LeanMachineLearning exposition

ProbabilityTheory.condDistrib_of_indepFun🔗

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condDistrib_of_indepFun🔗

LemmaProbabilityTheory.condDistrib_of_indepFun
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🔗theorem
ProbabilityTheory.condDistrib_of_indepFun.{u_1, u_2, u_5} {α : Type u_1}
  {β : Type u_2} {Ω : Type u_5} {mα : MeasurableSpace α}
  {μ : MeasureTheory.Measure α} {mβ : MeasurableSpace β}
  [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] {X : α → β}
  {Y : α → Ω} [MeasureTheory.IsFiniteMeasure μ] (h : IndepFun X Y μ)
  (hX : AEMeasurable X μ) (hY : AEMeasurable Y μ) :
  ⇑𝓛[Y | X; μ] =ᵐ[MeasureTheory.Measure.map X μ]
    ⇑(Kernel.const β (MeasureTheory.Measure.map Y μ))
ProbabilityTheory.condDistrib_of_indepFun.{u_1,
    u_2, u_5}
  {α : Type u_1} {β : Type u_2}
  {Ω : Type u_5} {mα : MeasurableSpace α}
  {μ : MeasureTheory.Measure α}
  {mβ : MeasurableSpace β}
  [MeasurableSpace Ω]
  [StandardBorelSpace Ω] [Nonempty Ω]
  {X : α → β} {Y : α → Ω}
  [MeasureTheory.IsFiniteMeasure μ]
  (h : IndepFun X Y μ)
  (hX : AEMeasurable X μ)
  (hY : AEMeasurable Y μ) :
  ⇑𝓛[Y | X;
        μ] =ᵐ[MeasureTheory.Measure.map X
      μ]
    ⇑(Kernel.const β
        (MeasureTheory.Measure.map Y μ))

Code

lemma condDistrib_of_indepFun (h : IndepFun X Y μ) (hX : AEMeasurable X μ) (hY : AEMeasurable Y μ) :
    condDistrib Y X μ =ᵐ[μ.map X] Kernel.const β (μ.map Y)
Used by (2)

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Proof
by
  refine condDistrib_ae_eq_of_measure_eq_compProd (μ := μ) X hY ?_
  simp only [Measure.compProd_const]
  exact (indepFun_iff_map_prod_eq_prod_map_map hX hY).mp h