ?_
    have : 0 < f x := h'x.2
    exact lt_irrefl 0 (this.trans_le hx.2.le)
  · exact hs.inter (hf measurableSet_Ioi)
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
⊢ ↑↑μ s = ↑↑μ (s ∩ f ⁻¹' {0}) + ↑↑μ (s ∩ f ⁻¹' Ioi 0)
[PROOFSTEP]
rw [← measure_union]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
⊢ ↑↑μ s = ↑↑μ (s ∩ f ⁻¹' {0} ∪ s ∩ f ⁻¹' Ioi 0)
[PROOFSTEP]
congr 1
[GOAL]
case e_a
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
⊢ s = s ∩ f ⁻¹' {0} ∪ s ∩ f ⁻¹' Ioi 0
[PROOFSTEP]
ext x
[GOAL]
case e_a.h
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
x : α
⊢ x ∈ s ↔ x ∈ s ∩ f ⁻¹' {0} ∪ s ∩ f ⁻¹' Ioi 0
[PROOFSTEP]
have : 0 = f x ∨ 0 < f x := eq_or_lt_of_le bot_le
[GOAL]
case e_a.h
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
x : α
this : 0 = f x ∨ 0 < f x
⊢ x ∈ s ↔ x ∈ s ∩ f ⁻¹' {0} ∪ s ∩ f ⁻¹' Ioi 0
[PROOFSTEP]
rw [eq_comm] at this
[GOAL]
case e_a.h
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
x : α
this : f x = 0 ∨ 0 < f x
⊢ x ∈ s ↔ x ∈ s ∩ f ⁻¹' {0} ∪ s ∩ f ⁻¹' Ioi 0
[PROOFSTEP]
simp only [← and_or_left, this, mem_singleton_iff, mem_inter_iff, and_true_iff, mem_union, mem_Ioi, mem_preimage]
[GOAL]
case hd
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
⊢ Disjoint (s ∩ f ⁻¹' {0}) (s ∩ f ⁻¹' Ioi 0)
[PROOFSTEP]
refine disjoint_left.2 fun x hx h'x => ?_
[GOAL]
case hd
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
x : α
hx : x ∈ s ∩ f ⁻¹' {0}
h'x : x ∈ s ∩ f ⁻¹' Ioi 0
⊢ False
[PROOFSTEP]
have : 0 < f x := h'x.2
[GOAL]
case hd
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
x : α
hx : x ∈ s ∩ f ⁻¹' {0}
h'x : x ∈ s ∩ f ⁻¹' Ioi 0
this : 0 < f x
⊢ False
[PROOFSTEP]
exact lt_irrefl 0 (this.trans_le hx.2.le)
[GOAL]
case h
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
⊢ MeasurableSet (s ∩ f ⁻¹' Ioi 0)
[PROOFSTEP]
exact hs.inter (hf measurableSet_Ioi)
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : ↑↑μ s = ↑↑μ (s ∩ f ⁻¹' {0}) + ↑↑μ (s ∩ f ⁻¹' Ioi 0)
⊢ ↑↑μ s = ↑↑μ (s ∩ f ⁻¹' {0}) + ↑↑μ (s ∩ f ⁻¹' {⊤}) + ∑' (n : ℤ), ↑↑μ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))
[PROOFSTEP]
have B : μ (s ∩ f ⁻¹' Ioi 0) = μ (s ∩ f ⁻¹' {∞}) + μ (s ∩ f ⁻¹' Ioo 0 ∞) :=
  by
  rw [← measure_union]
  · rw [← inter_union_distrib_left]
    congr
    ext x
    simp only [mem_singleton_iff, mem_union, mem_Ioo, mem_Ioi, mem_preimage]
    have H : f x = ∞ ∨ f x < ∞ := eq_or_lt_of_le le_top
    cases' H with H H
    · simp only [H, eq_self_iff_true, or_false_iff, WithTop.zero_lt_top, not_top_lt, and_false_iff]
    · simp only [H, H.ne, and_true_iff, false_or_iff]
  · refine disjoint_left.2 fun x hx h'x => ?_
    have : f x < ∞ := h'x.2.2
    exact lt_irrefl _ (this.trans_le (le_of_eq hx.2.symm))
  · exact hs.inter (hf measurableSet_Ioo)
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : ↑↑μ s = ↑↑μ (s ∩ f ⁻¹' {0}) + ↑↑μ (s ∩ f ⁻¹' Ioi 0)
⊢ ↑↑μ (s ∩ f ⁻¹' Ioi 0) = ↑↑μ (s ∩ f ⁻¹' {⊤}) + ↑↑μ (s ∩ f ⁻¹' Ioo 0 ⊤)
[PROOFSTEP]
rw [← measure_union]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : ↑↑μ s = ↑↑μ (s ∩ f ⁻¹' {0}) + ↑↑μ (s ∩ f ⁻¹' Ioi 0)
⊢ ↑↑μ (s ∩ f ⁻¹' Ioi 0) = ↑↑μ (s ∩ f ⁻¹' {⊤} ∪ s ∩ f ⁻¹' Ioo 0 ⊤)
[PROOFSTEP]
rw [← inter_union_distrib_left]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : ↑↑μ s = ↑↑μ (s ∩ f ⁻¹' {0}) + ↑↑μ (s ∩ f ⁻¹' Ioi 0)
⊢ ↑↑μ (s ∩ f ⁻¹' Ioi 0) = ↑↑μ (s ∩ (f ⁻¹' {⊤} ∪ f ⁻¹' Ioo 0 ⊤))
[PROOFSTEP]
congr
[GOAL]
case e_a.e_a
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : ↑↑μ s = ↑↑μ (s ∩ f ⁻¹' {0}) + ↑↑μ (s ∩ f ⁻¹' Ioi 0)
⊢ f ⁻¹' Ioi 0 = f ⁻¹' {⊤} ∪ f ⁻¹' Ioo 0 ⊤
[PROOFSTEP]
ext x
[GOAL]
case e_a.e_a.h
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : ↑↑μ s = ↑↑μ (s ∩ f ⁻¹' {0}) + ↑↑μ (s ∩ f ⁻¹' Ioi 0)
x : α
⊢ x ∈ f ⁻¹' Ioi 0 ↔ x ∈ f ⁻¹' {⊤} ∪ f ⁻¹' Ioo 0 ⊤
[PROOFSTEP]
simp only [mem_singleton_iff, mem_union, mem_Ioo, mem_Ioi, mem_preimage]
[GOAL]
case e_a.e_a.h
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : ↑↑μ s = ↑↑μ (s ∩ f ⁻¹' {0}) + ↑↑μ (s ∩ f ⁻¹' Ioi 0)
x : α
⊢ 0 < f x ↔ f x = ⊤ ∨ 0 < f x ∧ f x < ⊤
[PROOFSTEP]
have H : f x = ∞ ∨ f x < ∞ := eq_or_lt_of_le le_top
[GOAL]
case e_a.e_a.h
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : ↑↑μ s = ↑↑μ (s ∩ f ⁻¹' {0}) + ↑↑μ (s ∩ f ⁻¹' Ioi 0)
x : α
H : f x = ⊤ ∨ f x < ⊤
⊢ 0 < f x ↔ f x = ⊤ ∨ 0 < f x ∧ f x < ⊤
[PROOFSTEP]
cases' H with H H
[GOAL]
case e_a.e_a.h.inl
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : ↑↑μ s = ↑↑μ (s ∩ f ⁻¹' {0}) + ↑↑μ (s ∩ f ⁻¹' Ioi 0)
x : α
H : f x = ⊤
⊢ 0 < f x ↔ f x = ⊤ ∨ 0 < f x ∧ f x < ⊤
[PROOFSTEP]
simp only [H, eq_self_iff_true, or_false_iff, WithTop.zero_lt_top, not_top_lt, and_false_iff]
