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ProbabilityTheory.CondIndepFun.of_measurable_right🔗

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of_measurable_right🔗

LemmaProbabilityTheory.CondIndepFun.of_measurable_right
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🔗theorem
ProbabilityTheory.CondIndepFun.of_measurable_right.{u_1, u_2, u_3, u_4,
    u_6}
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4}
  {δ' : Type u_6} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
  {mγ : MeasurableSpace γ} {mδ : MeasurableSpace δ}
  {mδ' : MeasurableSpace δ'} [StandardBorelSpace δ'] [Nonempty δ']
  {μ : MeasureTheory.Measure α} {X : α → β} {hX : Measurable X}
  {Y : α → γ} {Z : α → δ} {Z' : α → δ'} [StandardBorelSpace α]
  [MeasureTheory.IsFiniteMeasure μ]
  (h_indep :
    CondIndepFun (MeasurableSpace.comap X inferInstance) ⋯ Y Z μ)
  (hZ_meas : Measurable Z') :
  CondIndepFun (MeasurableSpace.comap X inferInstance) ⋯ Y Z' μ
ProbabilityTheory.CondIndepFun.of_measurable_right.{u_1,
    u_2, u_3, u_4, u_6}
  {α : Type u_1} {β : Type u_2}
  {γ : Type u_3} {δ : Type u_4}
  {δ' : Type u_6} {mα : MeasurableSpace α}
  {mβ : MeasurableSpace β}
  {mγ : MeasurableSpace γ}
  {mδ : MeasurableSpace δ}
  {mδ' : MeasurableSpace δ'}
  [StandardBorelSpace δ'] [Nonempty δ']
  {μ : MeasureTheory.Measure α}
  {X : α → β} {hX : Measurable X}
  {Y : α → γ} {Z : α → δ} {Z' : α → δ'}
  [StandardBorelSpace α]
  [MeasureTheory.IsFiniteMeasure μ]
  (h_indep :
    CondIndepFun
      (MeasurableSpace.comap X
        inferInstance)
      ⋯ Y Z μ)
  (hZ_meas : Measurable Z') :
  CondIndepFun
    (MeasurableSpace.comap X
      inferInstance)
    ⋯ Y Z' μ

Code

lemma CondIndepFun.of_measurable_right
    (h_indep : Y ⟂ᵢ[X, hX; μ] Z) (hZ_meas : Measurable[mδ.comap Z] Z') :
    Y ⟂ᵢ[X, hX; μ] Z'
Used by (1)

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Proof
by
  obtain ⟨ψ, hψ_meas, h_eqZ⟩ : ∃ ψ, Measurable ψ ∧ Z' = ψ ∘ Z := hZ_meas.exists_eq_measurable_comp
  rw [h_eqZ]
  exact h_indep.comp measurable_id hψ_meas