Theorem: MeasureTheory.pdf.indepFun_iff_pdf_prod_eq_pdf_mul_pdf
File: Mathlib/Probability/Density.lean

Proof steps:
state_before: Ω : Type u_1
E : Type u_2
inst✝⁵ : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝⁴ : MeasurableSpace F
ν : Measure F
X : Ω → E
Y : Ω → F
inst✝³ : IsFiniteMeasure ℙ
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : HasPDF (fun ω => (X ω, Y ω)) ℙ (μ.prod ν)
⊢ IndepFun X Y ℙ ↔ pdf (fun ω => (X ω, Y ω)) ℙ (μ.prod ν) =ᶠ[ae (μ.prod ν)] fun z => pdf X ℙ μ z.1 * pdf Y ℙ ν z.2
state_after: Ω : Type u_1
E : Type u_2
inst✝⁵ : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝⁴ : MeasurableSpace F
ν : Measure F
X : Ω → E
Y : Ω → F
inst✝³ : IsFiniteMeasure ℙ
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : HasPDF (fun ω => (X ω, Y ω)) ℙ (μ.prod ν)
this : HasPDF X ℙ μ
⊢ IndepFun X Y ℙ ↔ pdf (fun ω => (X ω, Y ω)) ℙ (μ.prod ν) =ᶠ[ae (μ.prod ν)] fun z => pdf X ℙ μ z.1 * pdf Y ℙ ν z.2
tactic: have : HasPDF X ℙ μ := quasiMeasurePreserving_hasPDF' (μ := μ.prod ν)
    (HasPDF.aemeasurable (fun ω ↦ (X ω, Y ω)) ℙ (μ.prod ν)) quasiMeasurePreserving_fst
state_before: Ω : Type u_1
E : Type u_2
inst✝⁵ : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝⁴ : MeasurableSpace F
ν : Measure F
X : Ω → E
Y : Ω → F
inst✝³ : IsFiniteMeasure ℙ
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : HasPDF (fun ω => (X ω, Y ω)) ℙ (μ.prod ν)
this : HasPDF X ℙ μ
⊢ IndepFun X Y ℙ ↔ pdf (fun ω => (X ω, Y ω)) ℙ (μ.prod ν) =ᶠ[ae (μ.prod ν)] fun z => pdf X ℙ μ z.1 * pdf Y ℙ ν z.2
state_after: Ω : Type u_1
E : Type u_2
inst✝⁵ : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝⁴ : MeasurableSpace F
ν : Measure F
X : Ω → E
Y : Ω → F
inst✝³ : IsFiniteMeasure ℙ
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : HasPDF (fun ω => (X ω, Y ω)) ℙ (μ.prod ν)
this✝ : HasPDF X ℙ μ
this : HasPDF Y ℙ ν
⊢ IndepFun X Y ℙ ↔ pdf (fun ω => (X ω, Y ω)) ℙ (μ.prod ν) =ᶠ[ae (μ.prod ν)] fun z => pdf X ℙ μ z.1 * pdf Y ℙ ν z.2
tactic: have : HasPDF Y ℙ ν := quasiMeasurePreserving_hasPDF' (μ := μ.prod ν)
    (HasPDF.aemeasurable (fun ω ↦ (X ω, Y ω)) ℙ (μ.prod ν)) quasiMeasurePreserving_snd
state_before: Ω : Type u_1
E : Type u_2
inst✝⁵ : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝⁴ : MeasurableSpace F
ν : Measure F
X : Ω → E
Y : Ω → F
inst✝³ : IsFiniteMeasure ℙ
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : HasPDF (fun ω => (X ω, Y ω)) ℙ (μ.prod ν)
this✝ : HasPDF X ℙ μ
this : HasPDF Y ℙ ν
⊢ IndepFun X Y ℙ ↔ pdf (fun ω => (X ω, Y ω)) ℙ (μ.prod ν) =ᶠ[ae (μ.prod ν)] fun z => pdf X ℙ μ z.1 * pdf Y ℙ ν z.2
state_after: Ω : Type u_1
E : Type u_2