[GOAL]
case e_a.e_a.h.inr
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : ↑↑μ s = ↑↑μ (s ∩ f ⁻¹' {0}) + ↑↑μ (s ∩ f ⁻¹' Ioi 0)
x : α
H : f x < ⊤
⊢ 0 < f x ↔ f x = ⊤ ∨ 0 < f x ∧ f x < ⊤
[PROOFSTEP]
simp only [H, H.ne, and_true_iff, false_or_iff]
[GOAL]
case hd
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : ↑↑μ s = ↑↑μ (s ∩ f ⁻¹' {0}) + ↑↑μ (s ∩ f ⁻¹' Ioi 0)
⊢ Disjoint (s ∩ f ⁻¹' {⊤}) (s ∩ f ⁻¹' Ioo 0 ⊤)
[PROOFSTEP]
refine disjoint_left.2 fun x hx h'x => ?_
[GOAL]
case hd
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : ↑↑μ s = ↑↑μ (s ∩ f ⁻¹' {0}) + ↑↑μ (s ∩ f ⁻¹' Ioi 0)
x : α
hx : x ∈ s ∩ f ⁻¹' {⊤}
h'x : x ∈ s ∩ f ⁻¹' Ioo 0 ⊤
⊢ False
[PROOFSTEP]
have : f x < ∞ := h'x.2.2
[GOAL]
case hd
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : ↑↑μ s = ↑↑μ (s ∩ f ⁻¹' {0}) + ↑↑μ (s ∩ f ⁻¹' Ioi 0)
x : α
hx : x ∈ s ∩ f ⁻¹' {⊤}
h'x : x ∈ s ∩ f ⁻¹' Ioo 0 ⊤
this : f x < ⊤
⊢ False
[PROOFSTEP]
exact lt_irrefl _ (this.trans_le (le_of_eq hx.2.symm))
[GOAL]
case h
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : ↑↑μ s = ↑↑μ (s ∩ f ⁻¹' {0}) + ↑↑μ (s ∩ f ⁻¹' Ioi 0)
⊢ MeasurableSet (s ∩ f ⁻¹' Ioo 0 ⊤)
[PROOFSTEP]
exact hs.inter (hf measurableSet_Ioo)
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : ↑↑μ s = ↑↑μ (s ∩ f ⁻¹' {0}) + ↑↑μ (s ∩ f ⁻¹' Ioi 0)
B : ↑↑μ (s ∩ f ⁻¹' Ioi 0) = ↑↑μ (s ∩ f ⁻¹' {⊤}) + ↑↑μ (s ∩ f ⁻¹' Ioo 0 ⊤)
⊢ ↑↑μ s = ↑↑μ (s ∩ f ⁻¹' {0}) + ↑↑μ (s ∩ f ⁻¹' {⊤}) + ∑' (n : ℤ), ↑↑μ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))
[PROOFSTEP]
have C : μ (s ∩ f ⁻¹' Ioo 0 ∞) = ∑' n : ℤ, μ (s ∩ f ⁻¹' Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))) :=
  by
  rw [← measure_iUnion, ENNReal.Ioo_zero_top_eq_iUnion_Ico_zpow (ENNReal.one_lt_coe_iff.2 ht) ENNReal.coe_ne_top,
    preimage_iUnion, inter_iUnion]
  · intro i j
    simp only [Function.onFun]
    intro hij
    wlog h : i < j generalizing i j
    · exact (this hij.symm (hij.lt_or_lt.resolve_left h)).symm
    refine disjoint_left.2 fun x hx h'x => lt_irrefl (f x) ?_
    calc
      f x < (t : ℝ≥0∞) ^ (i + 1) := hx.2.2
      _ ≤ (t : ℝ≥0∞) ^ j := (ENNReal.zpow_le_of_le (ENNReal.one_le_coe_iff.2 ht.le) h)
      _ ≤ f x := h'x.2.1
  · intro n
    exact hs.inter (hf measurableSet_Ico)
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : ↑↑μ s = ↑↑μ (s ∩ f ⁻¹' {0}) + ↑↑μ (s ∩ f ⁻¹' Ioi 0)
B : ↑↑μ (s ∩ f ⁻¹' Ioi 0) = ↑↑μ (s ∩ f ⁻¹' {⊤}) + ↑↑μ (s ∩ f ⁻¹' Ioo 0 ⊤)
⊢ ↑↑μ (s ∩ f ⁻¹' Ioo 0 ⊤) = ∑' (n : ℤ), ↑↑μ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))
[PROOFSTEP]
rw [← measure_iUnion, ENNReal.Ioo_zero_top_eq_iUnion_Ico_zpow (ENNReal.one_lt_coe_iff.2 ht) ENNReal.coe_ne_top,
  preimage_iUnion, inter_iUnion]
[GOAL]
case hn
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : ↑↑μ s = ↑↑μ (s ∩ f ⁻¹' {0}) + ↑↑μ (s ∩ f ⁻¹' Ioi 0)
B : ↑↑μ (s ∩ f ⁻¹' Ioi 0) = ↑↑μ (s ∩ f ⁻¹' {⊤}) + ↑↑μ (s ∩ f ⁻¹' Ioo 0 ⊤)
⊢ Pairwise (Disjoint on fun n => s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))
[PROOFSTEP]
intro i j
[GOAL]
case hn
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : ↑↑μ s = ↑↑μ (s ∩ f ⁻¹' {0}) + ↑↑μ (s ∩ f ⁻¹' Ioi 0)
B : ↑↑μ (s ∩ f ⁻¹' Ioi 0) = ↑↑μ (s ∩ f ⁻¹' {⊤}) + ↑↑μ (s ∩ f ⁻¹' Ioo 0 ⊤)
i j : ℤ
⊢ i ≠ j → (Disjoint on fun n => s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) i j
[PROOFSTEP]
simp only [Function.onFun]
[GOAL]
case hn
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : ↑↑μ s = ↑↑μ (s ∩ f ⁻¹' {0}) + ↑↑μ (s ∩ f ⁻¹' Ioi 0)
B : ↑↑μ (s ∩ f ⁻¹' Ioi 0) = ↑↑μ (s ∩ f ⁻¹' {⊤}) + ↑↑μ (s ∩ f ⁻¹' Ioo 0 ⊤)
i j : ℤ
⊢ i ≠ j → Disjoint (s ∩ f ⁻¹' Ico (↑t ^ i) (↑t ^ (i + 1))) (s ∩ f ⁻¹' Ico (↑t ^ j) (↑t ^ (j + 1)))
[PROOFSTEP]
intro hij
[GOAL]
case hn
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : ↑↑μ s = ↑↑μ (s ∩ f ⁻¹' {0}) + ↑↑μ (s ∩ f ⁻¹' Ioi 0)
B : ↑↑μ (s ∩ f ⁻¹' Ioi 0) = ↑↑μ (s ∩ f ⁻¹' {⊤}) + ↑↑μ (s ∩ f ⁻¹' Ioo 0 ⊤)
i j : ℤ
hij : i ≠ j
⊢ Disjoint (s ∩ f ⁻¹' Ico (↑t ^ i) (↑t ^ (i + 1))) (s ∩ f ⁻¹' Ico (↑t ^ j) (↑t ^ (j + 1)))
[PROOFSTEP]
wlog h : i < j generalizing i j
[GOAL]
case hn.inr
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : ↑↑μ s = ↑↑μ (s ∩ f ⁻¹' {0}) + ↑↑μ (s ∩ f ⁻¹' Ioi 0)
B : ↑↑μ (s ∩ f ⁻¹' Ioi 0) = ↑↑μ (s ∩ f ⁻¹' {⊤}) + ↑↑μ (s ∩ f ⁻¹' Ioo 0 ⊤)
i j : ℤ
hij : i ≠ j
this :
  ∀ ⦃i j : ℤ⦄, i ≠ j → i < j → Disjoint (s ∩ f ⁻¹' Ico (↑t ^ i) (↑t ^ (i + 1))) (s ∩ f ⁻¹' Ico (↑t ^ j) (↑t ^ (j + 1)))
h : ¬i < j
⊢ Disjoint (s ∩ f ⁻¹' Ico (↑t ^ i) (↑t ^ (i + 1))) (s ∩ f ⁻¹' Ico (↑t ^ j) (↑t ^ (j + 1)))
[PROOFSTEP]
exact (this hij.symm (hij.lt_or_lt.resolve_left h)).symm
