[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ¬restrict s i ≤ restrict 0 i
⊢ MeasureTheory.SignedMeasure.ExistsOneDivLT s i (MeasureTheory.SignedMeasure.findExistsOneDivLT s i)
[PROOFSTEP]
rw [findExistsOneDivLT, dif_pos hi]
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ¬restrict s i ≤ restrict 0 i
⊢ MeasureTheory.SignedMeasure.ExistsOneDivLT s i (Nat.find (_ : ∃ n, MeasureTheory.SignedMeasure.ExistsOneDivLT s i n))
[PROOFSTEP]
convert Nat.find_spec (existsNatOneDivLTMeasure_of_not_negative hi)
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ¬restrict s i ≤ restrict 0 i
m : ℕ
hm : m < MeasureTheory.SignedMeasure.findExistsOneDivLT s i
⊢ ¬MeasureTheory.SignedMeasure.ExistsOneDivLT s i m
[PROOFSTEP]
rw [findExistsOneDivLT, dif_pos hi] at hm
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ¬restrict s i ≤ restrict 0 i
m : ℕ
hm : m < Nat.find (_ : ∃ n, MeasureTheory.SignedMeasure.ExistsOneDivLT s i n)
⊢ ¬MeasureTheory.SignedMeasure.ExistsOneDivLT s i m
[PROOFSTEP]
exact Nat.find_min _ hm
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ¬restrict s i ≤ restrict 0 i
⊢ MeasureTheory.SignedMeasure.someExistsOneDivLT s i ⊆ i ∧
    MeasurableSet (MeasureTheory.SignedMeasure.someExistsOneDivLT s i) ∧
      1 / (↑(MeasureTheory.SignedMeasure.findExistsOneDivLT s i) + 1) <
        ↑s (MeasureTheory.SignedMeasure.someExistsOneDivLT s i)
[PROOFSTEP]
rw [someExistsOneDivLT, dif_pos hi]
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ¬restrict s i ≤ restrict 0 i
⊢ Classical.choose
        (_ : MeasureTheory.SignedMeasure.ExistsOneDivLT s i (MeasureTheory.SignedMeasure.findExistsOneDivLT s i)) ⊆
      i ∧
    MeasurableSet
        (Classical.choose
          (_ : MeasureTheory.SignedMeasure.ExistsOneDivLT s i (MeasureTheory.SignedMeasure.findExistsOneDivLT s i))) ∧
      1 / (↑(MeasureTheory.SignedMeasure.findExistsOneDivLT s i) + 1) <
        ↑s
          (Classical.choose
            (_ : MeasureTheory.SignedMeasure.ExistsOneDivLT s i (MeasureTheory.SignedMeasure.findExistsOneDivLT s i)))
[PROOFSTEP]
exact Classical.choose_spec (findExistsOneDivLT_spec hi)
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
⊢ MeasureTheory.SignedMeasure.someExistsOneDivLT s i ⊆ i
[PROOFSTEP]
by_cases hi : ¬s ≤[i] 0
[GOAL]
case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ¬restrict s i ≤ restrict 0 i
⊢ MeasureTheory.SignedMeasure.someExistsOneDivLT s i ⊆ i
[PROOFSTEP]
exact
  let ⟨h, _⟩ := someExistsOneDivLT_spec hi
  h
[GOAL]
case neg
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ¬¬restrict s i ≤ restrict 0 i
⊢ MeasureTheory.SignedMeasure.someExistsOneDivLT s i ⊆ i
[PROOFSTEP]
rw [someExistsOneDivLT, dif_neg hi]
[GOAL]
case neg
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ¬¬restrict s i ≤ restrict 0 i
⊢ ∅ ⊆ i
[PROOFSTEP]
exact Set.empty_subset _
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
⊢ MeasurableSet (MeasureTheory.SignedMeasure.someExistsOneDivLT s i)
[PROOFSTEP]
by_cases hi : ¬s ≤[i] 0
[GOAL]
case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ¬restrict s i ≤ restrict 0 i
⊢ MeasurableSet (MeasureTheory.SignedMeasure.someExistsOneDivLT s i)
[PROOFSTEP]
exact
  let ⟨_, h, _⟩ := someExistsOneDivLT_spec hi
  h
[GOAL]
case neg
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ¬¬restrict s i ≤ restrict 0 i
⊢ MeasurableSet (MeasureTheory.SignedMeasure.someExistsOneDivLT s i)
[PROOFSTEP]
rw [someExistsOneDivLT, dif_neg hi]
[GOAL]
case neg
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ¬¬restrict s i ≤ restrict 0 i
⊢ MeasurableSet ∅
[PROOFSTEP]
exact MeasurableSet.empty
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
n : ℕ
⊢ MeasureTheory.SignedMeasure.restrictNonposSeq s i (Nat.succ n) =
    MeasureTheory.SignedMeasure.someExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
[PROOFSTEP]
rw [restrictNonposSeq]
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
n : ℕ
⊢ MeasureTheory.SignedMeasure.restrictNonposSeq s i n ⊆ i
[PROOFSTEP]
cases n
[GOAL]
case zero
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
⊢ MeasureTheory.SignedMeasure.restrictNonposSeq s i Nat.zero ⊆ i
[PROOFSTEP]
rw [restrictNonposSeq]
[GOAL]
case zero
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
⊢ MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ∅) ⊆ i
[PROOFSTEP]
exact someExistsOneDivLT_subset'
[GOAL]
case succ
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
n✝ : ℕ
⊢ MeasureTheory.SignedMeasure.restrictNonposSeq s i (Nat.succ n✝) ⊆ i
[PROOFSTEP]
rw [restrictNonposSeq]
[GOAL]
case succ
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
n✝ : ℕ
⊢ MeasureTheory.SignedMeasure.someExistsOneDivLT s
      (i \
        ⋃ (k : ℕ) (H : k ≤ n✝),
          let_fun this := (_ : k < Nat.succ n✝);
          MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ⊆
    i
[PROOFSTEP]
exact someExistsOneDivLT_subset'
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
n : ℕ
hn :
  ¬restrict s (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ 1 /
      (↑(MeasureTheory.SignedMeasure.findExistsOneDivLT s
            (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)) +
        1) <
    ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i (Nat.succ n))
[PROOFSTEP]
rw [restrictNonposSeq_succ]
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
n : ℕ
hn :
  ¬restrict s (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ 1 /
      (↑(MeasureTheory.SignedMeasure.findExistsOneDivLT s
            (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)) +
        1) <
    ↑s
      (MeasureTheory.SignedMeasure.someExistsOneDivLT s
        (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k))
[PROOFSTEP]
apply someExistsOneDivLT_lt hn
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₂ : ¬restrict s i ≤ restrict 0 i
n : ℕ
hn :
  ¬restrict s (i \ ⋃ (k : ℕ) (_ : k < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ (k : ℕ) (_ : k < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ 0 < ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i n)
[PROOFSTEP]
cases n with
| zero =>
  rw [restrictNonposSeq]; rw [← @Set.diff_empty _ i] at hi₂
  rcases someExistsOneDivLT_spec hi₂ with ⟨_, _, h⟩
  exact lt_trans Nat.one_div_pos_of_nat h
| succ n =>
  rw [restrictNonposSeq_succ]
  have h₁ : ¬s ≤[i \ ⋃ (k : ℕ) (_ : k ≤ n), restrictNonposSeq s i k] 0 :=
    by
    refine' mt (restrict_le_zero_subset _ _ (by simp [Nat.lt_succ_iff]; rfl)) hn
    convert measurable_of_not_restrict_le_zero _ hn using 3
    exact funext fun x => by rw [Nat.lt_succ_iff]
  rcases someExistsOneDivLT_spec h₁ with ⟨_, _, h⟩
  exact lt_trans Nat.one_div_pos_of_nat h
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₂ : ¬restrict s i ≤ restrict 0 i
n : ℕ
hn :
  ¬restrict s (i \ ⋃ (k : ℕ) (_ : k < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ (k : ℕ) (_ : k < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ 0 < ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i n)
[PROOFSTEP]
cases n with
| zero =>
  rw [restrictNonposSeq]; rw [← @Set.diff_empty _ i] at hi₂
  rcases someExistsOneDivLT_spec hi₂ with ⟨_, _, h⟩
  exact lt_trans Nat.one_div_pos_of_nat h
| succ n =>
  rw [restrictNonposSeq_succ]
  have h₁ : ¬s ≤[i \ ⋃ (k : ℕ) (_ : k ≤ n), restrictNonposSeq s i k] 0 :=
    by
    refine' mt (restrict_le_zero_subset _ _ (by simp [Nat.lt_succ_iff]; rfl)) hn
    convert measurable_of_not_restrict_le_zero _ hn using 3
    exact funext fun x => by rw [Nat.lt_succ_iff]
  rcases someExistsOneDivLT_spec h₁ with ⟨_, _, h⟩
  exact lt_trans Nat.one_div_pos_of_nat h
[GOAL]
case zero
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₂ : ¬restrict s i ≤ restrict 0 i
hn :
  ¬restrict s (i \ ⋃ (k : ℕ) (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ (k : ℕ) (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ 0 < ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i Nat.zero)
[PROOFSTEP]

| zero =>
  rw [restrictNonposSeq]; rw [← @Set.diff_empty _ i] at hi₂
  rcases someExistsOneDivLT_spec hi₂ with ⟨_, _, h⟩
  exact lt_trans Nat.one_div_pos_of_nat h
[GOAL]
case zero
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₂ : ¬restrict s i ≤ restrict 0 i
hn :
  ¬restrict s (i \ ⋃ (k : ℕ) (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ (k : ℕ) (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ 0 < ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i Nat.zero)
[PROOFSTEP]
rw [restrictNonposSeq]
[GOAL]
case zero
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₂ : ¬restrict s i ≤ restrict 0 i
hn :
  ¬restrict s (i \ ⋃ (k : ℕ) (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ (k : ℕ) (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ 0 < ↑s (MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ∅))
[PROOFSTEP]
rw [← @Set.diff_empty _ i] at hi₂
[GOAL]
case zero
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₂ : ¬restrict s (i \ ∅) ≤ restrict 0 (i \ ∅)
hn :
  ¬restrict s (i \ ⋃ (k : ℕ) (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ (k : ℕ) (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ 0 < ↑s (MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ∅))
[PROOFSTEP]
rcases someExistsOneDivLT_spec hi₂ with ⟨_, _, h⟩
[GOAL]
