Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: case indicator
Ω : Type u_1
Mf mΩ : MeasurableSpace Ω
μ : Measure Ω
hMf : Mf ≤ mΩ
c : ℝ≥0∞
T : Set Ω
h_meas_T : MeasurableSet T
h_ind : IndepSets {s | MeasurableSet s} {T} μ
h_mul_indicator : ∀ (g : Ω → ℝ≥0∞), Measurable g → Measurable fun a ↦ g a * T.indicator (fun x ↦ c) a
⊢ ∀ (c_1 : ℝ≥0∞) ⦃s : Set Ω⦄,
    MeasurableSet s →
      ∫⁻ (ω : Ω), s.indicator (fun x ↦ c_1) ω * T.indicator (fun x ↦ c) ω ∂μ =
        (∫⁻ (ω : Ω), s.indicator (fun x ↦ c_1) ω ∂μ) * ∫⁻ (ω : Ω), T.indicator (fun x ↦ c) ω ∂μ

pp: case add
Ω : Type u_1
Mf mΩ : MeasurableSpace Ω
μ : Measure Ω
hMf : Mf ≤ mΩ
c : ℝ≥0∞
T : Set Ω
h_meas_T : MeasurableSet T
h_ind : IndepSets {s | MeasurableSet s} {T} μ
h_mul_indicator : ∀ (g : Ω → ℝ≥0∞), Measurable g → Measurable fun a ↦ g a * T.indicator (fun x ↦ c) a
⊢ ∀ ⦃f g : Ω → ℝ≥0∞⦄,
    Disjoint (Function.support f) (Function.support g) →
      Measurable f →
        Measurable g →
          ∫⁻ (ω : Ω), f ω * T.indicator (fun x ↦ c) ω ∂μ =
              (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), T.indicator (fun x ↦ c) ω ∂μ →
            ∫⁻ (ω : Ω), g ω * T.indicator (fun x ↦ c) ω ∂μ =
                (∫⁻ (ω : Ω), g ω ∂μ) * ∫⁻ (ω : Ω), T.indicator (fun x ↦ c) ω ∂μ →
              ∫⁻ (ω : Ω), (f + g) ω * T.indicator (fun x ↦ c) ω ∂μ =
                (∫⁻ (ω : Ω), (f + g) ω ∂μ) * ∫⁻ (ω : Ω), T.indicator (fun x ↦ c) ω ∂μ

pp: case iSup
Ω : Type u_1
Mf mΩ : MeasurableSpace Ω
μ : Measure Ω
hMf : Mf ≤ mΩ
c : ℝ≥0∞
T : Set Ω
h_meas_T : MeasurableSet T
h_ind : IndepSets {s | MeasurableSet s} {T} μ
h_mul_indicator : ∀ (g : Ω → ℝ≥0∞), Measurable g → Measurable fun a ↦ g a * T.indicator (fun x ↦ c) a
⊢ ∀ ⦃f : ℕ → Ω → ℝ≥0∞⦄,
    (∀ (n : ℕ), Measurable (f n)) →
      Monotone f →
        (∀ (n : ℕ),
            ∫⁻ (ω : Ω), f n ω * T.indicator (fun x ↦ c) ω ∂μ =
              (∫⁻ (ω : Ω), f n ω ∂μ) * ∫⁻ (ω : Ω), T.indicator (fun x ↦ c) ω ∂μ) →
          ∫⁻ (ω : Ω), (fun x ↦ ⨆ n, f n x) ω * T.indicator (fun x ↦ c) ω ∂μ =
            (∫⁻ (ω : Ω), (fun x ↦ ⨆ n, f n x) ω ∂μ) * ∫⁻ (ω : Ω), T.indicator (fun x ↦ c) ω ∂μ

Tactic:
· intro c' s' h_meas_s'
    simp_rw [← inter_indicator_mul]
    rw [lintegral_indicator (MeasurableSet.inter (hMf _ h_meas_s') h_meas_T),
      lintegral_indicator (hMf _ h_meas_s'), lintegral_indicator h_meas_T]
    simp only [lintegral_const, univ_inter,
      MeasurableSet.univ, Measure.restrict_apply]
    rw [IndepSets_iff] at h_ind
    rw [mul_mul_mul_comm, h_ind s' T h_meas_s' (Set.mem_singleton _)]

Goal state after:
case add
Ω : Type u_1
Mf mΩ : MeasurableSpace Ω
μ : Measure Ω
hMf : Mf ≤ mΩ
c : ℝ≥0∞
T : Set Ω
h_meas_T : MeasurableSet T
h_ind : IndepSets {s | MeasurableSet s} {T} μ
h_mul_indicator : ∀ (g : Ω → ℝ≥0∞), Measurable g → Measurable fun a ↦ g a * T.indicator (fun x ↦ c) a
⊢ ∀ ⦃f g : Ω → ℝ≥0∞⦄,
    Disjoint (Function.support f) (Function.support g) →
      Measurable f →
        Measurable g →
          ∫⁻ (ω : Ω), f ω * T.indicator (fun x ↦ c) ω ∂μ =
              (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), T.indicator (fun x ↦ c) ω ∂μ →
            ∫⁻ (ω : Ω), g ω * T.indicator (fun x ↦ c) ω ∂μ =
                (∫⁻ (ω : Ω), g ω ∂μ) * ∫⁻ (ω : Ω), T.indicator (fun x ↦ c) ω ∂μ →
              ∫⁻ (ω : Ω), (f + g) ω * T.indicator (fun x ↦ c) ω ∂μ =
                (∫⁻ (ω : Ω), (f + g) ω ∂μ) * ∫⁻ (ω : Ω), T.indicator (fun x ↦ c) ω ∂μ

case iSup
Ω : Type u_1
Mf mΩ : MeasurableSpace Ω
μ : Measure Ω
hMf : Mf ≤ mΩ
c : ℝ≥0∞
T : Set Ω
h_meas_T : MeasurableSet T
h_ind : IndepSets {s | MeasurableSet s} {T} μ
h_mul_indicator : ∀ (g : Ω → ℝ≥0∞), Measurable g → Measurable fun a ↦ g a * T.indicator (fun x ↦ c) a
⊢ ∀ ⦃f : ℕ → Ω → ℝ≥0∞⦄,
    (∀ (n : ℕ), Measurable (f n)) →
      Monotone f →
        (∀ (n : ℕ),
            ∫⁻ (ω : Ω), f n ω * T.indicator (fun x ↦ c) ω ∂μ =
              (∫⁻ (ω : Ω), f n ω ∂μ) * ∫⁻ (ω : Ω), T.indicator (fun x ↦ c) ω ∂μ) →
          ∫⁻ (ω : Ω), (fun x ↦ ⨆ n, f n x) ω * T.indicator (fun x ↦ c) ω ∂μ =
            (∫⁻ (ω : Ω), (fun x ↦ ⨆ n, f n x) ω ∂μ) * ∫⁻ (ω : Ω), T.indicator (fun x ↦ c) ω ∂μ