-- Mathlib/MeasureTheory/Decomposition/SignedHahn.lean
theorem exists_compl_positive_negative (s : SignedMeasure α) :
    ∃ i : Set α, MeasurableSet i ∧ 0 ≤[i] s ∧ s ≤[iᶜ] 0

Proof state:
case intro.intro.intro
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf s.measureOfNegatives))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ s ≤[B] 0} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), s ≤[B n] 0
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
hA₁ : MeasurableSet A
hA₂ : s ≤[A] 0
hA₃ : ↑s A = sInf s.measureOfNegatives
⊢ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ Aᶜ → ↑0 j ≤ ↑s j

Tactic:
intro C _ hC₁

Result:
case intro.intro.intro
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf s.measureOfNegatives))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ s ≤[B] 0} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), s ≤[B n] 0
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
hA₁ : MeasurableSet A
hA₂ : s ≤[A] 0
hA₃ : ↑s A = sInf s.measureOfNegatives
C : Set α
a✝ : MeasurableSet C
hC₁ : C ⊆ Aᶜ
⊢ ↑0 C ≤ ↑s C

Proof state:
case intro.intro.intro
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf s.measureOfNegatives))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ s ≤[B] 0} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
⊢ ∃ i, MeasurableSet i ∧ 0 ≤[i] s ∧ s ≤[iᶜ] 0

Tactic:
have hB₂ : ∀ n, s ≤[B n] 0 := fun n => (hB n).1.2

Result:
case intro.intro.intro
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf s.measureOfNegatives))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ s ≤[B] 0} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), s ≤[B n] 0
⊢ ∃ i, MeasurableSet i ∧ 0 ≤[i] s ∧ s ≤[iᶜ] 0

Proof state:
case intro.intro.intro
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf s.measureOfNegatives))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ s ≤[B] 0} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), s ≤[B n] 0
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
hA₁ : MeasurableSet A
hA₂ : s ≤[A] 0
hA₃ : ↑s A = sInf s.measureOfNegatives
⊢ ∃ i, MeasurableSet i ∧ 0 ≤[i] s ∧ s ≤[iᶜ] 0

Tactic:
refine ⟨Aᶜ, hA₁.compl, ?_, (compl_compl A).symm ▸ hA₂⟩

Result:
case intro.intro.intro
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf s.measureOfNegatives))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ s ≤[B] 0} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), s ≤[B n] 0
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
hA₁ : MeasurableSet A
hA₂ : s ≤[A] 0
hA₃ : ↑s A = sInf s.measureOfNegatives
⊢ 0 ≤[Aᶜ] s

Proof state:
case a
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf s.measureOfNegatives))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ s ≤[B] 0} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), s ≤[B n] 0
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
hA₁ : MeasurableSet A
hA₂ : s ≤[A] 0
n : ℕ
this : s ≤[A] 0
⊢ ↑s ((⋃ i, B i) \ B n) ≤ 0

Tactic:
refine
  nonpos_of_restrict_le_zero _ (restrict_le_zero_subset _ ?_ Set.diff_subset this)

Result:
case a
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf s.measureOfNegatives))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ s ≤[B] 0} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), s ≤[B n] 0
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
hA₁ : MeasurableSet A
hA₂ : s ≤[A] 0
n : ℕ
this : s ≤[A] 0
⊢ MeasurableSet (⋃ i, B i)

Proof state:
case a
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf s.measureOfNegatives))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ s ≤[B] 0} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), s ≤[B n] 0
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
hA₁ : MeasurableSet A
hA₂ : s ≤[A] 0
n : ℕ
⊢ (fun x => ↑s ((⋃ i, B i) \ B n) + ↑s (B n)) n ≤ ↑s (B n)

Tactic:
refine add_le_of_nonpos_left ?_

Result:
case a
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf s.measureOfNegatives))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ s ≤[B] 0} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), s ≤[B n] 0
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
hA₁ : MeasurableSet A
hA₂ : s ≤[A] 0
n : ℕ
⊢ ↑s ((⋃ i, B i) \ B n) ≤ 0

Proof state:
case intro.intro.intro.intro.intro.intro.intro.refine_2
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf s.measureOfNegatives))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ s ≤[B] 0} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), s ≤[B n] 0
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
hA₁ : MeasurableSet A
hA₂ : s ≤[A] 0
hA₃ : ↑s A = sInf s.measureOfNegatives
C : Set α
a✝ : MeasurableSet C
hC₁ : C ⊆ Aᶜ
hC₂ : ↑s C < ↑0 C
D : Set α
hD₁ : MeasurableSet D
hD : D ⊆ C
hD₂ : s ≤[D] 0
hD₃ : ↑s D < 0
this : ↑s (A ∪ D) < sInf s.measureOfNegatives
⊢ s ≤[A ∪ D] 0

Tactic:
exact restrict_le_restrict_union _ _ hA₁ hA₂ hD₁ hD₂

Result:
no goals

Proof state:
case intro.intro.intro
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf s.measureOfNegatives))
hf₁ : ∀ (n : ℕ), f n ∈ s.measureOfNegatives
⊢ ∃ i, MeasurableSet i ∧ 0 ≤[i] s ∧ s ≤[iᶜ] 0

Tactic:
choose B hB using hf₁

Result:
case intro.intro.intro
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf s.measureOfNegatives))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ s ≤[B] 0} ∧ ↑s (B n) = f n
⊢ ∃ i, MeasurableSet i ∧ 0 ≤[i] s ∧ s ≤[iᶜ] 0

Proof state:
case intro.intro.intro
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf s.measureOfNegatives))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ s ≤[B] 0} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), s ≤[B n] 0
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
⊢ ∃ i, MeasurableSet i ∧ 0 ≤[i] s ∧ s ≤[iᶜ] 0

Tactic:
have hA₁ : MeasurableSet A := MeasurableSet.iUnion hB₁

Result:
case intro.intro.intro
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf s.measureOfNegatives))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ s ≤[B] 0} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), s ≤[B n] 0
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
hA₁ : MeasurableSet A
⊢ ∃ i, MeasurableSet i ∧ 0 ≤[i] s ∧ s ≤[iᶜ] 0