case zero.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₂ : ¬restrict s (i \ ∅) ≤ restrict 0 (i \ ∅)
hn :
  ¬restrict s (i \ ⋃ (k : ℕ) (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ (k : ℕ) (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
left✝¹ : MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ∅) ⊆ i \ ∅
left✝ : MeasurableSet (MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ∅))
h :
  1 / (↑(MeasureTheory.SignedMeasure.findExistsOneDivLT s (i \ ∅)) + 1) <
    ↑s (MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ∅))
⊢ 0 < ↑s (MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ∅))
[PROOFSTEP]
exact lt_trans Nat.one_div_pos_of_nat h
[GOAL]
case succ
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₂ : ¬restrict s i ≤ restrict 0 i
n : ℕ
hn :
  ¬restrict s (i \ ⋃ (k : ℕ) (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ (k : ℕ) (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ 0 < ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i (Nat.succ n))
[PROOFSTEP]

| succ n =>
  rw [restrictNonposSeq_succ]
  have h₁ : ¬s ≤[i \ ⋃ (k : ℕ) (_ : k ≤ n), restrictNonposSeq s i k] 0 :=
    by
    refine' mt (restrict_le_zero_subset _ _ (by simp [Nat.lt_succ_iff]; rfl)) hn
    convert measurable_of_not_restrict_le_zero _ hn using 3
    exact funext fun x => by rw [Nat.lt_succ_iff]
  rcases someExistsOneDivLT_spec h₁ with ⟨_, _, h⟩
  exact lt_trans Nat.one_div_pos_of_nat h
[GOAL]
case succ
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₂ : ¬restrict s i ≤ restrict 0 i
n : ℕ
hn :
  ¬restrict s (i \ ⋃ (k : ℕ) (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ (k : ℕ) (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ 0 < ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i (Nat.succ n))
[PROOFSTEP]
rw [restrictNonposSeq_succ]
[GOAL]
case succ
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₂ : ¬restrict s i ≤ restrict 0 i
n : ℕ
hn :
  ¬restrict s (i \ ⋃ (k : ℕ) (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ (k : ℕ) (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ 0 <
    ↑s
      (MeasureTheory.SignedMeasure.someExistsOneDivLT s
        (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k))
[PROOFSTEP]
have h₁ : ¬s ≤[i \ ⋃ (k : ℕ) (_ : k ≤ n), restrictNonposSeq s i k] 0 :=
  by
  refine' mt (restrict_le_zero_subset _ _ (by simp [Nat.lt_succ_iff]; rfl)) hn
  convert measurable_of_not_restrict_le_zero _ hn using 3
  exact funext fun x => by rw [Nat.lt_succ_iff]
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₂ : ¬restrict s i ≤ restrict 0 i
n : ℕ
hn :
  ¬restrict s (i \ ⋃ (k : ℕ) (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ (k : ℕ) (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ ¬restrict s (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
[PROOFSTEP]
refine' mt (restrict_le_zero_subset _ _ (by simp [Nat.lt_succ_iff]; rfl)) hn
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₂ : ¬restrict s i ≤ restrict 0 i
n : ℕ
hn :
  ¬restrict s (i \ ⋃ (k : ℕ) (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ (k : ℕ) (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ i \ ⋃ (k : ℕ) (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k ⊆
    i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k
[PROOFSTEP]
simp [Nat.lt_succ_iff]
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₂ : ¬restrict s i ≤ restrict 0 i
n : ℕ
hn :
  ¬restrict s (i \ ⋃ (k : ℕ) (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ (k : ℕ) (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k ⊆
    i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k
[PROOFSTEP]
rfl
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₂ : ¬restrict s i ≤ restrict 0 i
n : ℕ
hn :
  ¬restrict s (i \ ⋃ (k : ℕ) (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ (k : ℕ) (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ MeasurableSet (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
[PROOFSTEP]
convert measurable_of_not_restrict_le_zero _ hn using 3
[GOAL]
case h.e'_3.h.e'_4.h.e'_3
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₂ : ¬restrict s i ≤ restrict 0 i
n : ℕ
hn :
  ¬restrict s (i \ ⋃ (k : ℕ) (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ (k : ℕ) (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ (fun k => ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) = fun k =>
    ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k
[PROOFSTEP]
exact funext fun x => by rw [Nat.lt_succ_iff]
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₂ : ¬restrict s i ≤ restrict 0 i
n : ℕ
hn :
  ¬restrict s (i \ ⋃ (k : ℕ) (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ (k : ℕ) (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
x : ℕ
⊢ ⋃ (_ : x ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i x =
    ⋃ (_ : x < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i x
[PROOFSTEP]
rw [Nat.lt_succ_iff]
[GOAL]
case succ
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₂ : ¬restrict s i ≤ restrict 0 i
n : ℕ
hn :
  ¬restrict s (i \ ⋃ (k : ℕ) (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ (k : ℕ) (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
h₁ :
  ¬restrict s (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ 0 <
    ↑s
      (MeasureTheory.SignedMeasure.someExistsOneDivLT s
        (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k))
[PROOFSTEP]
rcases someExistsOneDivLT_spec h₁ with ⟨_, _, h⟩
[GOAL]
case succ.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₂ : ¬restrict s i ≤ restrict 0 i
n : ℕ
hn :
  ¬restrict s (i \ ⋃ (k : ℕ) (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ (k : ℕ) (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
h₁ :
  ¬restrict s (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
left✝¹ :
  MeasureTheory.SignedMeasure.someExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ⊆
    i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k
left✝ :
  MeasurableSet
    (MeasureTheory.SignedMeasure.someExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k))
h :
  1 /
      (↑(MeasureTheory.SignedMeasure.findExistsOneDivLT s
            (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)) +
        1) <
    ↑s
      (MeasureTheory.SignedMeasure.someExistsOneDivLT s
        (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k))
⊢ 0 <
    ↑s
      (MeasureTheory.SignedMeasure.someExistsOneDivLT s
        (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k))
[PROOFSTEP]
exact lt_trans Nat.one_div_pos_of_nat h
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
n : ℕ
⊢ MeasurableSet (MeasureTheory.SignedMeasure.restrictNonposSeq s i n)
[PROOFSTEP]
cases n
[GOAL]
case zero
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
⊢ MeasurableSet (MeasureTheory.SignedMeasure.restrictNonposSeq s i Nat.zero)
[PROOFSTEP]
rw [restrictNonposSeq]
[GOAL]
case zero
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
⊢ MeasurableSet (MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ∅))
[PROOFSTEP]
exact someExistsOneDivLT_measurableSet
[GOAL]
case succ
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
n✝ : ℕ
⊢ MeasurableSet (MeasureTheory.SignedMeasure.restrictNonposSeq s i (Nat.succ n✝))
[PROOFSTEP]
rw [restrictNonposSeq]
[GOAL]
case succ
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
n✝ : ℕ
⊢ MeasurableSet
    (MeasureTheory.SignedMeasure.someExistsOneDivLT s
      (i \
        ⋃ (k : ℕ) (H : k ≤ n✝),
          let_fun this := (_ : k < Nat.succ n✝);
          MeasureTheory.SignedMeasure.restrictNonposSeq s i k))
[PROOFSTEP]
exact someExistsOneDivLT_measurableSet
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
n m : ℕ
h : n < m
⊢ MeasureTheory.SignedMeasure.restrictNonposSeq s i n ∩ MeasureTheory.SignedMeasure.restrictNonposSeq s i m = ∅
[PROOFSTEP]
rw [Set.eq_empty_iff_forall_not_mem]
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
n m : ℕ
h : n < m
⊢ ∀ (x : α),
    ¬x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i n ∩ MeasureTheory.SignedMeasure.restrictNonposSeq s i m
[PROOFSTEP]
rintro x ⟨hx₁, hx₂⟩
[GOAL]
case intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
n m : ℕ
h : n < m
x : α
hx₁ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i n
hx₂ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i m
⊢ False
[PROOFSTEP]
cases m
[GOAL]
case intro.zero
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
n : ℕ
x : α
hx₁ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i n
h : n < Nat.zero
hx₂ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i Nat.zero
⊢ False
[PROOFSTEP]
rw [Nat.zero_eq] at h
[GOAL]
case intro.zero
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
n : ℕ
x : α
hx₁ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i n
h : n < 0
hx₂ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i Nat.zero
⊢ False
[PROOFSTEP]
linarith
[GOAL]
case intro.succ
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
n : ℕ
x : α
hx₁ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i n
n✝ : ℕ
h : n < Nat.succ n✝
hx₂ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i (Nat.succ n✝)
⊢ False
[PROOFSTEP]
rw [restrictNonposSeq] at hx₂
[GOAL]
case intro.succ
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