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : ↑↑μ s = ↑↑μ (s ∩ f ⁻¹' {0}) + ↑↑μ (s ∩ f ⁻¹' Ioi 0)
B : ↑↑μ (s ∩ f ⁻¹' Ioi 0) = ↑↑μ (s ∩ f ⁻¹' {⊤}) + ↑↑μ (s ∩ f ⁻¹' Ioo 0 ⊤)
i j : ℤ
hij : i ≠ j
h : i < j
⊢ Disjoint (s ∩ f ⁻¹' Ico (↑t ^ i) (↑t ^ (i + 1))) (s ∩ f ⁻¹' Ico (↑t ^ j) (↑t ^ (j + 1)))
[PROOFSTEP]
refine disjoint_left.2 fun x hx h'x => lt_irrefl (f x) ?_
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : ↑↑μ s = ↑↑μ (s ∩ f ⁻¹' {0}) + ↑↑μ (s ∩ f ⁻¹' Ioi 0)
B : ↑↑μ (s ∩ f ⁻¹' Ioi 0) = ↑↑μ (s ∩ f ⁻¹' {⊤}) + ↑↑μ (s ∩ f ⁻¹' Ioo 0 ⊤)
i j : ℤ
hij : i ≠ j
h : i < j
x : α
hx : x ∈ s ∩ f ⁻¹' Ico (↑t ^ i) (↑t ^ (i + 1))
h'x : x ∈ s ∩ f ⁻¹' Ico (↑t ^ j) (↑t ^ (j + 1))
⊢ f x < f x
[PROOFSTEP]
calc
  f x < (t : ℝ≥0∞) ^ (i + 1) := hx.2.2
  _ ≤ (t : ℝ≥0∞) ^ j := (ENNReal.zpow_le_of_le (ENNReal.one_le_coe_iff.2 ht.le) h)
  _ ≤ f x := h'x.2.1
[GOAL]
case h
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : ↑↑μ s = ↑↑μ (s ∩ f ⁻¹' {0}) + ↑↑μ (s ∩ f ⁻¹' Ioi 0)
B : ↑↑μ (s ∩ f ⁻¹' Ioi 0) = ↑↑μ (s ∩ f ⁻¹' {⊤}) + ↑↑μ (s ∩ f ⁻¹' Ioo 0 ⊤)
⊢ ∀ (i : ℤ), MeasurableSet (s ∩ f ⁻¹' Ico (↑t ^ i) (↑t ^ (i + 1)))
[PROOFSTEP]
intro n
[GOAL]
case h
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : ↑↑μ s = ↑↑μ (s ∩ f ⁻¹' {0}) + ↑↑μ (s ∩ f ⁻¹' Ioi 0)
B : ↑↑μ (s ∩ f ⁻¹' Ioi 0) = ↑↑μ (s ∩ f ⁻¹' {⊤}) + ↑↑μ (s ∩ f ⁻¹' Ioo 0 ⊤)
n : ℤ
⊢ MeasurableSet (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))
[PROOFSTEP]
exact hs.inter (hf measurableSet_Ico)
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t✝ u : Set α
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : ↑↑μ s = ↑↑μ (s ∩ f ⁻¹' {0}) + ↑↑μ (s ∩ f ⁻¹' Ioi 0)
B : ↑↑μ (s ∩ f ⁻¹' Ioi 0) = ↑↑μ (s ∩ f ⁻¹' {⊤}) + ↑↑μ (s ∩ f ⁻¹' Ioo 0 ⊤)
C : ↑↑μ (s ∩ f ⁻¹' Ioo 0 ⊤) = ∑' (n : ℤ), ↑↑μ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))
⊢ ↑↑μ s = ↑↑μ (s ∩ f ⁻¹' {0}) + ↑↑μ (s ∩ f ⁻¹' {⊤}) + ∑' (n : ℤ), ↑↑μ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))
[PROOFSTEP]
rw [A, B, C, add_assoc]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝³ : PseudoEMetricSpace α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
inst✝ : MeasurableSpace β
x : α
ε : ℝ≥0∞
μ : Measure α
s : Set α
hs : ∃ R, R > 0 ∧ ↑↑μ (cthickening R s) ≠ ⊤
⊢ Tendsto (fun r => ↑↑μ (cthickening r s)) (𝓝 0) (𝓝 (↑↑μ (closure s)))
[PROOFSTEP]
have A : Tendsto (fun r => μ (cthickening r s)) (𝓝[Ioi 0] 0) (𝓝 (μ (closure s))) :=
  by
  rw [closure_eq_iInter_cthickening]
  exact
    tendsto_measure_biInter_gt (fun r _ => isClosed_cthickening.measurableSet) (fun i j _ ij => cthickening_mono ij _)
      hs
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝³ : PseudoEMetricSpace α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
inst✝ : MeasurableSpace β
x : α
ε : ℝ≥0∞
μ : Measure α
s : Set α
hs : ∃ R, R > 0 ∧ ↑↑μ (cthickening R s) ≠ ⊤
⊢ Tendsto (fun r => ↑↑μ (cthickening r s)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (closure s)))
[PROOFSTEP]
rw [closure_eq_iInter_cthickening]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝³ : PseudoEMetricSpace α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
inst✝ : MeasurableSpace β
x : α
ε : ℝ≥0∞
μ : Measure α
s : Set α
hs : ∃ R, R > 0 ∧ ↑↑μ (cthickening R s) ≠ ⊤
⊢ Tendsto (fun r => ↑↑μ (cthickening r s)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (⋂ (δ : ℝ) (_ : 0 < δ), cthickening δ s)))
[PROOFSTEP]
exact
  tendsto_measure_biInter_gt (fun r _ => isClosed_cthickening.measurableSet) (fun i j _ ij => cthickening_mono ij _) hs
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝³ : PseudoEMetricSpace α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
inst✝ : MeasurableSpace β
x : α
ε : ℝ≥0∞
μ : Measure α
s : Set α
hs : ∃ R, R > 0 ∧ ↑↑μ (cthickening R s) ≠ ⊤
A : Tendsto (fun r => ↑↑μ (cthickening r s)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (closure s)))
⊢ Tendsto (fun r => ↑↑μ (cthickening r s)) (𝓝 0) (𝓝 (↑↑μ (closure s)))
[PROOFSTEP]
have B : Tendsto (fun r => μ (cthickening r s)) (𝓝[Iic 0] 0) (𝓝 (μ (closure s))) :=
  by
  apply Tendsto.congr' _ tendsto_const_nhds
  filter_upwards [self_mem_nhdsWithin (α := ℝ)] with _ hr
  rw [cthickening_of_nonpos hr]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝³ : PseudoEMetricSpace α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
inst✝ : MeasurableSpace β
x : α
ε : ℝ≥0∞
μ : Measure α
s : Set α
hs : ∃ R, R > 0 ∧ ↑↑μ (cthickening R s) ≠ ⊤
A : Tendsto (fun r => ↑↑μ (cthickening r s)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (closure s)))
⊢ Tendsto (fun r => ↑↑μ (cthickening r s)) (𝓝[Iic 0] 0) (𝓝 (↑↑μ (closure s)))
[PROOFSTEP]