inst✝⁵ : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝⁴ : MeasurableSpace F
ν : Measure F
X : Ω → E
Y : Ω → F
inst✝³ : IsFiniteMeasure ℙ
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : HasPDF (fun ω => (X ω, Y ω)) ℙ (μ.prod ν)
this✝ : HasPDF X ℙ μ
this : HasPDF Y ℙ ν
h₀ : (map X ℙ).prod (map Y ℙ) = (μ.prod ν).withDensity fun z => pdf X ℙ μ z.1 * pdf Y ℙ ν z.2
⊢ IndepFun X Y ℙ ↔ pdf (fun ω => (X ω, Y ω)) ℙ (μ.prod ν) =ᶠ[ae (μ.prod ν)] fun z => pdf X ℙ μ z.1 * pdf Y ℙ ν z.2
tactic: have h₀ : (ℙ.map X).prod (ℙ.map Y) =
      (μ.prod ν).withDensity fun z ↦ pdf X ℙ μ z.1 * pdf Y ℙ ν z.2 :=
    prod_eq fun s t hs ht ↦ by rw [withDensity_apply _ (hs.prod ht), ← prod_restrict,
      lintegral_prod_mul (measurable_pdf X ℙ μ).aemeasurable (measurable_pdf Y ℙ ν).aemeasurable,
      map_eq_setLIntegral_pdf X ℙ μ hs, map_eq_setLIntegral_pdf Y ℙ ν ht]
state_before: Ω : Type u_1
E : Type u_2
inst✝⁵ : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝⁴ : MeasurableSpace F
ν : Measure F
X : Ω → E
Y : Ω → F
inst✝³ : IsFiniteMeasure ℙ
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : HasPDF (fun ω => (X ω, Y ω)) ℙ (μ.prod ν)
this✝ : HasPDF X ℙ μ
this : HasPDF Y ℙ ν
h₀ : (map X ℙ).prod (map Y ℙ) = (μ.prod ν).withDensity fun z => pdf X ℙ μ z.1 * pdf Y ℙ ν z.2
⊢ IndepFun X Y ℙ ↔ pdf (fun ω => (X ω, Y ω)) ℙ (μ.prod ν) =ᶠ[ae (μ.prod ν)] fun z => pdf X ℙ μ z.1 * pdf Y ℙ ν z.2
state_after: case hmf
Ω : Type u_1
E : Type u_2
inst✝⁵ : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝⁴ : MeasurableSpace F
ν : Measure F
X : Ω → E
Y : Ω → F
inst✝³ : IsFiniteMeasure ℙ
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : HasPDF (fun ω => (X ω, Y ω)) ℙ (μ.prod ν)
this✝ : HasPDF X ℙ μ
this : HasPDF Y ℙ ν
h₀ : (map X ℙ).prod (map Y ℙ) = (μ.prod ν).withDensity fun z => pdf X ℙ μ z.1 * pdf Y ℙ ν z.2
⊢ AEMeasurable (fun z => pdf X ℙ μ z.1 * pdf Y ℙ ν z.2) (μ.prod ν)
tactic: rw [indepFun_iff_map_prod_eq_prod_map_map (HasPDF.aemeasurable X ℙ μ) (HasPDF.aemeasurable Y ℙ ν),
    ← eq_of_map_eq_withDensity, h₀]
state_before: case hmf
Ω : Type u_1
E : Type u_2
inst✝⁵ : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝⁴ : MeasurableSpace F
ν : Measure F
X : Ω → E
Y : Ω → F
inst✝³ : IsFiniteMeasure ℙ
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : HasPDF (fun ω => (X ω, Y ω)) ℙ (μ.prod ν)
this✝ : HasPDF X ℙ μ
this : HasPDF Y ℙ ν
h₀ : (map X ℙ).prod (map Y ℙ) = (μ.prod ν).withDensity fun z => pdf X ℙ μ z.1 * pdf Y ℙ ν z.2
⊢ AEMeasurable (fun z => pdf X ℙ μ z.1 * pdf Y ℙ ν z.2) (μ.prod ν)
state_after: no goals
tactic: exact (((measurable_pdf X ℙ μ).comp measurable_fst).mul
    ((measurable_pdf Y ℙ ν).comp measurable_snd)).aemeasurable