n : ℕ
x : α
hx₁ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i n
n✝ : ℕ
h : n < Nat.succ n✝
hx₂ :
  x ∈
    MeasureTheory.SignedMeasure.someExistsOneDivLT s
      (i \
        ⋃ (k : ℕ) (H : k ≤ n✝),
          let_fun this := (_ : k < Nat.succ n✝);
          MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ False
[PROOFSTEP]
exact (someExistsOneDivLT_subset hx₂).2 (Set.mem_iUnion.2 ⟨n, Set.mem_iUnion.2 ⟨Nat.lt_succ_iff.mp h, hx₁⟩⟩)
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
⊢ Pairwise (Disjoint on MeasureTheory.SignedMeasure.restrictNonposSeq s i)
[PROOFSTEP]
intro n m h
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
n m : ℕ
h : n ≠ m
⊢ (Disjoint on MeasureTheory.SignedMeasure.restrictNonposSeq s i) n m
[PROOFSTEP]
rw [Function.onFun, Set.disjoint_iff_inter_eq_empty]
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
n m : ℕ
h : n ≠ m
⊢ MeasureTheory.SignedMeasure.restrictNonposSeq s i n ∩ MeasureTheory.SignedMeasure.restrictNonposSeq s i m = ∅
[PROOFSTEP]
rcases lt_or_gt_of_ne h with (h | h)
[GOAL]
case inl
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
n m : ℕ
h✝ : n ≠ m
h : n < m
⊢ MeasureTheory.SignedMeasure.restrictNonposSeq s i n ∩ MeasureTheory.SignedMeasure.restrictNonposSeq s i m = ∅
[PROOFSTEP]
rw [restrictNonposSeq_disjoint' h]
[GOAL]
case inr
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
n m : ℕ
h✝ : n ≠ m
h : n > m
⊢ MeasureTheory.SignedMeasure.restrictNonposSeq s i n ∩ MeasureTheory.SignedMeasure.restrictNonposSeq s i m = ∅
[PROOFSTEP]
rw [Set.inter_comm, restrictNonposSeq_disjoint' h]
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
hn :
  ¬∀ (n : ℕ),
      ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
          restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
[PROOFSTEP]
by_cases s ≤[i] 0
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
hn :
  ¬∀ (n : ℕ),
      ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
          restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
[PROOFSTEP]
by_cases s ≤[i] 0
[GOAL]
case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
hn :
  ¬∀ (n : ℕ),
      ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
          restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h : restrict s i ≤ restrict 0 i
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
[PROOFSTEP]
exact ⟨i, hi₁, Set.Subset.refl _, h, hi₂⟩
[GOAL]
case neg
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
hn :
  ¬∀ (n : ℕ),
      ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
          restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h : ¬restrict s i ≤ restrict 0 i
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
[PROOFSTEP]
push_neg at hn
[GOAL]
case neg
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
[PROOFSTEP]
set k := Nat.find hn
[GOAL]
case neg
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
[PROOFSTEP]
have hk₂ : s ≤[i \ ⋃ l < k, restrictNonposSeq s i l] 0 := Nat.find_spec hn
[GOAL]
case neg
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
[PROOFSTEP]
have hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), restrictNonposSeq s i l) :=
  MeasurableSet.iUnion fun _ => MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
[GOAL]
case neg
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
[PROOFSTEP]
refine' ⟨i \ ⋃ l < k, restrictNonposSeq s i l, hi₁.diff hmeas, Set.diff_subset _ _, hk₂, _⟩
[GOAL]
case neg
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ ↑s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) < 0
[PROOFSTEP]
rw [of_diff hmeas hi₁, s.of_disjoint_iUnion_nat]
[GOAL]
case neg
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ ↑s i - ∑' (i_1 : ℕ), ↑s (⋃ (_ : i_1 < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1) < 0
[PROOFSTEP]
have h₁ : ∀ l < k, 0 ≤ s (restrictNonposSeq s i l) := by
  intro l hl
  refine' le_of_lt (measure_of_restrictNonposSeq h _ _)
  refine' mt (restrict_le_zero_subset _ (hi₁.diff _) (Set.Subset.refl _)) (Nat.find_min hn hl)
  exact MeasurableSet.iUnion fun _ => MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ ∀ (l : ℕ), l < k → 0 ≤ ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
[PROOFSTEP]
intro l hl
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
l : ℕ
hl : l < k
⊢ 0 ≤ ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
[PROOFSTEP]
refine' le_of_lt (measure_of_restrictNonposSeq h _ _)
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
l : ℕ
hl : l < k
⊢ ¬restrict s (i \ ⋃ (k : ℕ) (_ : k < l), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ (k : ℕ) (_ : k < l), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
[PROOFSTEP]
refine' mt (restrict_le_zero_subset _ (hi₁.diff _) (Set.Subset.refl _)) (Nat.find_min hn hl)
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
l : ℕ
hl : l < k
⊢ MeasurableSet (⋃ (k : ℕ) (_ : k < l), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
[PROOFSTEP]
exact MeasurableSet.iUnion fun _ => MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
[GOAL]
case neg
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ∀ (l : ℕ), l < k → 0 ≤ ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ ↑s i - ∑' (i_1 : ℕ), ↑s (⋃ (_ : i_1 < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1) < 0
[PROOFSTEP]
suffices 0 ≤ ∑' l : ℕ, s (⋃ _ : l < k, restrictNonposSeq s i l)
  by
  rw [sub_neg]
  exact lt_of_lt_of_le hi₂ this
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ∀ (l : ℕ), l < k → 0 ≤ ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
this : 0 ≤ ∑' (l : ℕ), ↑s (⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ ↑s i - ∑' (i_1 : ℕ), ↑s (⋃ (_ : i_1 < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1) < 0
[PROOFSTEP]
rw [sub_neg]
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ∀ (l : ℕ), l < k → 0 ≤ ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
this : 0 ≤ ∑' (l : ℕ), ↑s (⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ ↑s i < ∑' (i_1 : ℕ), ↑s (⋃ (_ : i_1 < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1)
[PROOFSTEP]
exact lt_of_lt_of_le hi₂ this
[GOAL]
case neg
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ∀ (l : ℕ), l < k → 0 ≤ ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ 0 ≤ ∑' (l : ℕ), ↑s (⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
[PROOFSTEP]
refine' tsum_nonneg _
[GOAL]
case neg
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ∀ (l : ℕ), l < k → 0 ≤ ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ ∀ (i_1 : ℕ), 0 ≤ ↑s (⋃ (_ : i_1 < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1)
[PROOFSTEP]
intro l
[GOAL]
case neg
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ∀ (l : ℕ), l < k → 0 ≤ ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
l : ℕ
⊢ 0 ≤ ↑s (⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
[PROOFSTEP]
by_cases l < k
[GOAL]
case neg
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ∀ (l : ℕ), l < k → 0 ≤ ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
l : ℕ
⊢ 0 ≤ ↑s (⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
[PROOFSTEP]
by_cases l < k
[GOAL]
case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h✝ : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ∀ (l : ℕ), l < k → 0 ≤ ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
l : ℕ
h : l < k
⊢ 0 ≤ ↑s (⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
[PROOFSTEP]
convert h₁ _ h
[GOAL]
case h.e'_4.h.e'_7
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h✝ : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ∀ (l : ℕ), l < k → 0 ≤ ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
l : ℕ
h : l < k
⊢ ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l =
    MeasureTheory.SignedMeasure.restrictNonposSeq s i l
[PROOFSTEP]
ext x
[GOAL]
case h.e'_4.h.e'_7.h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h✝ : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ∀ (l : ℕ), l < k → 0 ≤ ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
l : ℕ
h : l < k
x : α
⊢ x ∈ ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l ↔
    x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i l
[PROOFSTEP]
rw [Set.mem_iUnion, exists_prop, and_iff_right_iff_imp]
[GOAL]
case h.e'_4.h.e'_7.h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h✝ : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ∀ (l : ℕ), l < k → 0 ≤ ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
l : ℕ
h : l < k
x : α
⊢ x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i l → l < k
[PROOFSTEP]
exact fun _ => h
[GOAL]
case neg
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h✝ : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ∀ (l : ℕ), l < k → 0 ≤ ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
l : ℕ
h : ¬l < k
⊢ 0 ≤ ↑s (⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
[PROOFSTEP]
convert le_of_eq s.empty.symm
[GOAL]
case h.e'_4.h.e'_7
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h✝ : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ∀ (l : ℕ), l < k → 0 ≤ ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
l : ℕ
h : ¬l < k
⊢ ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l = ∅
[PROOFSTEP]
ext
[GOAL]