apply Tendsto.congr' _ tendsto_const_nhds
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝³ : PseudoEMetricSpace α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
inst✝ : MeasurableSpace β
x : α
ε : ℝ≥0∞
μ : Measure α
s : Set α
hs : ∃ R, R > 0 ∧ ↑↑μ (cthickening R s) ≠ ⊤
A : Tendsto (fun r => ↑↑μ (cthickening r s)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (closure s)))
⊢ (fun x => ↑↑μ (closure s)) =ᶠ[𝓝[Iic 0] 0] fun r => ↑↑μ (cthickening r s)
[PROOFSTEP]
filter_upwards [self_mem_nhdsWithin (α := ℝ)] with _ hr
[GOAL]
case h
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝³ : PseudoEMetricSpace α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
inst✝ : MeasurableSpace β
x : α
ε : ℝ≥0∞
μ : Measure α
s : Set α
hs : ∃ R, R > 0 ∧ ↑↑μ (cthickening R s) ≠ ⊤
A : Tendsto (fun r => ↑↑μ (cthickening r s)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (closure s)))
a✝ : ℝ
hr : a✝ ∈ Iic 0
⊢ ↑↑μ (closure s) = ↑↑μ (cthickening a✝ s)
[PROOFSTEP]
rw [cthickening_of_nonpos hr]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝³ : PseudoEMetricSpace α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
inst✝ : MeasurableSpace β
x : α
ε : ℝ≥0∞
μ : Measure α
s : Set α
hs : ∃ R, R > 0 ∧ ↑↑μ (cthickening R s) ≠ ⊤
A : Tendsto (fun r => ↑↑μ (cthickening r s)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (closure s)))
B : Tendsto (fun r => ↑↑μ (cthickening r s)) (𝓝[Iic 0] 0) (𝓝 (↑↑μ (closure s)))
⊢ Tendsto (fun r => ↑↑μ (cthickening r s)) (𝓝 0) (𝓝 (↑↑μ (closure s)))
[PROOFSTEP]
convert B.sup A
[GOAL]
case h.e'_4
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝³ : PseudoEMetricSpace α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
inst✝ : MeasurableSpace β
x : α
ε : ℝ≥0∞
μ : Measure α
s : Set α
hs : ∃ R, R > 0 ∧ ↑↑μ (cthickening R s) ≠ ⊤
A : Tendsto (fun r => ↑↑μ (cthickening r s)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (closure s)))
B : Tendsto (fun r => ↑↑μ (cthickening r s)) (𝓝[Iic 0] 0) (𝓝 (↑↑μ (closure s)))
⊢ 𝓝 0 = 𝓝[Iic 0] 0 ⊔ 𝓝[Ioi 0] 0
[PROOFSTEP]
exact (nhds_left_sup_nhds_right' 0).symm
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝³ : PseudoEMetricSpace α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
inst✝ : MeasurableSpace β
x : α
ε : ℝ≥0∞
μ : Measure α
s : Set α
hs : ∃ R, R > 0 ∧ ↑↑μ (cthickening R s) ≠ ⊤
h's : IsClosed s
⊢ Tendsto (fun r => ↑↑μ (cthickening r s)) (𝓝 0) (𝓝 (↑↑μ s))
[PROOFSTEP]
convert tendsto_measure_cthickening hs
[GOAL]
case h.e'_5.h.e'_3.h.e'_3
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝³ : PseudoEMetricSpace α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
inst✝ : MeasurableSpace β
x : α
ε : ℝ≥0∞
μ : Measure α
s : Set α
hs : ∃ R, R > 0 ∧ ↑↑μ (cthickening R s) ≠ ⊤
h's : IsClosed s
⊢ s = closure s
[PROOFSTEP]
exact h's.closure_eq.symm
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝³ : PseudoEMetricSpace α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
inst✝ : MeasurableSpace β
x : α
ε : ℝ≥0∞
μ : Measure α
s : Set α
hs : ∃ R, R > 0 ∧ ↑↑μ (thickening R s) ≠ ⊤
⊢ Tendsto (fun r => ↑↑μ (thickening r s)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (closure s)))
[PROOFSTEP]
rw [closure_eq_iInter_thickening]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝³ : PseudoEMetricSpace α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
inst✝ : MeasurableSpace β
x : α
ε : ℝ≥0∞
μ : Measure α
s : Set α
hs : ∃ R, R > 0 ∧ ↑↑μ (thickening R s) ≠ ⊤
⊢ Tendsto (fun r => ↑↑μ (thickening r s)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (⋂ (δ : ℝ) (_ : 0 < δ), thickening δ s)))
[PROOFSTEP]
exact tendsto_measure_biInter_gt (fun r _ => isOpen_thickening.measurableSet) (fun i j _ ij => thickening_mono ij _) hs
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝³ : PseudoEMetricSpace α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
inst✝ : MeasurableSpace β
x : α
ε : ℝ≥0∞
μ : Measure α
s : Set α
hs : ∃ R, R > 0 ∧ ↑↑μ (thickening R s) ≠ ⊤
h's : IsClosed s
⊢ Tendsto (fun r => ↑↑μ (thickening r s)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ s))
[PROOFSTEP]
convert tendsto_measure_thickening hs
[GOAL]
case h.e'_5.h.e'_3.h.e'_3
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝³ : PseudoEMetricSpace α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
inst✝ : MeasurableSpace β
x : α
ε : ℝ≥0∞
μ : Measure α
s : Set α
hs : ∃ R, R > 0 ∧ ↑↑μ (thickening R s) ≠ ⊤
h's : IsClosed s
⊢ s = closure s
[PROOFSTEP]
exact h's.closure_eq.symm
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
⊢ borel ℝ = MeasurableSpace.generateFrom (⋃ (a : ℚ), {Iio ↑a})
[PROOFSTEP]
