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ProbabilityTheory.ae_eq_of_hasCondDistrib_deterministic🔗

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ae_eq_of_hasCondDistrib_deterministic🔗

LemmaProbabilityTheory.ae_eq_of_hasCondDistrib_deterministic
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🔗theorem
ProbabilityTheory.ae_eq_of_hasCondDistrib_deterministic.{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} [MeasurableEq Ω]
  [MeasureTheory.SFinite μ] {f : β → Ω} (hf : Measurable f)
  (hX : AEMeasurable X μ) (hY : AEMeasurable Y μ)
  (h : HasCondDistrib Y X (Kernel.deterministic f hf) μ) : Y =ᵐ[μ] f ∘ X
ProbabilityTheory.ae_eq_of_hasCondDistrib_deterministic.{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} [MeasurableEq Ω]
  [MeasureTheory.SFinite μ] {f : β → Ω}
  (hf : Measurable f)
  (hX : AEMeasurable X μ)
  (hY : AEMeasurable Y μ)
  (h :
    HasCondDistrib Y X
      (Kernel.deterministic f hf) μ) :
  Y =ᵐ[μ] f ∘ X

Code

lemma ae_eq_of_hasCondDistrib_deterministic [MeasurableEq Ω] [SFinite μ] {f : β → Ω}
    (hf : Measurable f) (hX : AEMeasurable X μ)
    (hY : AEMeasurable Y μ) (h : HasCondDistrib Y X (Kernel.deterministic f hf) μ) :
    Y =ᵐ[μ] f ∘ X
Body uses (1)
Used by (1)

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Proof
by
  refine ae_eq_of_map_prodMk_eq hf hX hY ?_
  rw [h.map_eq, Measure.compProd_deterministic,
    AEMeasurable.map_map_of_aemeasurable (by fun_prop) (by fun_prop)]
  rfl