case h.e'_4.h.e'_7.h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h✝ : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ∀ (l : ℕ), l < k → 0 ≤ ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
l : ℕ
h : ¬l < k
x✝ : α
⊢ x✝ ∈ ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l ↔ x✝ ∈ ∅
[PROOFSTEP]
simp only [exists_prop, Set.mem_empty_iff_false, Set.mem_iUnion, not_and, iff_false_iff]
[GOAL]
case h.e'_4.h.e'_7.h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h✝ : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ∀ (l : ℕ), l < k → 0 ≤ ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
l : ℕ
h : ¬l < k
x✝ : α
⊢ l < Nat.find hn → ¬x✝ ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i l
[PROOFSTEP]
exact fun h' => False.elim (h h')
[GOAL]
case neg.hf₁
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ ∀ (i_1 : ℕ), MeasurableSet (⋃ (_ : i_1 < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1)
[PROOFSTEP]
intro
[GOAL]
case neg.hf₁
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
i✝ : ℕ
⊢ MeasurableSet (⋃ (_ : i✝ < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i✝)
[PROOFSTEP]
exact MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
[GOAL]
case neg.hf₂
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ Pairwise (Disjoint on fun l => ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
[PROOFSTEP]
intro a b hab
[GOAL]
case neg.hf₂
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
a b : ℕ
hab : a ≠ b
⊢ (Disjoint on fun l => ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) a b
[PROOFSTEP]
refine' Set.disjoint_iUnion_left.mpr fun _ => _
[GOAL]
case neg.hf₂
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
a b : ℕ
hab : a ≠ b
x✝ : a < k
⊢ Disjoint (MeasureTheory.SignedMeasure.restrictNonposSeq s i a)
    ((fun l => ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) b)
[PROOFSTEP]
refine' Set.disjoint_iUnion_right.mpr fun _ => _
[GOAL]
case neg.hf₂
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
a b : ℕ
hab : a ≠ b
x✝¹ : a < k
x✝ : b < k
⊢ Disjoint (MeasureTheory.SignedMeasure.restrictNonposSeq s i a) (MeasureTheory.SignedMeasure.restrictNonposSeq s i b)
[PROOFSTEP]
exact restrictNonposSeq_disjoint hab
[GOAL]
case neg
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l ⊆ i
[PROOFSTEP]
apply Set.iUnion_subset
[GOAL]
case neg.h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ ∀ (i_1 : ℕ), ⋃ (_ : i_1 < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1 ⊆ i
[PROOFSTEP]
intro a x
[GOAL]
case neg.h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
a : ℕ
x : α
⊢ x ∈ ⋃ (_ : a < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i a → x ∈ i
[PROOFSTEP]
simp only [and_imp, exists_prop, Set.mem_iUnion]
[GOAL]
case neg.h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
a : ℕ
x : α
⊢ a < Nat.find hn → x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i a → x ∈ i
[PROOFSTEP]
intro _ hx
[GOAL]
case neg.h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
a : ℕ
x : α
a✝ : a < Nat.find hn
hx : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i a
⊢ x ∈ i
[PROOFSTEP]
exact restrictNonposSeq_subset _ hx
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
[PROOFSTEP]
have hi₁ : MeasurableSet i := by_contradiction fun h => ne_of_lt hi <| s.not_measurable h
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
[PROOFSTEP]
by_cases s ≤[i] 0
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
[PROOFSTEP]
by_cases s ≤[i] 0
[GOAL]
case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : restrict s i ≤ restrict 0 i
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
[PROOFSTEP]
exact ⟨i, hi₁, Set.Subset.refl _, h, hi⟩
[GOAL]
case neg
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
[PROOFSTEP]
by_cases hn : ∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, restrictNonposSeq s i l] 0
[GOAL]
case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
case neg
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ¬∀ (n : ℕ),
      ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
          restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
[PROOFSTEP]
swap
[GOAL]
case neg
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ¬∀ (n : ℕ),
      ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
          restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
[PROOFSTEP]
exact exists_subset_restrict_nonpos' hi₁ hi hn
[GOAL]
case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
[PROOFSTEP]
set A := i \ ⋃ l, restrictNonposSeq s i l with hA
[GOAL]
case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
[PROOFSTEP]
set bdd : ℕ → ℕ := fun n => findExistsOneDivLT s (i \ ⋃ k ≤ n, restrictNonposSeq s i k)
[GOAL]
case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
[PROOFSTEP]
have hn' : ∀ n : ℕ, ¬s ≤[i \ ⋃ l ≤ n, restrictNonposSeq s i l] 0 :=
  by
  intro n
  convert hn (n + 1) using 5 <;>
    · ext l
      simp only [exists_prop, Set.mem_iUnion, and_congr_left_iff]
      exact fun _ => Nat.lt_succ_iff.symm
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
[PROOFSTEP]
intro n
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
n : ℕ
⊢ ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
[PROOFSTEP]
convert hn (n + 1) using 5
[GOAL]
case h.e'_1.h.e'_3.h.e'_7.h.e'_4.h.e'_3
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
n : ℕ
⊢ (fun l => ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) = fun l =>
    ⋃ (_ : l < n + 1), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
[PROOFSTEP]
ext l
[GOAL]
case h.e'_1.h.e'_3.h.e'_7.h.e'_4.h.e'_3.h.h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
n l : ℕ
x✝ : α
⊢ x✝ ∈ ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l ↔
    x✝ ∈ ⋃ (_ : l < n + 1), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
[PROOFSTEP]
simp only [exists_prop, Set.mem_iUnion, and_congr_left_iff]
[GOAL]
case h.e'_1.h.e'_3.h.e'_7.h.e'_4.h.e'_3.h.h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
n l : ℕ
x✝ : α
⊢ x✝ ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i l → (l ≤ n ↔ l < n + 1)
[PROOFSTEP]
exact fun _ => Nat.lt_succ_iff.symm
[GOAL]
case h.e'_1.h.e'_4.h.e'_7.h.e'_4.h.e'_3
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
n : ℕ
⊢ (fun l => ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) = fun l =>
    ⋃ (_ : l < n + 1), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
[PROOFSTEP]
ext l
[GOAL]
case h.e'_1.h.e'_4.h.e'_7.h.e'_4.h.e'_3.h.h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
n l : ℕ
x✝ : α
⊢ x✝ ∈ ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l ↔
    x✝ ∈ ⋃ (_ : l < n + 1), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
[PROOFSTEP]
simp only [exists_prop, Set.mem_iUnion, and_congr_left_iff]
[GOAL]
case h.e'_1.h.e'_4.h.e'_7.h.e'_4.h.e'_3.h.h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
n l : ℕ
x✝ : α
⊢ x✝ ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i l → (l ≤ n ↔ l < n + 1)
[PROOFSTEP]
exact fun _ => Nat.lt_succ_iff.symm
[GOAL]
case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
[PROOFSTEP]
have h₁ : s i = s A + ∑' l, s (restrictNonposSeq s i l) :=
  by
  rw [hA, ← s.of_disjoint_iUnion_nat, add_comm, of_add_of_diff]
  exact MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
  exacts [hi₁, Set.iUnion_subset fun _ => restrictNonposSeq_subset _, fun _ => restrictNonposSeq_measurableSet _,
    restrictNonposSeq_disjoint]
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
[PROOFSTEP]
rw [hA, ← s.of_disjoint_iUnion_nat, add_comm, of_add_of_diff]
[GOAL]
case hA
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ MeasurableSet (⋃ (i_1 : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1)
case hB
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ MeasurableSet i
case h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ ⋃ (i_1 : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1 ⊆ i
case hf₁
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ ∀ (i_1 : ℕ), MeasurableSet (MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1)
case hf₂
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ Pairwise (Disjoint on fun l => MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
[PROOFSTEP]
exact MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
[GOAL]
case hB
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ MeasurableSet i
case h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ ⋃ (i_1 : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1 ⊆ i
case hf₁
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ ∀ (i_1 : ℕ), MeasurableSet (MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1)
case hf₂
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ Pairwise (Disjoint on fun l => MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
[PROOFSTEP]
exacts [hi₁, Set.iUnion_subset fun _ => restrictNonposSeq_subset _, fun _ => restrictNonposSeq_measurableSet _,
  restrictNonposSeq_disjoint]
[GOAL]
case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
[PROOFSTEP]
have h₂ : s A ≤ s i := by
  rw [h₁]
  apply le_add_of_nonneg_right
  exact tsum_nonneg fun n => le_of_lt (measure_of_restrictNonposSeq h _ (hn n))
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ ↑s A ≤ ↑s i
