Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f g : Ω → ℝ≥0∞
X Y : Ω → ℝ
h_meas_f : AEMeasurable f
h_meas_g : AEMeasurable g
h_indep_fun : IndepFun f g
fg_ae : f * g =ᵐ[μ] AEMeasurable.mk f h_meas_f * AEMeasurable.mk g h_meas_g
⊢ (∫⁻ (ω : Ω), (f * g) ω ∂μ) = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), g ω ∂μ
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f g : Ω → ℝ≥0∞
X Y : Ω → ℝ
h_meas_f : AEMeasurable f
h_meas_g : AEMeasurable g
h_indep_fun : IndepFun f g
⊢ (∫⁻ (ω : Ω), (f * g) ω ∂μ) = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), g ω ∂μ
have fg_ae : f * g =ᵐ[μ] h_meas_f.mk _ * h_meas_g.mk _ := h_meas_f.ae_eq_mk.mul h_meas_g.ae_eq_mk
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f g : Ω → ℝ≥0∞
X Y : Ω → ℝ
h_meas_f : AEMeasurable f
h_meas_g : AEMeasurable g
h_indep_fun : IndepFun f g
fg_ae : f * g =ᵐ[μ] AEMeasurable.mk f h_meas_f * AEMeasurable.mk g h_meas_g
⊢ (∫⁻ (a : Ω), (AEMeasurable.mk f h_meas_f * AEMeasurable.mk g h_meas_g) a ∂μ) =
    (∫⁻ (a : Ω), AEMeasurable.mk f h_meas_f a ∂μ) * ∫⁻ (a : Ω), AEMeasurable.mk g h_meas_g a ∂μ
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f g : Ω → ℝ≥0∞
X Y : Ω → ℝ
h_meas_f : AEMeasurable f
h_meas_g : AEMeasurable g
h_indep_fun : IndepFun f g
fg_ae : f * g =ᵐ[μ] AEMeasurable.mk f h_meas_f * AEMeasurable.mk g h_meas_g
⊢ (∫⁻ (ω : Ω), (f * g) ω ∂μ) = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), g ω ∂μ
rw [lintegral_congr_ae h_meas_f.ae_eq_mk, lintegral_congr_ae h_meas_g.ae_eq_mk,
  lintegral_congr_ae fg_ae]
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f g : Ω → ℝ≥0∞
X Y : Ω → ℝ
h_meas_f : AEMeasurable f
h_meas_g : AEMeasurable g
h_indep_fun : IndepFun f g
fg_ae : f * g =ᵐ[μ] AEMeasurable.mk f h_meas_f * AEMeasurable.mk g h_meas_g
⊢ IndepFun (AEMeasurable.mk f h_meas_f) (AEMeasurable.mk g h_meas_g)
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f g : Ω → ℝ≥0∞
X Y : Ω → ℝ
h_meas_f : AEMeasurable f
h_meas_g : AEMeasurable g
h_indep_fun : IndepFun f g
fg_ae : f * g =ᵐ[μ] AEMeasurable.mk f h_meas_f * AEMeasurable.mk g h_meas_g
⊢ (∫⁻ (a : Ω), (AEMeasurable.mk f h_meas_f * AEMeasurable.mk g h_meas_g) a ∂μ) =
    (∫⁻ (a : Ω), AEMeasurable.mk f h_meas_f a ∂μ) * ∫⁻ (a : Ω), AEMeasurable.mk g h_meas_g a ∂μ
apply lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun h_meas_f.measurable_mk
    h_meas_g.measurable_mk
no goals
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f g : Ω → ℝ≥0∞
X Y : Ω → ℝ
h_meas_f : AEMeasurable f
h_meas_g : AEMeasurable g
h_indep_fun : IndepFun f g
fg_ae : f * g =ᵐ[μ] AEMeasurable.mk f h_meas_f * AEMeasurable.mk g h_meas_g
⊢ IndepFun (AEMeasurable.mk f h_meas_f) (AEMeasurable.mk g h_meas_g)
exact h_indep_fun.ae_eq h_meas_f.ae_eq_mk h_meas_g.ae_eq_mk