rw [borel_eq_generateFrom_Iio]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
⊢ MeasurableSpace.generateFrom (range Iio) = MeasurableSpace.generateFrom (⋃ (a : ℚ), {Iio ↑a})
[PROOFSTEP]
refine
  le_antisymm (generateFrom_le ?_)
    (generateFrom_mono <| iUnion_subset fun q ↦ singleton_subset_iff.mpr <| mem_range_self _)
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
⊢ ∀ (t : Set ℝ), t ∈ range Iio → MeasurableSet t
[PROOFSTEP]
rintro _ ⟨a, rfl⟩
[GOAL]
case intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
a : ℝ
⊢ MeasurableSet (Iio a)
[PROOFSTEP]
have : IsLUB (range ((↑) : ℚ → ℝ) ∩ Iio a) a := by
  simp [isLUB_iff_le_iff, mem_upperBounds, ← le_iff_forall_rat_lt_imp_le]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
a : ℝ
⊢ IsLUB (range Rat.cast ∩ Iio a) a
[PROOFSTEP]
simp [isLUB_iff_le_iff, mem_upperBounds, ← le_iff_forall_rat_lt_imp_le]
[GOAL]
case intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
a : ℝ
this : IsLUB (range Rat.cast ∩ Iio a) a
⊢ MeasurableSet (Iio a)
[PROOFSTEP]
rw [← this.biUnion_Iio_eq, ← image_univ, ← image_inter_preimage, univ_inter, biUnion_image]
[GOAL]
case intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
a : ℝ
this : IsLUB (range Rat.cast ∩ Iio a) a
⊢ MeasurableSet (⋃ (y : ℚ) (_ : y ∈ Rat.cast ⁻¹' Iio a), Iio ↑y)
[PROOFSTEP]
exact MeasurableSet.biUnion (to_countable _) fun b _ => GenerateMeasurable.basic (Iio (b : ℝ)) (by simp)
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
a : ℝ
this : IsLUB (range Rat.cast ∩ Iio a) a
b : ℚ
x✝ : b ∈ Rat.cast ⁻¹' Iio a
⊢ Iio ↑b ∈ ⋃ (a : ℚ), {Iio ↑a}
[PROOFSTEP]
simp
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
⊢ borel ℝ = MeasurableSpace.generateFrom (⋃ (a : ℚ), {Ioi ↑a})
[PROOFSTEP]
rw [borel_eq_generateFrom_Ioi]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
⊢ MeasurableSpace.generateFrom (range Ioi) = MeasurableSpace.generateFrom (⋃ (a : ℚ), {Ioi ↑a})
[PROOFSTEP]
refine
  le_antisymm (generateFrom_le ?_)
    (generateFrom_mono <| iUnion_subset fun q ↦ singleton_subset_iff.mpr <| mem_range_self _)
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
⊢ ∀ (t : Set ℝ), t ∈ range Ioi → MeasurableSet t
[PROOFSTEP]
rintro _ ⟨a, rfl⟩
[GOAL]
case intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
a : ℝ
⊢ MeasurableSet (Ioi a)
[PROOFSTEP]
have : IsGLB (range ((↑) : ℚ → ℝ) ∩ Ioi a) a := by
  simp [isGLB_iff_le_iff, mem_lowerBounds, ← le_iff_forall_lt_rat_imp_le]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
a : ℝ
⊢ IsGLB (range Rat.cast ∩ Ioi a) a
[PROOFSTEP]
simp [isGLB_iff_le_iff, mem_lowerBounds, ← le_iff_forall_lt_rat_imp_le]
[GOAL]
case intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
a : ℝ
this : IsGLB (range Rat.cast ∩ Ioi a) a
⊢ MeasurableSet (Ioi a)
[PROOFSTEP]
rw [← this.biUnion_Ioi_eq, ← image_univ, ← image_inter_preimage, univ_inter, biUnion_image]
[GOAL]
case intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
a : ℝ
this : IsGLB (range Rat.cast ∩ Ioi a) a
⊢ MeasurableSet (⋃ (y : ℚ) (_ : y ∈ Rat.cast ⁻¹' Ioi a), Ioi ↑y)
[PROOFSTEP]
exact MeasurableSet.biUnion (to_countable _) fun b _ => GenerateMeasurable.basic (Ioi (b : ℝ)) (by simp)
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
a : ℝ
this : IsGLB (range Rat.cast ∩ Ioi a) a
b : ℚ
x✝ : b ∈ Rat.cast ⁻¹' Ioi a
⊢ Ioi ↑b ∈ ⋃ (a : ℚ), {Ioi ↑a}
[PROOFSTEP]
simp
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
⊢ borel ℝ = MeasurableSpace.generateFrom (⋃ (a : ℚ), {Iic ↑a})
[PROOFSTEP]
rw [borel_eq_generateFrom_Ioi_rat, iUnion_singleton_eq_range, iUnion_singleton_eq_range]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
⊢ MeasurableSpace.generateFrom (range fun a => Ioi ↑a) = MeasurableSpace.generateFrom (range fun a => Iic ↑a)
[PROOFSTEP]
refine le_antisymm (generateFrom_le ?_) (generateFrom_le ?_) <;> rintro _ ⟨q, rfl⟩ <;> dsimp only <;> [rw [← compl_Iic];
  rw [← compl_Ioi]]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
⊢ MeasurableSpace.generateFrom (range fun a => Ioi ↑a) = MeasurableSpace.generateFrom (range fun a => Iic ↑a)
[PROOFSTEP]
refine le_antisymm (generateFrom_le ?_) (generateFrom_le ?_)
[GOAL]
case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
⊢ ∀ (t : Set ℝ), (t ∈ range fun a => Ioi ↑a) → MeasurableSet t
[PROOFSTEP]
rintro _ ⟨q, rfl⟩
[GOAL]
case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
⊢ ∀ (t : Set ℝ), (t ∈ range fun a => Iic ↑a) → MeasurableSet t
[PROOFSTEP]
rintro _ ⟨q, rfl⟩
[GOAL]
case refine_1.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
q : ℚ
⊢ MeasurableSet ((fun a => Ioi ↑a) q)
[PROOFSTEP]