[PROOFSTEP]
rw [h₁]
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ ↑s A ≤ ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
[PROOFSTEP]
apply le_add_of_nonneg_right
[GOAL]
case h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ 0 ≤ ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
[PROOFSTEP]
exact tsum_nonneg fun n => le_of_lt (measure_of_restrictNonposSeq h _ (hn n))
[GOAL]
case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
[PROOFSTEP]
have h₃' : Summable fun n => (1 / (bdd n + 1) : ℝ) :=
  by
  have : Summable fun l => s (restrictNonposSeq s i l) :=
    HasSum.summable (s.m_iUnion (fun _ => restrictNonposSeq_measurableSet _) restrictNonposSeq_disjoint)
  refine' summable_of_nonneg_of_le (fun n => _) (fun n => _) (Summable.comp_injective this Nat.succ_injective)
  · exact le_of_lt Nat.one_div_pos_of_nat
  · exact le_of_lt (restrictNonposSeq_lt n (hn' n))
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
⊢ Summable fun n => 1 / (↑(bdd n) + 1)
[PROOFSTEP]
have : Summable fun l => s (restrictNonposSeq s i l) :=
  HasSum.summable (s.m_iUnion (fun _ => restrictNonposSeq_measurableSet _) restrictNonposSeq_disjoint)
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
this : Summable fun l => ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ Summable fun n => 1 / (↑(bdd n) + 1)
[PROOFSTEP]
refine' summable_of_nonneg_of_le (fun n => _) (fun n => _) (Summable.comp_injective this Nat.succ_injective)
[GOAL]
case refine'_1
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
this : Summable fun l => ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
n : ℕ
⊢ 0 ≤ 1 / (↑(bdd n) + 1)
[PROOFSTEP]
exact le_of_lt Nat.one_div_pos_of_nat
[GOAL]
case refine'_2
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
this : Summable fun l => ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
n : ℕ
⊢ 1 / (↑(bdd n) + 1) ≤ ((fun l => ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)) ∘ Nat.succ) n
[PROOFSTEP]
exact le_of_lt (restrictNonposSeq_lt n (hn' n))
[GOAL]
case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
[PROOFSTEP]
have h₃ : Tendsto (fun n => (bdd n : ℝ) + 1) atTop atTop :=
  by
  simp only [one_div] at h₃'
  exact Summable.tendsto_atTop_of_pos h₃' fun n => Nat.cast_add_one_pos (bdd n)
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
⊢ Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
[PROOFSTEP]
simp only [one_div] at h₃'
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' :
  Summable fun n =>
    (↑(MeasureTheory.SignedMeasure.findExistsOneDivLT s
            (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)) +
        1)⁻¹
⊢ Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
[PROOFSTEP]
exact Summable.tendsto_atTop_of_pos h₃' fun n => Nat.cast_add_one_pos (bdd n)
[GOAL]
case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
[PROOFSTEP]
have h₄ : Tendsto (fun n => (bdd n : ℝ)) atTop atTop := by convert atTop.tendsto_atTop_add_const_right (-1) h₃; simp
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
⊢ Tendsto (fun n => ↑(bdd n)) atTop atTop
[PROOFSTEP]
convert atTop.tendsto_atTop_add_const_right (-1) h₃
[GOAL]
case h.e'_3.h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
x✝ : ℕ
⊢ ↑(bdd x✝) = ↑(bdd x✝) + 1 + -1
[PROOFSTEP]
simp
[GOAL]
case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
[PROOFSTEP]
have A_meas : MeasurableSet A := hi₁.diff (MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _)
[GOAL]
case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop
A_meas : MeasurableSet A
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
[PROOFSTEP]
refine' ⟨A, A_meas, Set.diff_subset _ _, _, h₂.trans_lt hi⟩
[GOAL]
case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop
A_meas : MeasurableSet A
⊢ restrict s A ≤ restrict 0 A
[PROOFSTEP]
by_contra hnn
[GOAL]
case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop
A_meas : MeasurableSet A
hnn : ¬restrict s A ≤ restrict 0 A
⊢ False
[PROOFSTEP]
rw [restrict_le_restrict_iff _ _ A_meas] at hnn
[GOAL]
case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop
A_meas : MeasurableSet A
hnn : ¬∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ A → ↑s j ≤ ↑0 j
⊢ False
[PROOFSTEP]
push_neg at hnn
[GOAL]
case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop
A_meas : MeasurableSet A
hnn :
  Exists fun ⦃j⦄ =>
    MeasurableSet j ∧ j ⊆ i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l ∧ ↑0 j < ↑s j
⊢ False
[PROOFSTEP]
obtain ⟨E, hE₁, hE₂, hE₃⟩ := hnn
[GOAL]
case pos.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop
A_meas : MeasurableSet A
E : Set α
hE₁ : MeasurableSet E
hE₂ : E ⊆ i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hE₃ : ↑0 E < ↑s E
⊢ False
[PROOFSTEP]
have : ∃ k, 1 ≤ bdd k ∧ 1 / (bdd k : ℝ) < s E :=
  by
  rw [tendsto_atTop_atTop] at h₄
  obtain ⟨k, hk⟩ := h₄ (max (1 / s E + 1) 1)
  refine' ⟨k, _, _⟩
  · have hle := le_of_max_le_right (hk k le_rfl)
    norm_cast at hle
  · have : 1 / s E < bdd k := by linarith only [le_of_max_le_left (hk k le_rfl)]
    rw [one_div] at this ⊢
    rwa [inv_lt (lt_trans (inv_pos.2 hE₃) this) hE₃]
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop
A_meas : MeasurableSet A
E : Set α
hE₁ : MeasurableSet E
hE₂ : E ⊆ i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hE₃ : ↑0 E < ↑s E
⊢ ∃ k, 1 ≤ bdd k ∧ 1 / ↑(bdd k) < ↑s E
[PROOFSTEP]
rw [tendsto_atTop_atTop] at h₄
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : ∀ (b : ℝ), ∃ i, ∀ (a : ℕ), i ≤ a → b ≤ ↑(bdd a)
A_meas : MeasurableSet A
E : Set α
hE₁ : MeasurableSet E
hE₂ : E ⊆ i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hE₃ : ↑0 E < ↑s E
⊢ ∃ k, 1 ≤ bdd k ∧ 1 / ↑(bdd k) < ↑s E
[PROOFSTEP]
obtain ⟨k, hk⟩ := h₄ (max (1 / s E + 1) 1)
[GOAL]
case intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : ∀ (b : ℝ), ∃ i, ∀ (a : ℕ), i ≤ a → b ≤ ↑(bdd a)
A_meas : MeasurableSet A
E : Set α
hE₁ : MeasurableSet E
hE₂ : E ⊆ i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hE₃ : ↑0 E < ↑s E
k : ℕ
hk : ∀ (a : ℕ), k ≤ a → max (1 / ↑s E + 1) 1 ≤ ↑(bdd a)
⊢ ∃ k, 1 ≤ bdd k ∧ 1 / ↑(bdd k) < ↑s E
[PROOFSTEP]
refine' ⟨k, _, _⟩
[GOAL]
case intro.refine'_1
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : ∀ (b : ℝ), ∃ i, ∀ (a : ℕ), i ≤ a → b ≤ ↑(bdd a)
A_meas : MeasurableSet A
E : Set α
hE₁ : MeasurableSet E
hE₂ : E ⊆ i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hE₃ : ↑0 E < ↑s E
k : ℕ
hk : ∀ (a : ℕ), k ≤ a → max (1 / ↑s E + 1) 1 ≤ ↑(bdd a)
⊢ 1 ≤ bdd k
[PROOFSTEP]
have hle := le_of_max_le_right (hk k le_rfl)
[GOAL]
case intro.refine'_1
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : ∀ (b : ℝ), ∃ i, ∀ (a : ℕ), i ≤ a → b ≤ ↑(bdd a)
A_meas : MeasurableSet A
E : Set α
hE₁ : MeasurableSet E
hE₂ : E ⊆ i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hE₃ : ↑0 E < ↑s E
k : ℕ
hk : ∀ (a : ℕ), k ≤ a → max (1 / ↑s E + 1) 1 ≤ ↑(bdd a)
hle : 1 ≤ ↑(bdd k)
⊢ 1 ≤ bdd k
[PROOFSTEP]
norm_cast at hle
[GOAL]
case intro.refine'_2
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : ∀ (b : ℝ), ∃ i, ∀ (a : ℕ), i ≤ a → b ≤ ↑(bdd a)
A_meas : MeasurableSet A
E : Set α
hE₁ : MeasurableSet E
hE₂ : E ⊆ i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hE₃ : ↑0 E < ↑s E
k : ℕ
hk : ∀ (a : ℕ), k ≤ a → max (1 / ↑s E + 1) 1 ≤ ↑(bdd a)
⊢ 1 / ↑(bdd k) < ↑s E
[PROOFSTEP]
have : 1 / s E < bdd k := by linarith only [le_of_max_le_left (hk k le_rfl)]
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : ∀ (b : ℝ), ∃ i, ∀ (a : ℕ), i ≤ a → b ≤ ↑(bdd a)
A_meas : MeasurableSet A
E : Set α
hE₁ : MeasurableSet E
hE₂ : E ⊆ i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hE₃ : ↑0 E < ↑s E
k : ℕ
hk : ∀ (a : ℕ), k ≤ a → max (1 / ↑s E + 1) 1 ≤ ↑(bdd a)
⊢ 1 / ↑s E < ↑(bdd k)
[PROOFSTEP]
linarith only [le_of_max_le_left (hk k le_rfl)]
[GOAL]
case intro.refine'_2
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : ∀ (b : ℝ), ∃ i, ∀ (a : ℕ), i ≤ a → b ≤ ↑(bdd a)
A_meas : MeasurableSet A
E : Set α
hE₁ : MeasurableSet E
hE₂ : E ⊆ i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hE₃ : ↑0 E < ↑s E
k : ℕ
hk : ∀ (a : ℕ), k ≤ a → max (1 / ↑s E + 1) 1 ≤ ↑(bdd a)
this : 1 / ↑s E < ↑(bdd k)
⊢ 1 / ↑(bdd k) < ↑s E
[PROOFSTEP]
rw [one_div] at this ⊢
[GOAL]
case intro.refine'_2
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : ∀ (b : ℝ), ∃ i, ∀ (a : ℕ), i ≤ a → b ≤ ↑(bdd a)
A_meas : MeasurableSet A
E : Set α
hE₁ : MeasurableSet E
hE₂ : E ⊆ i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hE₃ : ↑0 E < ↑s E
k : ℕ
hk : ∀ (a : ℕ), k ≤ a → max (1 / ↑s E + 1) 1 ≤ ↑(bdd a)
this : (↑s E)⁻¹ < ↑(bdd k)
⊢ (↑(bdd k))⁻¹ < ↑s E
[PROOFSTEP]
rwa [inv_lt (lt_trans (inv_pos.2 hE₃) this) hE₃]
[GOAL]
case pos.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop
A_meas : MeasurableSet A
E : Set α
hE₁ : MeasurableSet E
hE₂ : E ⊆ i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hE₃ : ↑0 E < ↑s E
this : ∃ k, 1 ≤ bdd k ∧ 1 / ↑(bdd k) < ↑s E
⊢ False
[PROOFSTEP]
obtain ⟨k, hk₁, hk₂⟩ := this
[GOAL]
case pos.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop
A_meas : MeasurableSet A
E : Set α
hE₁ : MeasurableSet E
hE₂ : E ⊆ i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hE₃ : ↑0 E < ↑s E
k : ℕ
hk₁ : 1 ≤ bdd k
hk₂ : 1 / ↑(bdd k) < ↑s E
⊢ False
[PROOFSTEP]
