Module: Mathlib.Probability.Independence.Integration

Goal state before:
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∞
h_meas_f : ∀ (n : ℕ), Measurable (f n)
h_mono_f : Monotone f
h_ind_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
h_measM_f : ∀ (n : ℕ), Measurable (f n)
⊢ ⨆ n, ∫⁻ (a : Ω), f n a * T.indicator (fun x ↦ c) a ∂μ =
    ⨆ i, (∫⁻ (a : Ω), f i a ∂μ) * ∫⁻ (ω : Ω), T.indicator (fun x ↦ c) ω ∂μ

pp: case iSup.hf
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∞
h_meas_f : ∀ (n : ℕ), Measurable (f n)
h_mono_f : Monotone f
h_ind_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
h_measM_f : ∀ (n : ℕ), Measurable (f n)
⊢ ∀ (n : ℕ), Measurable fun ω ↦ f n ω * T.indicator (fun x ↦ c) ω

pp: case iSup.h_mono
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∞
h_meas_f : ∀ (n : ℕ), Measurable (f n)
h_mono_f : Monotone f
h_ind_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
h_measM_f : ∀ (n : ℕ), Measurable (f n)
⊢ Monotone fun i ω ↦ f i ω * T.indicator (fun x ↦ c) ω

Tactic:
]

Goal state after:
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∞
h_meas_f : ∀ (n : ℕ), Measurable (f n)
h_mono_f : Monotone f
h_ind_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
h_measM_f : ∀ (n : ℕ), Measurable (f n)
⊢ ⨆ n, ∫⁻ (a : Ω), f n a * T.indicator (fun x ↦ c) a ∂μ =
    ⨆ i, (∫⁻ (a : Ω), f i a ∂μ) * ∫⁻ (ω : Ω), T.indicator (fun x ↦ c) ω ∂μ

case iSup.hf
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∞
h_meas_f : ∀ (n : ℕ), Measurable (f n)
h_mono_f : Monotone f
h_ind_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
h_measM_f : ∀ (n : ℕ), Measurable (f n)
⊢ ∀ (n : ℕ), Measurable fun ω ↦ f n ω * T.indicator (fun x ↦ c) ω

case iSup.h_mono
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∞
h_meas_f : ∀ (n : ℕ), Measurable (f n)
h_mono_f : Monotone f
h_ind_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
h_measM_f : ∀ (n : ℕ), Measurable (f n)
⊢ Monotone fun i ω ↦ f i ω * T.indicator (fun x ↦ c) ω