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ProbabilityTheory.CondIndepFun.of_measurable_left🔗

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of_measurable_left🔗

LemmaProbabilityTheory.CondIndepFun.of_measurable_left
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🔗theorem
ProbabilityTheory.CondIndepFun.of_measurable_left.{u_1, u_2, u_3, u_4,
    u_5}
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4}
  {γ' : Type u_5} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
  {mγ : MeasurableSpace γ} {mδ : MeasurableSpace δ}
  {mγ' : MeasurableSpace γ'} [StandardBorelSpace γ'] [Nonempty γ']
  {μ : MeasureTheory.Measure α} {X : α → β} {hX : Measurable X}
  {Y : α → γ} {Z : α → δ} {Y' : α → γ'} [StandardBorelSpace α]
  [MeasureTheory.IsFiniteMeasure μ]
  (h_indep :
    CondIndepFun (MeasurableSpace.comap X inferInstance) ⋯ Y Z μ)
  (hY_meas : Measurable Y') :
  CondIndepFun (MeasurableSpace.comap X inferInstance) ⋯ Y' Z μ
ProbabilityTheory.CondIndepFun.of_measurable_left.{u_1,
    u_2, u_3, u_4, u_5}
  {α : Type u_1} {β : Type u_2}
  {γ : Type u_3} {δ : Type u_4}
  {γ' : Type u_5} {mα : MeasurableSpace α}
  {mβ : MeasurableSpace β}
  {mγ : MeasurableSpace γ}
  {mδ : MeasurableSpace δ}
  {mγ' : MeasurableSpace γ'}
  [StandardBorelSpace γ'] [Nonempty γ']
  {μ : MeasureTheory.Measure α}
  {X : α → β} {hX : Measurable X}
  {Y : α → γ} {Z : α → δ} {Y' : α → γ'}
  [StandardBorelSpace α]
  [MeasureTheory.IsFiniteMeasure μ]
  (h_indep :
    CondIndepFun
      (MeasurableSpace.comap X
        inferInstance)
      ⋯ Y Z μ)
  (hY_meas : Measurable Y') :
  CondIndepFun
    (MeasurableSpace.comap X
      inferInstance)
    ⋯ Y' Z μ

Code

lemma CondIndepFun.of_measurable_left
    (h_indep : Y ⟂ᵢ[X, hX; μ] Z) (hY_meas : Measurable[mγ.comap Y] Y') :
    Y' ⟂ᵢ[X, hX; μ] Z

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Proof
by
  obtain ⟨φ, hφ_meas, h_eqY⟩ : ∃ φ, Measurable φ ∧ Y' = φ ∘ Y := hY_meas.exists_eq_measurable_comp
  rw [h_eqY]
  exact h_indep.comp hφ_meas measurable_id