have hA' : A ⊆ i \ ⋃ l ≤ k, restrictNonposSeq s i l :=
  by
  apply Set.diff_subset_diff_right
  intro x; simp only [Set.mem_iUnion]
  rintro ⟨n, _, hn₂⟩
  exact ⟨n, hn₂⟩
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop
A_meas : MeasurableSet A
E : Set α
hE₁ : MeasurableSet E
hE₂ : E ⊆ i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hE₃ : ↑0 E < ↑s E
k : ℕ
hk₁ : 1 ≤ bdd k
hk₂ : 1 / ↑(bdd k) < ↑s E
⊢ A ⊆ i \ ⋃ (l : ℕ) (_ : l ≤ k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
[PROOFSTEP]
apply Set.diff_subset_diff_right
[GOAL]
case h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop
A_meas : MeasurableSet A
E : Set α
hE₁ : MeasurableSet E
hE₂ : E ⊆ i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hE₃ : ↑0 E < ↑s E
k : ℕ
hk₁ : 1 ≤ bdd k
hk₂ : 1 / ↑(bdd k) < ↑s E
⊢ ⋃ (l : ℕ) (_ : l ≤ k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l ⊆
    ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
[PROOFSTEP]
intro x
[GOAL]
case h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop
A_meas : MeasurableSet A
E : Set α
hE₁ : MeasurableSet E
hE₂ : E ⊆ i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hE₃ : ↑0 E < ↑s E
k : ℕ
hk₁ : 1 ≤ bdd k
hk₂ : 1 / ↑(bdd k) < ↑s E
x : α
⊢ x ∈ ⋃ (l : ℕ) (_ : l ≤ k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l →
    x ∈ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
[PROOFSTEP]
simp only [Set.mem_iUnion]
[GOAL]
case h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop
A_meas : MeasurableSet A
E : Set α
hE₁ : MeasurableSet E
hE₂ : E ⊆ i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hE₃ : ↑0 E < ↑s E
k : ℕ
hk₁ : 1 ≤ bdd k
hk₂ : 1 / ↑(bdd k) < ↑s E
x : α
⊢ (∃ i_1 i_2, x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1) →
    ∃ i_1, x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1
[PROOFSTEP]
rintro ⟨n, _, hn₂⟩
[GOAL]
case h.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop
A_meas : MeasurableSet A
E : Set α
hE₁ : MeasurableSet E
hE₂ : E ⊆ i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hE₃ : ↑0 E < ↑s E
k : ℕ
hk₁ : 1 ≤ bdd k
hk₂ : 1 / ↑(bdd k) < ↑s E
x : α
n : ℕ
w✝ : n ≤ k
hn₂ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i n
⊢ ∃ i_1, x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1
[PROOFSTEP]
exact ⟨n, hn₂⟩
[GOAL]
case pos.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop
A_meas : MeasurableSet A
E : Set α
hE₁ : MeasurableSet E
hE₂ : E ⊆ i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hE₃ : ↑0 E < ↑s E
k : ℕ
hk₁ : 1 ≤ bdd k
hk₂ : 1 / ↑(bdd k) < ↑s E
hA' : A ⊆ i \ ⋃ (l : ℕ) (_ : l ≤ k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
⊢ False
[PROOFSTEP]
refine' findExistsOneDivLT_min (hn' k) (Nat.sub_lt hk₁ Nat.zero_lt_one) ⟨E, Set.Subset.trans hE₂ hA', hE₁, _⟩
[GOAL]
case pos.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop
A_meas : MeasurableSet A
E : Set α
hE₁ : MeasurableSet E
hE₂ : E ⊆ i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hE₃ : ↑0 E < ↑s E
k : ℕ
hk₁ : 1 ≤ bdd k
hk₂ : 1 / ↑(bdd k) < ↑s E
hA' : A ⊆ i \ ⋃ (l : ℕ) (_ : l ≤ k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
⊢ 1 /
      (↑(MeasureTheory.SignedMeasure.findExistsOneDivLT s
              (i \ ⋃ (l : ℕ) (_ : l ≤ k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) -
            1) +
        1) <
    ↑s E
[PROOFSTEP]
convert hk₂
[GOAL]
case h.e'_3.h.e'_6
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop
A_meas : MeasurableSet A
E : Set α
hE₁ : MeasurableSet E
hE₂ : E ⊆ i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hE₃ : ↑0 E < ↑s E
k : ℕ
hk₁ : 1 ≤ bdd k
hk₂ : 1 / ↑(bdd k) < ↑s E
hA' : A ⊆ i \ ⋃ (l : ℕ) (_ : l ≤ k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
⊢ ↑(MeasureTheory.SignedMeasure.findExistsOneDivLT s
            (i \ ⋃ (l : ℕ) (_ : l ≤ k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) -
          1) +
      1 =
    ↑(bdd k)
[PROOFSTEP]
norm_cast
[GOAL]
case h.e'_3.h.e'_6
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ (k : ℕ) (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ (l : ℕ) (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop
A_meas : MeasurableSet A
E : Set α
hE₁ : MeasurableSet E
hE₂ : E ⊆ i \ ⋃ (l : ℕ), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hE₃ : ↑0 E < ↑s E
k : ℕ
hk₁ : 1 ≤ bdd k
hk₂ : 1 / ↑(bdd k) < ↑s E
hA' : A ⊆ i \ ⋃ (l : ℕ) (_ : l ≤ k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
⊢ MeasureTheory.SignedMeasure.findExistsOneDivLT s
          (i \ ⋃ (l : ℕ) (_ : l ≤ k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) -
        1 +
      1 =
    bdd k
[PROOFSTEP]
exact tsub_add_cancel_of_le hk₁
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
⊢ BddBelow (measureOfNegatives s)
[PROOFSTEP]
simp_rw [BddBelow, Set.Nonempty, mem_lowerBounds]
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
⊢ ∃ x, ∀ (x_1 : ℝ), x_1 ∈ measureOfNegatives s → x ≤ x_1
[PROOFSTEP]
by_contra' h
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x
⊢ False
[PROOFSTEP]
have h' : ∀ n : ℕ, ∃ y : ℝ, y ∈ s.measureOfNegatives ∧ y < -n := fun n => h (-n)
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x
h' : ∀ (n : ℕ), ∃ y, y ∈ measureOfNegatives s ∧ y < -↑n
⊢ False
[PROOFSTEP]
choose f hf using h'
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n
⊢ False
[PROOFSTEP]
have hf' : ∀ n : ℕ, ∃ B, MeasurableSet B ∧ s ≤[B] 0 ∧ s B < -n :=
  by
  intro n
  rcases hf n with ⟨⟨B, ⟨hB₁, hBr⟩, hB₂⟩, hlt⟩
  exact ⟨B, hB₁, hBr, hB₂.symm ▸ hlt⟩
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n
⊢ ∀ (n : ℕ), ∃ B, MeasurableSet B ∧ restrict s B ≤ restrict 0 B ∧ ↑s B < -↑n
[PROOFSTEP]
intro n
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n
n : ℕ
⊢ ∃ B, MeasurableSet B ∧ restrict s B ≤ restrict 0 B ∧ ↑s B < -↑n
[PROOFSTEP]
rcases hf n with ⟨⟨B, ⟨hB₁, hBr⟩, hB₂⟩, hlt⟩
[GOAL]
case intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n
n : ℕ
hlt : f n < -↑n
B : Set α
hB₂ : ↑s B = f n
hB₁ : MeasurableSet B
hBr : restrict s B ≤ restrict 0 B
⊢ ∃ B, MeasurableSet B ∧ restrict s B ≤ restrict 0 B ∧ ↑s B < -↑n
[PROOFSTEP]
exact ⟨B, hB₁, hBr, hB₂.symm ▸ hlt⟩
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n
hf' : ∀ (n : ℕ), ∃ B, MeasurableSet B ∧ restrict s B ≤ restrict 0 B ∧ ↑s B < -↑n
⊢ False
[PROOFSTEP]
choose B hmeas hr h_lt using hf'
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n
B : ℕ → Set α
hmeas : ∀ (n : ℕ), MeasurableSet (B n)
hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n
⊢ False
[PROOFSTEP]
set A := ⋃ n, B n with hA
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n
B : ℕ → Set α
hmeas : ∀ (n : ℕ), MeasurableSet (B n)
hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
⊢ False
[PROOFSTEP]
have hfalse : ∀ n : ℕ, s A ≤ -n := by
  intro n
  refine' le_trans _ (le_of_lt (h_lt _))
  rw [hA, ← Set.diff_union_of_subset (Set.subset_iUnion _ n), of_union Set.disjoint_sdiff_left _ (hmeas n)]
  · refine' add_le_of_nonpos_left _
    have : s ≤[A] 0 := restrict_le_restrict_iUnion _ _ hmeas hr
    refine' nonpos_of_restrict_le_zero _ (restrict_le_zero_subset _ _ (Set.diff_subset _ _) this)
    exact MeasurableSet.iUnion hmeas
  · exact (MeasurableSet.iUnion hmeas).diff (hmeas n)
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n
B : ℕ → Set α
hmeas : ∀ (n : ℕ), MeasurableSet (B n)
hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
⊢ ∀ (n : ℕ), ↑s A ≤ -↑n
[PROOFSTEP]
intro n
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n
B : ℕ → Set α
hmeas : ∀ (n : ℕ), MeasurableSet (B n)
hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
n : ℕ
⊢ ↑s A ≤ -↑n
[PROOFSTEP]
refine' le_trans _ (le_of_lt (h_lt _))
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n
B : ℕ → Set α
hmeas : ∀ (n : ℕ), MeasurableSet (B n)
hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
n : ℕ
⊢ ↑s A ≤ ↑s (B n)
[PROOFSTEP]
rw [hA, ← Set.diff_union_of_subset (Set.subset_iUnion _ n), of_union Set.disjoint_sdiff_left _ (hmeas n)]
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n
B : ℕ → Set α
hmeas : ∀ (n : ℕ), MeasurableSet (B n)
hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
n : ℕ
⊢ ↑s ((⋃ (i : ℕ), B i) \ B n) + ↑s (B n) ≤ ↑s (B n)
[PROOFSTEP]
refine' add_le_of_nonpos_left _
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n
B : ℕ → Set α
hmeas : ∀ (n : ℕ), MeasurableSet (B n)
hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
n : ℕ
⊢ ↑s ((⋃ (i : ℕ), B i) \ B n) ≤ 0
[PROOFSTEP]
have : s ≤[A] 0 := restrict_le_restrict_iUnion _ _ hmeas hr
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n
B : ℕ → Set α
hmeas : ∀ (n : ℕ), MeasurableSet (B n)
hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
n : ℕ
this : restrict s A ≤ restrict 0 A
⊢ ↑s ((⋃ (i : ℕ), B i) \ B n) ≤ 0
[PROOFSTEP]
refine' nonpos_of_restrict_le_zero _ (restrict_le_zero_subset _ _ (Set.diff_subset _ _) this)
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n
B : ℕ → Set α
hmeas : ∀ (n : ℕ), MeasurableSet (B n)
hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
n : ℕ
this : restrict s A ≤ restrict 0 A
⊢ MeasurableSet (⋃ (i : ℕ), B i)
[PROOFSTEP]
exact MeasurableSet.iUnion hmeas
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n
B : ℕ → Set α
hmeas : ∀ (n : ℕ), MeasurableSet (B n)
hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
