Module: Mathlib.Probability.Martingale.OptionalStopping

Goal state before:
pp: Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
f : ℕ → Ω → ℝ
inst✝ : IsFiniteMeasure μ
hsub : Submartingale f 𝒢 μ
hnonneg : 0 ≤ f
ε : ℝ≥0
n : ℕ
this :
  ε • μ {ω | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k ↦ f k ω} +
      ENNReal.ofReal (∫ (ω : Ω) in {ω | ((range (n + 1)).sup' ⋯ fun k ↦ f k ω) < ↑ε}, f n ω ∂μ) ≤
    ENNReal.ofReal (∫ (x : Ω), f n x ∂μ)
⊢ ENNReal.ofReal (∫ (ω : Ω), f n ω ∂μ) =
    ENNReal.ofReal (∫ (ω : Ω) in {ω | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k ↦ f k ω}, f n ω ∂μ) +
      ENNReal.ofReal (∫ (ω : Ω) in {ω | ((range (n + 1)).sup' ⋯ fun k ↦ f k ω) < ↑ε}, f n ω ∂μ)

Tactic:
rw [← ENNReal.ofReal_add, ← setIntegral_union]

Goal state after:
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
f : ℕ → Ω → ℝ
inst✝ : IsFiniteMeasure μ
hsub : Submartingale f 𝒢 μ
hnonneg : 0 ≤ f
ε : ℝ≥0
n : ℕ
this :
  ε • μ {ω | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k ↦ f k ω} +
      ENNReal.ofReal (∫ (ω : Ω) in {ω | ((range (n + 1)).sup' ⋯ fun k ↦ f k ω) < ↑ε}, f n ω ∂μ) ≤
    ENNReal.ofReal (∫ (x : Ω), f n x ∂μ)
⊢ ENNReal.ofReal (∫ (ω : Ω), f n ω ∂μ) =
    ENNReal.ofReal
      (∫ (x : Ω) in {ω | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k ↦ f k ω} ∪ {ω | ((range (n + 1)).sup' ⋯ fun k ↦ f k ω) < ↑ε},
        f n x ∂μ)

case hst
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
f : ℕ → Ω → ℝ
inst✝ : IsFiniteMeasure μ
hsub : Submartingale f 𝒢 μ
hnonneg : 0 ≤ f
ε : ℝ≥0
n : ℕ
this :
  ε • μ {ω | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k ↦ f k ω} +
      ENNReal.ofReal (∫ (ω : Ω) in {ω | ((range (n + 1)).sup' ⋯ fun k ↦ f k ω) < ↑ε}, f n ω ∂μ) ≤
    ENNReal.ofReal (∫ (x : Ω), f n x ∂μ)
⊢ Disjoint {ω | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k ↦ f k ω} {ω | ((range (n + 1)).sup' ⋯ fun k ↦ f k ω) < ↑ε}

case ht
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
f : ℕ → Ω → ℝ
inst✝ : IsFiniteMeasure μ
hsub : Submartingale f 𝒢 μ
hnonneg : 0 ≤ f
ε : ℝ≥0
n : ℕ
this :
  ε • μ {ω | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k ↦ f k ω} +
      ENNReal.ofReal (∫ (ω : Ω) in {ω | ((range (n + 1)).sup' ⋯ fun k ↦ f k ω) < ↑ε}, f n ω ∂μ) ≤
    ENNReal.ofReal (∫ (x : Ω), f n x ∂μ)
⊢ MeasurableSet {ω | ((range (n + 1)).sup' ⋯ fun k ↦ f k ω) < ↑ε}

case hfs
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
f : ℕ → Ω → ℝ
inst✝ : IsFiniteMeasure μ
hsub : Submartingale f 𝒢 μ
hnonneg : 0 ≤ f
ε : ℝ≥0
n : ℕ
this :
  ε • μ {ω | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k ↦ f k ω} +
      ENNReal.ofReal (∫ (ω : Ω) in {ω | ((range (n + 1)).sup' ⋯ fun k ↦ f k ω) < ↑ε}, f n ω ∂μ) ≤
    ENNReal.ofReal (∫ (x : Ω), f n x ∂μ)
⊢ IntegrableOn (f n) {ω | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k ↦ f k ω} μ

case hft
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
f : ℕ → Ω → ℝ
inst✝ : IsFiniteMeasure μ
hsub : Submartingale f 𝒢 μ
hnonneg : 0 ≤ f
ε : ℝ≥0
n : ℕ
this :
  ε • μ {ω | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k ↦ f k ω} +
      ENNReal.ofReal (∫ (ω : Ω) in {ω | ((range (n + 1)).sup' ⋯ fun k ↦ f k ω) < ↑ε}, f n ω ∂μ) ≤
    ENNReal.ofReal (∫ (x : Ω), f n x ∂μ)
⊢ IntegrableOn (f n) {ω | ((range (n + 1)).sup' ⋯ fun k ↦ f k ω) < ↑ε} μ

case hp
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
f : ℕ → Ω → ℝ
inst✝ : IsFiniteMeasure μ
hsub : Submartingale f 𝒢 μ
hnonneg : 0 ≤ f
ε : ℝ≥0
n : ℕ
this :
  ε • μ {ω | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k ↦ f k ω} +
      ENNReal.ofReal (∫ (ω : Ω) in {ω | ((range (n + 1)).sup' ⋯ fun k ↦ f k ω) < ↑ε}, f n ω ∂μ) ≤
    ENNReal.ofReal (∫ (x : Ω), f n x ∂μ)
⊢ 0 ≤ ∫ (ω : Ω) in {ω | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k ↦ f k ω}, f n ω ∂μ

case hq
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
f : ℕ → Ω → ℝ
inst✝ : IsFiniteMeasure μ
hsub : Submartingale f 𝒢 μ
hnonneg : 0 ≤ f
ε : ℝ≥0
n : ℕ
this :
  ε • μ {ω | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k ↦ f k ω} +
      ENNReal.ofReal (∫ (ω : Ω) in {ω | ((range (n + 1)).sup' ⋯ fun k ↦ f k ω) < ↑ε}, f n ω ∂μ) ≤
    ENNReal.ofReal (∫ (x : Ω), f n x ∂μ)
⊢ 0 ≤ ∫ (ω : Ω) in {ω | ((range (n + 1)).sup' ⋯ fun k ↦ f k ω) < ↑ε}, f n ω ∂μ