dsimp only
[GOAL]
case refine_2.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
q : ℚ
⊢ MeasurableSet ((fun a => Iic ↑a) q)
[PROOFSTEP]
dsimp only
[GOAL]
case refine_1.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
q : ℚ
⊢ MeasurableSet (Ioi ↑q)
[PROOFSTEP]
rw [← compl_Iic]
[GOAL]
case refine_2.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
q : ℚ
⊢ MeasurableSet (Iic ↑q)
[PROOFSTEP]
rw [← compl_Ioi]
[GOAL]
case refine_1.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
q : ℚ
⊢ MeasurableSet (Iic ↑q)ᶜ
[PROOFSTEP]
exact MeasurableSet.compl (GenerateMeasurable.basic _ (mem_range_self q))
[GOAL]
case refine_2.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
q : ℚ
⊢ MeasurableSet (Ioi ↑q)ᶜ
[PROOFSTEP]
exact MeasurableSet.compl (GenerateMeasurable.basic _ (mem_range_self q))
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
⊢ borel ℝ = MeasurableSpace.generateFrom (⋃ (a : ℚ), {Ici ↑a})
[PROOFSTEP]
rw [borel_eq_generateFrom_Iio_rat, iUnion_singleton_eq_range, iUnion_singleton_eq_range]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
⊢ MeasurableSpace.generateFrom (range fun a => Iio ↑a) = MeasurableSpace.generateFrom (range fun a => Ici ↑a)
[PROOFSTEP]
refine le_antisymm (generateFrom_le ?_) (generateFrom_le ?_) <;> rintro _ ⟨q, rfl⟩ <;> dsimp only <;> [rw [← compl_Ici];
  rw [← compl_Iio]]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
⊢ MeasurableSpace.generateFrom (range fun a => Iio ↑a) = MeasurableSpace.generateFrom (range fun a => Ici ↑a)
[PROOFSTEP]
refine le_antisymm (generateFrom_le ?_) (generateFrom_le ?_)
[GOAL]
case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
⊢ ∀ (t : Set ℝ), (t ∈ range fun a => Iio ↑a) → MeasurableSet t
[PROOFSTEP]
rintro _ ⟨q, rfl⟩
[GOAL]
case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
⊢ ∀ (t : Set ℝ), (t ∈ range fun a => Ici ↑a) → MeasurableSet t
[PROOFSTEP]
rintro _ ⟨q, rfl⟩
[GOAL]
case refine_1.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
q : ℚ
⊢ MeasurableSet ((fun a => Iio ↑a) q)
[PROOFSTEP]
dsimp only
[GOAL]
case refine_2.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
q : ℚ
⊢ MeasurableSet ((fun a => Ici ↑a) q)
[PROOFSTEP]
dsimp only
[GOAL]
case refine_1.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
q : ℚ
⊢ MeasurableSet (Iio ↑q)
[PROOFSTEP]
rw [← compl_Ici]
[GOAL]
case refine_2.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
q : ℚ
⊢ MeasurableSet (Ici ↑q)
[PROOFSTEP]
rw [← compl_Iio]
[GOAL]
case refine_1.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
q : ℚ
⊢ MeasurableSet (Ici ↑q)ᶜ
[PROOFSTEP]
exact MeasurableSet.compl (GenerateMeasurable.basic _ (mem_range_self q))
[GOAL]
case refine_2.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
q : ℚ
⊢ MeasurableSet (Iio ↑q)ᶜ
[PROOFSTEP]
exact MeasurableSet.compl (GenerateMeasurable.basic _ (mem_range_self q))
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
⊢ IsPiSystem (⋃ (a : ℚ) (b : ℚ) (_ : a < b), {Ioo ↑a ↑b})
[PROOFSTEP]
convert isPiSystem_Ioo ((↑) : ℚ → ℝ) ((↑) : ℚ → ℝ)
[GOAL]
case h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
⊢ ⋃ (a : ℚ) (b : ℚ) (_ : a < b), {Ioo ↑a ↑b} = {S | ∃ l u, ↑l < ↑u ∧ Ioo ↑l ↑u = S}
[PROOFSTEP]
ext x
[GOAL]
case h.e'_2.h
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
x : Set ℝ
⊢ x ∈ ⋃ (a : ℚ) (b : ℚ) (_ : a < b), {Ioo ↑a ↑b} ↔ x ∈ {S | ∃ l u, ↑l < ↑u ∧ Ioo ↑l ↑u = S}
[PROOFSTEP]
simp [eq_comm]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
⊢ IsPiSystem (⋃ (a : ℚ), {Iio ↑a})
[PROOFSTEP]
convert isPiSystem_image_Iio (((↑) : ℚ → ℝ) '' univ)
[GOAL]
case h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
⊢ ⋃ (a : ℚ), {Iio ↑a} = Iio '' (Rat.cast '' univ)
[PROOFSTEP]
ext x
[GOAL]
case h.e'_2.h
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
x : Set ℝ
⊢ x ∈ ⋃ (a : ℚ), {Iio ↑a} ↔ x ∈ Iio '' (Rat.cast '' univ)
[PROOFSTEP]
simp only [iUnion_singleton_eq_range, mem_range, image_univ, mem_image, exists_exists_eq_and]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
⊢ IsPiSystem (⋃ (a : ℚ), {Ioi ↑a})
[PROOFSTEP]
convert isPiSystem_image_Ioi (((↑) : ℚ → ℝ) '' univ)
[GOAL]
case h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
⊢ ⋃ (a : ℚ), {Ioi ↑a} = Ioi '' (Rat.cast '' univ)
[PROOFSTEP]
ext x
[GOAL]
case h.e'_2.h
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
x : Set ℝ
⊢ x ∈ ⋃ (a : ℚ), {Ioi ↑a} ↔ x ∈ Ioi '' (Rat.cast '' univ)
[PROOFSTEP]
simp only [iUnion_singleton_eq_range, mem_range, image_univ, mem_image, exists_exists_eq_and]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
⊢ IsPiSystem (⋃ (a : ℚ), {Iic ↑a})