n : ℕ
⊢ MeasurableSet ((⋃ (i : ℕ), B i) \ B n)
[PROOFSTEP]
exact (MeasurableSet.iUnion hmeas).diff (hmeas n)
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n
B : ℕ → Set α
hmeas : ∀ (n : ℕ), MeasurableSet (B n)
hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
hfalse : ∀ (n : ℕ), ↑s A ≤ -↑n
⊢ False
[PROOFSTEP]
rcases exists_nat_gt (-s A) with ⟨n, hn⟩
[GOAL]
case intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n
B : ℕ → Set α
hmeas : ∀ (n : ℕ), MeasurableSet (B n)
hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
hfalse : ∀ (n : ℕ), ↑s A ≤ -↑n
n : ℕ
hn : -↑s A < ↑n
⊢ False
[PROOFSTEP]
exact lt_irrefl _ ((neg_lt.1 hn).trans_le (hfalse n))
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i j : Set α
s : SignedMeasure α
⊢ ∃ i, MeasurableSet i ∧ restrict 0 i ≤ restrict s i ∧ restrict s iᶜ ≤ restrict 0 iᶜ
[PROOFSTEP]
obtain ⟨f, _, hf₂, hf₁⟩ := exists_seq_tendsto_sInf ⟨0, @zero_mem_measureOfNegatives _ _ s⟩ bddBelow_measureOfNegatives
[GOAL]
case intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i j : Set α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
hf₁ : ∀ (n : ℕ), f n ∈ measureOfNegatives s
⊢ ∃ i, MeasurableSet i ∧ restrict 0 i ≤ restrict s i ∧ restrict s iᶜ ≤ restrict 0 iᶜ
[PROOFSTEP]
choose B hB using hf₁
[GOAL]
case intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i j : Set α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
⊢ ∃ i, MeasurableSet i ∧ restrict 0 i ≤ restrict s i ∧ restrict s iᶜ ≤ restrict 0 iᶜ
[PROOFSTEP]
have hB₁ : ∀ n, MeasurableSet (B n) := fun n => (hB n).1.1
[GOAL]
case intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i j : Set α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
⊢ ∃ i, MeasurableSet i ∧ restrict 0 i ≤ restrict s i ∧ restrict s iᶜ ≤ restrict 0 iᶜ
[PROOFSTEP]
have hB₂ : ∀ n, s ≤[B n] 0 := fun n => (hB n).1.2
[GOAL]
case intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i j : Set α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
⊢ ∃ i, MeasurableSet i ∧ restrict 0 i ≤ restrict s i ∧ restrict s iᶜ ≤ restrict 0 iᶜ
[PROOFSTEP]
set A := ⋃ n, B n with hA
[GOAL]
case intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i j : Set α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
⊢ ∃ i, MeasurableSet i ∧ restrict 0 i ≤ restrict s i ∧ restrict s iᶜ ≤ restrict 0 iᶜ
[PROOFSTEP]
have hA₁ : MeasurableSet A := MeasurableSet.iUnion hB₁
[GOAL]
case intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i j : Set α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
hA₁ : MeasurableSet A
⊢ ∃ i, MeasurableSet i ∧ restrict 0 i ≤ restrict s i ∧ restrict s iᶜ ≤ restrict 0 iᶜ
[PROOFSTEP]
have hA₂ : s ≤[A] 0 := restrict_le_restrict_iUnion _ _ hB₁ hB₂
[GOAL]
case intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i j : Set α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
hA₁ : MeasurableSet A
hA₂ : restrict s A ≤ restrict 0 A
⊢ ∃ i, MeasurableSet i ∧ restrict 0 i ≤ restrict s i ∧ restrict s iᶜ ≤ restrict 0 iᶜ
[PROOFSTEP]
have hA₃ : s A = sInf s.measureOfNegatives := by
  apply le_antisymm
  · refine' le_of_tendsto_of_tendsto tendsto_const_nhds hf₂ (eventually_of_forall fun n => _)
    rw [← (hB n).2, hA, ← Set.diff_union_of_subset (Set.subset_iUnion _ n), of_union Set.disjoint_sdiff_left _ (hB₁ n)]
    · refine' add_le_of_nonpos_left _
      have : s ≤[A] 0 :=
        restrict_le_restrict_iUnion _ _ hB₁ fun m =>
          let ⟨_, h⟩ := (hB m).1
          h
      refine' nonpos_of_restrict_le_zero _ (restrict_le_zero_subset _ _ (Set.diff_subset _ _) this)
      exact MeasurableSet.iUnion hB₁
    · exact (MeasurableSet.iUnion hB₁).diff (hB₁ n)
  · exact csInf_le bddBelow_measureOfNegatives ⟨A, ⟨hA₁, hA₂⟩, rfl⟩
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i j : Set α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
hA₁ : MeasurableSet A
hA₂ : restrict s A ≤ restrict 0 A
⊢ ↑s A = sInf (measureOfNegatives s)
[PROOFSTEP]
apply le_antisymm
[GOAL]
case a
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i j : Set α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
hA₁ : MeasurableSet A
hA₂ : restrict s A ≤ restrict 0 A
⊢ ↑s A ≤ sInf (measureOfNegatives s)
[PROOFSTEP]
refine' le_of_tendsto_of_tendsto tendsto_const_nhds hf₂ (eventually_of_forall fun n => _)
[GOAL]
case a
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i j : Set α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
hA₁ : MeasurableSet A
hA₂ : restrict s A ≤ restrict 0 A
n : ℕ
⊢ (fun x => ↑s A) n ≤ f n
[PROOFSTEP]
rw [← (hB n).2, hA, ← Set.diff_union_of_subset (Set.subset_iUnion _ n), of_union Set.disjoint_sdiff_left _ (hB₁ n)]
[GOAL]
case a
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i j : Set α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
hA₁ : MeasurableSet A
hA₂ : restrict s A ≤ restrict 0 A
n : ℕ
⊢ (fun x => ↑s ((⋃ (i : ℕ), B i) \ B n) + ↑s (B n)) n ≤ ↑s (B n)
[PROOFSTEP]
refine' add_le_of_nonpos_left _
[GOAL]
case a
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i j : Set α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
hA₁ : MeasurableSet A
hA₂ : restrict s A ≤ restrict 0 A
n : ℕ
⊢ ↑s ((⋃ (i : ℕ), B i) \ B n) ≤ 0
[PROOFSTEP]
have : s ≤[A] 0 :=
  restrict_le_restrict_iUnion _ _ hB₁ fun m =>
    let ⟨_, h⟩ := (hB m).1
    h
[GOAL]
case a
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i j : Set α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
hA₁ : MeasurableSet A
hA₂ : restrict s A ≤ restrict 0 A
n : ℕ
this : restrict s A ≤ restrict 0 A
⊢ ↑s ((⋃ (i : ℕ), B i) \ B n) ≤ 0
[PROOFSTEP]
refine' nonpos_of_restrict_le_zero _ (restrict_le_zero_subset _ _ (Set.diff_subset _ _) this)
[GOAL]
case a
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i j : Set α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
hA₁ : MeasurableSet A
hA₂ : restrict s A ≤ restrict 0 A
n : ℕ
this : restrict s A ≤ restrict 0 A
⊢ MeasurableSet (⋃ (i : ℕ), B i)
[PROOFSTEP]
exact MeasurableSet.iUnion hB₁
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i j : Set α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
hA₁ : MeasurableSet A
hA₂ : restrict s A ≤ restrict 0 A
n : ℕ
⊢ MeasurableSet ((⋃ (i : ℕ), B i) \ B n)
[PROOFSTEP]
exact (MeasurableSet.iUnion hB₁).diff (hB₁ n)
[GOAL]
case a
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i j : Set α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
hA₁ : MeasurableSet A
hA₂ : restrict s A ≤ restrict 0 A
⊢ sInf (measureOfNegatives s) ≤ ↑s A
[PROOFSTEP]
exact csInf_le bddBelow_measureOfNegatives ⟨A, ⟨hA₁, hA₂⟩, rfl⟩
[GOAL]
case intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i j : Set α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
hA₁ : MeasurableSet A
hA₂ : restrict s A ≤ restrict 0 A
hA₃ : ↑s A = sInf (measureOfNegatives s)
⊢ ∃ i, MeasurableSet i ∧ restrict 0 i ≤ restrict s i ∧ restrict s iᶜ ≤ restrict 0 iᶜ
[PROOFSTEP]
refine' ⟨Aᶜ, hA₁.compl, _, (compl_compl A).symm ▸ hA₂⟩
[GOAL]
case intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i j : Set α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
hA₁ : MeasurableSet A
hA₂ : restrict s A ≤ restrict 0 A
hA₃ : ↑s A = sInf (measureOfNegatives s)
⊢ restrict 0 Aᶜ ≤ restrict s Aᶜ
[PROOFSTEP]
rw [restrict_le_restrict_iff _ _ hA₁.compl]
[GOAL]
case intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i j : Set α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
hA₁ : MeasurableSet A
hA₂ : restrict s A ≤ restrict 0 A
hA₃ : ↑s A = sInf (measureOfNegatives s)
⊢ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ Aᶜ → ↑0 j ≤ ↑s j
[PROOFSTEP]
intro C _ hC₁
[GOAL]
case intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i j : Set α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
hA₁ : MeasurableSet A
hA₂ : restrict s A ≤ restrict 0 A
hA₃ : ↑s A = sInf (measureOfNegatives s)
C : Set α
a✝ : MeasurableSet C
hC₁ : C ⊆ Aᶜ
⊢ ↑0 C ≤ ↑s C
[PROOFSTEP]
by_contra' hC₂
[GOAL]
case intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i j : Set α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
hA₁ : MeasurableSet A
hA₂ : restrict s A ≤ restrict 0 A
hA₃ : ↑s A = sInf (measureOfNegatives s)
C : Set α
a✝ : MeasurableSet C
hC₁ : C ⊆ Aᶜ
hC₂ : ↑s C < ↑0 C
⊢ False
[PROOFSTEP]
rcases exists_subset_restrict_nonpos hC₂ with ⟨D, hD₁, hD, hD₂, hD₃⟩
[GOAL]
case intro.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i j : Set α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
hA₁ : MeasurableSet A
hA₂ : restrict s A ≤ restrict 0 A
hA₃ : ↑s A = sInf (measureOfNegatives s)
C : Set α
a✝ : MeasurableSet C
hC₁ : C ⊆ Aᶜ
hC₂ : ↑s C < ↑0 C
D : Set α
hD₁ : MeasurableSet D
hD : D ⊆ C
hD₂ : restrict s D ≤ restrict 0 D
hD₃ : ↑s D < 0
⊢ False
[PROOFSTEP]
have : s (A ∪ D) < sInf s.measureOfNegatives :=
  by
  rw [← hA₃, of_union (Set.disjoint_of_subset_right (Set.Subset.trans hD hC₁) disjoint_compl_right) hA₁ hD₁]
  linarith
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i j : Set α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
hA₁ : MeasurableSet A
hA₂ : restrict s A ≤ restrict 0 A
hA₃ : ↑s A = sInf (measureOfNegatives s)
C : Set α
a✝ : MeasurableSet C
hC₁ : C ⊆ Aᶜ
hC₂ : ↑s C < ↑0 C
D : Set α
hD₁ : MeasurableSet D
hD : D ⊆ C
hD₂ : restrict s D ≤ restrict 0 D
