-- Mathlib/Probability/Integration.lean
theorem lintegral_mul_eq_lintegral_mul_lintegral_of_independent_measurableSpace
    {Mf Mg mΩ : MeasurableSpace Ω} {μ : Measure Ω} (hMf : Mf ≤ mΩ) (hMg : Mg ≤ mΩ)
    (h_ind : Indep Mf Mg μ) (h_meas_f : Measurable[Mf] f) (h_meas_g : Measurable[Mg] g) :
    ∫⁻ ω, f ω * g ω ∂μ = (∫⁻ ω, f ω ∂μ) * ∫⁻ ω, g ω ∂μ

Proof state:
case h_ind
Ω : Type u_1
f : Ω → ℝ≥0∞
Mf Mg mΩ : MeasurableSpace Ω
μ : Measure Ω
hMf : Mf ≤ mΩ
hMg : Mg ≤ mΩ
h_ind : Indep Mf Mg μ
h_meas_f : Measurable f
h_measM_f : Measurable f
c : ℝ≥0∞
s : Set Ω
h_s : MeasurableSet s
⊢ ∫⁻ (ω : Ω), f ω * s.indicator (fun x => c) ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), s.indicator (fun x => c) ω ∂μ

Tactic:
apply lintegral_mul_indicator_eq_lintegral_mul_lintegral_indicator hMf _ (hMg _ h_s) _ h_meas_f

Result:
Ω : Type u_1
f : Ω → ℝ≥0∞
Mf Mg mΩ : MeasurableSpace Ω
μ : Measure Ω
hMf : Mf ≤ mΩ
hMg : Mg ≤ mΩ
h_ind : Indep Mf Mg μ
h_meas_f : Measurable f
h_measM_f : Measurable f
c : ℝ≥0∞
s : Set Ω
h_s : MeasurableSet s
⊢ IndepSets {s | MeasurableSet s} {s} μ

Proof state:
case h_add
Ω : Type u_1
f : Ω → ℝ≥0∞
Mf Mg mΩ : MeasurableSpace Ω
μ : Measure Ω
hMf : Mf ≤ mΩ
hMg : Mg ≤ mΩ
h_ind : Indep Mf Mg μ
h_meas_f : Measurable f
h_measM_f : Measurable f
f' g : Ω → ℝ≥0∞
a✝¹ : Disjoint (Function.support f') (Function.support g)
h_measMg_f' : Measurable f'
a✝ : Measurable g
h_ind_f' : ∫⁻ (ω : Ω), f ω * f' ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), f' ω ∂μ
h_ind_g' : ∫⁻ (ω : Ω), f ω * g ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), g ω ∂μ
⊢ ∫⁻ (ω : Ω), f ω * (f' + g) ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), (f' + g) ω ∂μ

Tactic:
have h_measM_f' : Measurable f' := h_measMg_f'.mono hMg le_rfl

Result:
case h_add
Ω : Type u_1
f : Ω → ℝ≥0∞
Mf Mg mΩ : MeasurableSpace Ω
μ : Measure Ω
hMf : Mf ≤ mΩ
hMg : Mg ≤ mΩ
h_ind : Indep Mf Mg μ
h_meas_f : Measurable f
h_measM_f : Measurable f
f' g : Ω → ℝ≥0∞
a✝¹ : Disjoint (Function.support f') (Function.support g)
h_measMg_f' : Measurable f'
a✝ : Measurable g
h_ind_f' : ∫⁻ (ω : Ω), f ω * f' ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), f' ω ∂μ
h_ind_g' : ∫⁻ (ω : Ω), f ω * g ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), g ω ∂μ
h_measM_f' : Measurable f'
⊢ ∫⁻ (ω : Ω), f ω * (f' + g) ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), (f' + g) ω ∂μ

Proof state:
case h_ind
Ω : Type u_1
f : Ω → ℝ≥0∞
Mf Mg mΩ : MeasurableSpace Ω
μ : Measure Ω
hMf : Mf ≤ mΩ
hMg : Mg ≤ mΩ
h_ind : Indep Mf Mg μ
h_meas_f : Measurable f
h_measM_f : Measurable f
⊢ ∀ (c : ℝ≥0∞) ⦃s : Set Ω⦄,
    MeasurableSet s →
      ∫⁻ (ω : Ω), f ω * s.indicator (fun x => c) ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), s.indicator (fun x => c) ω ∂μ

Tactic:
intro c s h_s

Result:
case h_ind
Ω : Type u_1
f : Ω → ℝ≥0∞
Mf Mg mΩ : MeasurableSpace Ω
μ : Measure Ω
hMf : Mf ≤ mΩ
hMg : Mg ≤ mΩ
h_ind : Indep Mf Mg μ
h_meas_f : Measurable f
h_measM_f : Measurable f
c : ℝ≥0∞
s : Set Ω
h_s : MeasurableSet s
⊢ ∫⁻ (ω : Ω), f ω * s.indicator (fun x => c) ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), s.indicator (fun x => c) ω ∂μ

Proof state:
case h_add
Ω : Type u_1
f : Ω → ℝ≥0∞
Mf Mg mΩ : MeasurableSpace Ω
μ : Measure Ω
hMf : Mf ≤ mΩ
hMg : Mg ≤ mΩ
h_ind : Indep Mf Mg μ
h_meas_f : Measurable f
h_measM_f : Measurable f
f' g : Ω → ℝ≥0∞
a✝¹ : Disjoint (Function.support f') (Function.support g)
h_measMg_f' : Measurable f'
a✝ : Measurable g
h_ind_f' : ∫⁻ (ω : Ω), f ω * f' ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), f' ω ∂μ
h_ind_g' : ∫⁻ (ω : Ω), f ω * g ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), g ω ∂μ
h_measM_f' : Measurable f'
⊢ ∫⁻ (ω : Ω), f ω * (f' + g) ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), (f' + g) ω ∂μ