[PROOFSTEP]
convert isPiSystem_image_Iic (((↑) : ℚ → ℝ) '' univ)
[GOAL]
case h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
⊢ ⋃ (a : ℚ), {Iic ↑a} = Iic '' (Rat.cast '' univ)
[PROOFSTEP]
ext x
[GOAL]
case h.e'_2.h
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
x : Set ℝ
⊢ x ∈ ⋃ (a : ℚ), {Iic ↑a} ↔ x ∈ Iic '' (Rat.cast '' univ)
[PROOFSTEP]
simp only [iUnion_singleton_eq_range, mem_range, image_univ, mem_image, exists_exists_eq_and]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
⊢ IsPiSystem (⋃ (a : ℚ), {Ici ↑a})
[PROOFSTEP]
convert isPiSystem_image_Ici (((↑) : ℚ → ℝ) '' univ)
[GOAL]
case h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
⊢ ⋃ (a : ℚ), {Ici ↑a} = Ici '' (Rat.cast '' univ)
[PROOFSTEP]
ext x
[GOAL]
case h.e'_2.h
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
x : Set ℝ
⊢ x ∈ ⋃ (a : ℚ), {Ici ↑a} ↔ x ∈ Ici '' (Rat.cast '' univ)
[PROOFSTEP]
simp only [iUnion_singleton_eq_range, mem_range, image_univ, mem_image, exists_exists_eq_and]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
μ : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
n : ℕ
⊢ (fun n => Ioo (-(↑n + 1)) (↑n + 1)) n ∈ ⋃ (a : ℚ) (b : ℚ) (_ : a < b), {Ioo ↑a ↑b}
[PROOFSTEP]
simp only [mem_iUnion, mem_singleton_iff]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
μ : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
n : ℕ
⊢ ∃ i i_1 h, Ioo (-(↑n + 1)) (↑n + 1) = Ioo ↑i ↑i_1
[PROOFSTEP]
refine'
  ⟨-(n + 1 : ℕ), n + 1, _, by simp⟩
    -- TODO: norm_cast fails here?
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
μ : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
n : ℕ
⊢ Ioo (-(↑n + 1)) (↑n + 1) = Ioo ↑(-↑(n + 1)) ↑(↑n + 1)
[PROOFSTEP]
simp
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
μ : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
n : ℕ
⊢ -↑(n + 1) < ↑n + 1
[PROOFSTEP]
exact (neg_nonpos.2 (@Nat.cast_nonneg ℚ _ (n + 1))).trans_lt n.cast_add_one_pos
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
μ ν : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
h : ∀ (a b : ℚ), ↑↑μ (Ioo ↑a ↑b) = ↑↑ν (Ioo ↑a ↑b)
⊢ ∀ (s : Set ℝ), s ∈ ⋃ (a : ℚ) (b : ℚ) (_ : a < b), {Ioo ↑a ↑b} → ↑↑μ s = ↑↑ν s
[PROOFSTEP]
simp only [mem_iUnion, mem_singleton_iff]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
μ ν : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
h : ∀ (a b : ℚ), ↑↑μ (Ioo ↑a ↑b) = ↑↑ν (Ioo ↑a ↑b)
⊢ ∀ (s : Set ℝ), (∃ i i_1 h, s = Ioo ↑i ↑i_1) → ↑↑μ s = ↑↑ν s
[PROOFSTEP]
rintro _ ⟨a, b, -, rfl⟩
[GOAL]
case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
μ ν : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
h : ∀ (a b : ℚ), ↑↑μ (Ioo ↑a ↑b) = ↑↑ν (Ioo ↑a ↑b)
a b : ℚ
⊢ ↑↑μ (Ioo ↑a ↑b) = ↑↑ν (Ioo ↑a ↑b)
[PROOFSTEP]
apply h
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝ : MeasurableSpace α
f : α → ℝ≥0
h : Measurable fun x => ↑(f x)
⊢ Measurable f
[PROOFSTEP]
simpa only [Real.toNNReal_coe] using h.real_toNNReal
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝ : MeasurableSpace α
f : α → ℝ≥0
μ : Measure α
h : AEMeasurable fun x => ↑(f x)
⊢ AEMeasurable f
[PROOFSTEP]
simpa only [Real.toNNReal_coe] using h.real_toNNReal
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝ : MeasurableSpace α
⊢ MeasurableMul₂ ℝ≥0∞
[PROOFSTEP]
refine' ⟨measurable_of_measurable_nnreal_nnreal _ _ _⟩
[GOAL]
case refine'_1
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝ : MeasurableSpace α
⊢ Measurable fun p => (↑p.fst, ↑p.snd).fst * (↑p.fst, ↑p.snd).snd
[PROOFSTEP]
simp only [← ENNReal.coe_mul, measurable_mul.coe_nnreal_ennreal]
[GOAL]
case refine'_2
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝ : MeasurableSpace α
⊢ Measurable fun r => (⊤, ↑r).fst * (⊤, ↑r).snd
[PROOFSTEP]
simp only [ENNReal.top_mul', ENNReal.coe_eq_zero]
[GOAL]
case refine'_2
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝ : MeasurableSpace α
⊢ Measurable fun r => if r = 0 then 0 else ⊤
[PROOFSTEP]
exact measurable_const.piecewise (measurableSet_singleton _) measurable_const
[GOAL]
case refine'_3
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝ : MeasurableSpace α
⊢ Measurable fun r => (↑r, ⊤).fst * (↑r, ⊤).snd
[PROOFSTEP]
simp only [ENNReal.mul_top', ENNReal.coe_eq_zero]
[GOAL]
case refine'_3
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝ : MeasurableSpace α
⊢ Measurable fun r => if r = 0 then 0 else ⊤
[PROOFSTEP]
exact measurable_const.piecewise (measurableSet_singleton _) measurable_const
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝ : MeasurableSpace α
⊢ Measurable fun p => p.fst - p.snd
[PROOFSTEP]