hD₃ : ↑s D < 0
⊢ ↑s (A ∪ D) < sInf (measureOfNegatives s)
[PROOFSTEP]
rw [← hA₃, of_union (Set.disjoint_of_subset_right (Set.Subset.trans hD hC₁) disjoint_compl_right) hA₁ hD₁]
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i j : Set α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
hA₁ : MeasurableSet A
hA₂ : restrict s A ≤ restrict 0 A
hA₃ : ↑s A = sInf (measureOfNegatives s)
C : Set α
a✝ : MeasurableSet C
hC₁ : C ⊆ Aᶜ
hC₂ : ↑s C < ↑0 C
D : Set α
hD₁ : MeasurableSet D
hD : D ⊆ C
hD₂ : restrict s D ≤ restrict 0 D
hD₃ : ↑s D < 0
⊢ ↑s A + ↑s D < ↑s A
[PROOFSTEP]
linarith
[GOAL]
case intro.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i j : Set α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
hA₁ : MeasurableSet A
hA₂ : restrict s A ≤ restrict 0 A
hA₃ : ↑s A = sInf (measureOfNegatives s)
C : Set α
a✝ : MeasurableSet C
hC₁ : C ⊆ Aᶜ
hC₂ : ↑s C < ↑0 C
D : Set α
hD₁ : MeasurableSet D
hD : D ⊆ C
hD₂ : restrict s D ≤ restrict 0 D
hD₃ : ↑s D < 0
this : ↑s (A ∪ D) < sInf (measureOfNegatives s)
⊢ False
[PROOFSTEP]
refine' not_le.2 this _
[GOAL]
case intro.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i j : Set α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
hA₁ : MeasurableSet A
hA₂ : restrict s A ≤ restrict 0 A
hA₃ : ↑s A = sInf (measureOfNegatives s)
C : Set α
a✝ : MeasurableSet C
hC₁ : C ⊆ Aᶜ
hC₂ : ↑s C < ↑0 C
D : Set α
hD₁ : MeasurableSet D
hD : D ⊆ C
hD₂ : restrict s D ≤ restrict 0 D
hD₃ : ↑s D < 0
this : ↑s (A ∪ D) < sInf (measureOfNegatives s)
⊢ sInf (measureOfNegatives s) ≤ ↑s (A ∪ D)
[PROOFSTEP]
refine' csInf_le bddBelow_measureOfNegatives ⟨A ∪ D, ⟨_, _⟩, rfl⟩
[GOAL]
case intro.intro.intro.intro.intro.intro.intro.refine'_1
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i j : Set α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
hA₁ : MeasurableSet A
hA₂ : restrict s A ≤ restrict 0 A
hA₃ : ↑s A = sInf (measureOfNegatives s)
C : Set α
a✝ : MeasurableSet C
hC₁ : C ⊆ Aᶜ
hC₂ : ↑s C < ↑0 C
D : Set α
hD₁ : MeasurableSet D
hD : D ⊆ C
hD₂ : restrict s D ≤ restrict 0 D
hD₃ : ↑s D < 0
this : ↑s (A ∪ D) < sInf (measureOfNegatives s)
⊢ MeasurableSet (A ∪ D)
[PROOFSTEP]
exact hA₁.union hD₁
[GOAL]
case intro.intro.intro.intro.intro.intro.intro.refine'_2
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i j : Set α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ (n : ℕ), B n
hA : A = ⋃ (n : ℕ), B n
hA₁ : MeasurableSet A
hA₂ : restrict s A ≤ restrict 0 A
hA₃ : ↑s A = sInf (measureOfNegatives s)
C : Set α
a✝ : MeasurableSet C
hC₁ : C ⊆ Aᶜ
hC₂ : ↑s C < ↑0 C
D : Set α
hD₁ : MeasurableSet D
hD : D ⊆ C
hD₂ : restrict s D ≤ restrict 0 D
hD₃ : ↑s D < 0
this : ↑s (A ∪ D) < sInf (measureOfNegatives s)
⊢ restrict s (A ∪ D) ≤ restrict 0 (A ∪ D)
[PROOFSTEP]
exact restrict_le_restrict_union _ _ hA₁ hA₂ hD₁ hD₂
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : restrict 0 i ≤ restrict s i ∧ restrict s iᶜ ≤ restrict 0 iᶜ
hj' : restrict 0 j ≤ restrict s j ∧ restrict s jᶜ ≤ restrict 0 jᶜ
⊢ ↑s (i ∆ j) = 0 ∧ ↑s (iᶜ ∆ jᶜ) = 0
[PROOFSTEP]
rw [restrict_le_restrict_iff s 0, restrict_le_restrict_iff 0 s] at hi' hj'
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' :
  (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → ↑0 j_1 ≤ ↑s j_1) ∧
    ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1
⊢ ↑s (i ∆ j) = 0 ∧ ↑s (iᶜ ∆ jᶜ) = 0
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' : restrict 0 j ≤ restrict s j ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1
⊢ MeasurableSet j
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : restrict 0 i ≤ restrict s i ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' : restrict 0 j ≤ restrict s j ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1
⊢ MeasurableSet i
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : restrict 0 i ≤ restrict s i ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' : restrict 0 j ≤ restrict s j ∧ restrict s jᶜ ≤ restrict 0 jᶜ
⊢ MeasurableSet jᶜ
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : restrict 0 i ≤ restrict s i ∧ restrict s iᶜ ≤ restrict 0 iᶜ
hj' : restrict 0 j ≤ restrict s j ∧ restrict s jᶜ ≤ restrict 0 jᶜ
⊢ MeasurableSet iᶜ
[PROOFSTEP]
constructor
[GOAL]
case left
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' :
  (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → ↑0 j_1 ≤ ↑s j_1) ∧
    ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1
⊢ ↑s (i ∆ j) = 0
[PROOFSTEP]
rw [Set.symmDiff_def, Set.diff_eq_compl_inter, Set.diff_eq_compl_inter, of_union,
  le_antisymm (hi'.2 (hi.compl.inter hj) (Set.inter_subset_left _ _))
    (hj'.1 (hi.compl.inter hj) (Set.inter_subset_right _ _)),
  le_antisymm (hj'.2 (hj.compl.inter hi) (Set.inter_subset_left _ _))
    (hi'.1 (hj.compl.inter hi) (Set.inter_subset_right _ _)),
  zero_apply, zero_apply, zero_add]
[GOAL]
case left.h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' :
  (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → ↑0 j_1 ≤ ↑s j_1) ∧
    ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1
⊢ Disjoint (jᶜ ∩ i) (iᶜ ∩ j)
[PROOFSTEP]
exact
  Set.disjoint_of_subset_left (Set.inter_subset_left _ _)
    (Set.disjoint_of_subset_right (Set.inter_subset_right _ _) (disjoint_comm.1 (IsCompl.disjoint isCompl_compl)))
[GOAL]
case left.hA
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' :
  (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → ↑0 j_1 ≤ ↑s j_1) ∧
    ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1
⊢ MeasurableSet (jᶜ ∩ i)
[PROOFSTEP]
exact hj.compl.inter hi
[GOAL]
case left.hB
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' :
  (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → ↑0 j_1 ≤ ↑s j_1) ∧
    ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1
⊢ MeasurableSet (iᶜ ∩ j)
[PROOFSTEP]
exact hi.compl.inter hj
[GOAL]
case right
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' :
  (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → ↑0 j_1 ≤ ↑s j_1) ∧
    ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1
⊢ ↑s (iᶜ ∆ jᶜ) = 0
[PROOFSTEP]
rw [Set.symmDiff_def, Set.diff_eq_compl_inter, Set.diff_eq_compl_inter, compl_compl, compl_compl, of_union,
  le_antisymm (hi'.2 (hj.inter hi.compl) (Set.inter_subset_right _ _))
    (hj'.1 (hj.inter hi.compl) (Set.inter_subset_left _ _)),
  le_antisymm (hj'.2 (hi.inter hj.compl) (Set.inter_subset_right _ _))
    (hi'.1 (hi.inter hj.compl) (Set.inter_subset_left _ _)),
  zero_apply, zero_apply, zero_add]
[GOAL]
case right.h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' :
  (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → ↑0 j_1 ≤ ↑s j_1) ∧
    ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1
⊢ Disjoint (j ∩ iᶜ) (i ∩ jᶜ)
[PROOFSTEP]
exact
  Set.disjoint_of_subset_left (Set.inter_subset_left _ _)
    (Set.disjoint_of_subset_right (Set.inter_subset_right _ _) (IsCompl.disjoint isCompl_compl))
[GOAL]
case right.hA
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' :
  (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → ↑0 j_1 ≤ ↑s j_1) ∧
    ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1
⊢ MeasurableSet (j ∩ iᶜ)
[PROOFSTEP]
exact hj.inter hi.compl
[GOAL]
case right.hB
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' :
  (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → ↑0 j_1 ≤ ↑s j_1) ∧
    ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1
⊢ MeasurableSet (i ∩ jᶜ)
[PROOFSTEP]
exact hi.inter hj.compl
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' : restrict 0 j ≤ restrict s j ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1
⊢ MeasurableSet j
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : restrict 0 i ≤ restrict s i ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' : restrict 0 j ≤ restrict s j ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1
⊢ MeasurableSet i
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : restrict 0 i ≤ restrict s i ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' : restrict 0 j ≤ restrict s j ∧ restrict s jᶜ ≤ restrict 0 jᶜ
⊢ MeasurableSet jᶜ
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : restrict 0 i ≤ restrict s i ∧ restrict s iᶜ ≤ restrict 0 iᶜ
hj' : restrict 0 j ≤ restrict s j ∧ restrict s jᶜ ≤ restrict 0 jᶜ
⊢ MeasurableSet iᶜ
[PROOFSTEP]
all_goals measurability
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' : restrict 0 j ≤ restrict s j ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1
⊢ MeasurableSet j
[PROOFSTEP]
measurability
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : restrict 0 i ≤ restrict s i ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' : restrict 0 j ≤ restrict s j ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1
⊢ MeasurableSet i
[PROOFSTEP]
measurability
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : restrict 0 i ≤ restrict s i ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' : restrict 0 j ≤ restrict s j ∧ restrict s jᶜ ≤ restrict 0 jᶜ
⊢ MeasurableSet jᶜ
[PROOFSTEP]
measurability
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : restrict 0 i ≤ restrict s i ∧ restrict s iᶜ ≤ restrict 0 iᶜ
hj' : restrict 0 j ≤ restrict s j ∧ restrict s jᶜ ≤ restrict 0 jᶜ
⊢ MeasurableSet iᶜ
[PROOFSTEP]
measurability