Tactic:
simp_rw [Pi.add_apply, left_distrib]

Result:
case h_add
Ω : Type u_1
f : Ω → ℝ≥0∞
Mf Mg mΩ : MeasurableSpace Ω
μ : Measure Ω
hMf : Mf ≤ mΩ
hMg : Mg ≤ mΩ
h_ind : Indep Mf Mg μ
h_meas_f : Measurable f
h_measM_f : Measurable f
f' g : Ω → ℝ≥0∞
a✝¹ : Disjoint (Function.support f') (Function.support g)
h_measMg_f' : Measurable f'
a✝ : Measurable g
h_ind_f' : ∫⁻ (ω : Ω), f ω * f' ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), f' ω ∂μ
h_ind_g' : ∫⁻ (ω : Ω), f ω * g ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), g ω ∂μ
h_measM_f' : Measurable f'
⊢ ∫⁻ (ω : Ω), f ω * f' ω + f ω * g ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), f' ω + g ω ∂μ

Proof state:
case h_iSup.h_mono
Ω : Type u_1
f : Ω → ℝ≥0∞
Mf Mg mΩ : MeasurableSpace Ω
μ : Measure Ω
hMf : Mf ≤ mΩ
hMg : Mg ≤ mΩ
h_ind : Indep Mf Mg μ
h_meas_f : Measurable f
h_measM_f : Measurable f
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 ω * f' n ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), f' n ω ∂μ
h_measM_f' : ∀ (n : ℕ), Measurable (f' n)
⊢ Monotone fun i ω => f ω * f' i ω

Tactic:
exact fun n m (h_le : n ≤ m) a => mul_le_mul_left' (h_mono_f' h_le a) _

Result:
no goals

Proof state:
Ω : Type u_1
f : Ω → ℝ≥0∞
Mf Mg mΩ : MeasurableSpace Ω
μ : Measure Ω
hMf : Mf ≤ mΩ
hMg : Mg ≤ mΩ
h_ind : Indep Mf Mg μ
h_meas_f : Measurable f
h_measM_f : Measurable f
c : ℝ≥0∞
s : Set Ω
h_s : MeasurableSet s
⊢ IndepSets {s | MeasurableSet s} {s} μ

Tactic:
apply indepSets_of_indepSets_of_le_right h_ind

Result:
Ω : Type u_1
f : Ω → ℝ≥0∞
Mf Mg mΩ : MeasurableSpace Ω
μ : Measure Ω
hMf : Mf ≤ mΩ
hMg : Mg ≤ mΩ
h_ind : Indep Mf Mg μ
h_meas_f : Measurable f
h_measM_f : Measurable f
c : ℝ≥0∞
s : Set Ω
h_s : MeasurableSet s
⊢ {s} ⊆ {s | MeasurableSet s}

Proof state:
case h_add
Ω : Type u_1
f : Ω → ℝ≥0∞
Mf Mg mΩ : MeasurableSpace Ω
μ : Measure Ω
hMf : Mf ≤ mΩ
hMg : Mg ≤ mΩ
h_ind : Indep Mf Mg μ
h_meas_f : Measurable f
h_measM_f : Measurable f
⊢ ∀ ⦃f_1 g : Ω → ℝ≥0∞⦄,
    Disjoint (Function.support f_1) (Function.support g) →
      Measurable f_1 →
        Measurable g →
          ∫⁻ (ω : Ω), f ω * f_1 ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), f_1 ω ∂μ →
            ∫⁻ (ω : Ω), f ω * g ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), g ω ∂μ →
              ∫⁻ (ω : Ω), f ω * (f_1 + g) ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), (f_1 + g) ω ∂μ

Tactic:
intro f' g _ h_measMg_f' _ h_ind_f' h_ind_g'

Result:
case h_add
Ω : Type u_1
f : Ω → ℝ≥0∞
Mf Mg mΩ : MeasurableSpace Ω
μ : Measure Ω
hMf : Mf ≤ mΩ
hMg : Mg ≤ mΩ
h_ind : Indep Mf Mg μ
h_meas_f : Measurable f
h_measM_f : Measurable f
f' g : Ω → ℝ≥0∞
a✝¹ : Disjoint (Function.support f') (Function.support g)
h_measMg_f' : Measurable f'
a✝ : Measurable g
h_ind_f' : ∫⁻ (ω : Ω), f ω * f' ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), f' ω ∂μ
h_ind_g' : ∫⁻ (ω : Ω), f ω * g ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), g ω ∂μ
⊢ ∫⁻ (ω : Ω), f ω * (f' + g) ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), (f' + g) ω ∂μ

Proof state:
Ω : Type u_1
f : Ω → ℝ≥0∞
Mf Mg mΩ : MeasurableSpace Ω
μ : Measure Ω
hMf : Mf ≤ mΩ
hMg : Mg ≤ mΩ
h_ind : Indep Mf Mg μ
h_meas_f : Measurable f
h_measM_f : Measurable f
c : ℝ≥0∞
s : Set Ω
h_s : MeasurableSet s
⊢ {s} ⊆ {s | MeasurableSet s}

Tactic:
rwa [singleton_subset_iff]

Result:
no goals