apply measurable_of_measurable_nnreal_nnreal
[GOAL]
case h₁
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝ : MeasurableSpace α
⊢ Measurable fun p => (↑p.fst, ↑p.snd).fst - (↑p.fst, ↑p.snd).snd
[PROOFSTEP]
simp [← WithTop.coe_sub]
[GOAL]
case h₂
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝ : MeasurableSpace α
⊢ Measurable fun r => (⊤, ↑r).fst - (⊤, ↑r).snd
[PROOFSTEP]
simp [← WithTop.coe_sub]
[GOAL]
case h₃
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝ : MeasurableSpace α
⊢ Measurable fun r => (↑r, ⊤).fst - (↑r, ⊤).snd
[PROOFSTEP]
simp [← WithTop.coe_sub]
[GOAL]
case h₁
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝ : MeasurableSpace α
⊢ Measurable fun p => ↑p.fst - ↑p.snd
[PROOFSTEP]
exact continuous_sub.measurable.coe_nnreal_ennreal
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹ : MeasurableSpace α
ι : Type u_6
inst✝ : Countable ι
f : ι → α → ℝ≥0∞
h : ∀ (i : ι), Measurable (f i)
⊢ Measurable fun x => ∑' (i : ι), f i x
[PROOFSTEP]
simp_rw [ENNReal.tsum_eq_iSup_sum]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹ : MeasurableSpace α
ι : Type u_6
inst✝ : Countable ι
f : ι → α → ℝ≥0∞
h : ∀ (i : ι), Measurable (f i)
⊢ Measurable fun x => ⨆ (s : Finset ι), ∑ i in s, f i x
[PROOFSTEP]
apply measurable_iSup
[GOAL]
case hf
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹ : MeasurableSpace α
ι : Type u_6
inst✝ : Countable ι
f : ι → α → ℝ≥0∞
h : ∀ (i : ι), Measurable (f i)
⊢ ∀ (i : Finset ι), Measurable fun b => ∑ i in i, f i b
[PROOFSTEP]
exact fun s => s.measurable_sum fun i _ => h i
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹ : MeasurableSpace α
ι : Type u_6
inst✝ : Countable ι
f : ι → α → ℝ≥0∞
h : ∀ (i : ι), Measurable (f i)
⊢ Measurable (∑' (i : ι), f i)
[PROOFSTEP]
convert Measurable.ennreal_tsum h with x
[GOAL]
case h.e'_5.h
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹ : MeasurableSpace α
ι : Type u_6
inst✝ : Countable ι
f : ι → α → ℝ≥0∞
h : ∀ (i : ι), Measurable (f i)
x : α
⊢ tsum (fun i => f i) x = ∑' (i : ι), f i x
[PROOFSTEP]
exact tsum_apply (Pi.summable.2 fun _ => ENNReal.summable)
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹ : MeasurableSpace α
ι : Type u_6
inst✝ : Countable ι
f : ι → α → ℝ≥0
h : ∀ (i : ι), Measurable (f i)
⊢ Measurable fun x => ∑' (i : ι), f i x
[PROOFSTEP]
simp_rw [NNReal.tsum_eq_toNNReal_tsum]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹ : MeasurableSpace α
ι : Type u_6
inst✝ : Countable ι
f : ι → α → ℝ≥0
h : ∀ (i : ι), Measurable (f i)
⊢ Measurable fun x => ENNReal.toNNReal (∑' (b : ι), ↑(f b x))
[PROOFSTEP]
exact (Measurable.ennreal_tsum fun i => (h i).coe_nnreal_ennreal).ennreal_toNNReal
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹ : MeasurableSpace α
ι : Type u_6
inst✝ : Countable ι
f : ι → α → ℝ≥0∞
μ : Measure α
h : ∀ (i : ι), AEMeasurable (f i)
⊢ AEMeasurable fun x => ∑' (i : ι), f i x
[PROOFSTEP]
simp_rw [ENNReal.tsum_eq_iSup_sum]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹ : MeasurableSpace α
ι : Type u_6
inst✝ : Countable ι
f : ι → α → ℝ≥0∞
μ : Measure α
h : ∀ (i : ι), AEMeasurable (f i)
⊢ AEMeasurable fun x => ⨆ (s : Finset ι), ∑ i in s, f i x
[PROOFSTEP]
apply aemeasurable_iSup
[GOAL]
case hf
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α
inst✝¹ : MeasurableSpace α
ι : Type u_6
inst✝ : Countable ι
f : ι → α → ℝ≥0∞
μ : Measure α
h : ∀ (i : ι), AEMeasurable (f i)
⊢ ∀ (i : Finset ι), AEMeasurable fun b => ∑ i in i, f i b
[PROOFSTEP]
exact fun s => Finset.aemeasurable_sum s fun i _ => h i
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α✝
inst✝² : MeasurableSpace α✝
α : Type u_6
inst✝¹ : MeasurableSpace α
ι : Type u_7
inst✝ : Countable ι
f : ι → α → ℝ≥0
μ : Measure α
h : ∀ (i : ι), AEMeasurable (f i)
⊢ AEMeasurable fun x => ∑' (i : ι), f i x
[PROOFSTEP]
simp_rw [NNReal.tsum_eq_toNNReal_tsum]
[GOAL]
α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι✝ : Sort y
s t u : Set α✝
inst✝² : MeasurableSpace α✝
α : Type u_6
inst✝¹ : MeasurableSpace α
ι : Type u_7
inst✝ : Countable ι
f : ι → α → ℝ≥0
μ : Measure α
h : ∀ (i : ι), AEMeasurable (f i)
⊢ AEMeasurable fun x => ENNReal.toNNReal (∑' (b : ι), ↑(f b x))
[PROOFSTEP]
exact (AEMeasurable.ennreal_tsum fun i => (h i).coe_nnreal_ennreal).ennreal_toNNReal
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝ : MeasurableSpace α
f : EReal → α
h : Measurable fun p => f ↑p
⊢ Set.Finite {⊥, ⊤}
[PROOFSTEP]
simp
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝ : MeasurableSpace α
⊢ Measurable fun p => EReal.toReal ↑p
[PROOFSTEP]
simpa using measurable_id