LeanMachineLearning exposition

ProbabilityTheory.HasCondDistrib.indepFun_of_const🔗

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indepFun_of_const🔗

LemmaProbabilityTheory.HasCondDistrib.indepFun_of_const
Details

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🔗theorem
ProbabilityTheory.HasCondDistrib.indepFun_of_const.{u_1, u_2, u_4}
  {α : Type u_1} {β : Type u_2} {Ω : Type u_4} {mα : MeasurableSpace α}
  {mβ : MeasurableSpace β} {mΩ : MeasurableSpace Ω}
  {μ : MeasureTheory.Measure α} {X : α → β} {Y : α → Ω}
  [MeasureTheory.IsProbabilityMeasure μ] {Q : MeasureTheory.Measure Ω}
  [MeasureTheory.SFinite Q]
  (h : HasCondDistrib Y X (Kernel.const β Q) μ) : IndepFun X Y μ
ProbabilityTheory.HasCondDistrib.indepFun_of_const.{u_1,
    u_2, u_4}
  {α : Type u_1} {β : Type u_2}
  {Ω : Type u_4} {mα : MeasurableSpace α}
  {mβ : MeasurableSpace β}
  {mΩ : MeasurableSpace Ω}
  {μ : MeasureTheory.Measure α}
  {X : α → β} {Y : α → Ω}
  [MeasureTheory.IsProbabilityMeasure μ]
  {Q : MeasureTheory.Measure Ω}
  [MeasureTheory.SFinite Q]
  (h :
    HasCondDistrib Y X (Kernel.const β Q)
      μ) :
  IndepFun X Y μ

Code

lemma HasCondDistrib.indepFun_of_const [IsProbabilityMeasure μ] {Q : Measure Ω} [SFinite Q]
    (h : HasCondDistrib Y X (Kernel.const β Q) μ) :
    IndepFun X Y μ

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Proof
by
  rw [indepFun_iff_map_prod_eq_prod_map_map h.aemeasurable_fst h.aemeasurable_snd, h.map_eq,
    h.hasLaw_of_const.map_eq, Measure.compProd_const]