LeanMachineLearning

ProbabilityTheory.cond_of_condIndepFun🔗

Lemma

No docstring.

🔗theorem
ProbabilityTheory.cond_of_condIndepFun.{u_1, u_2, u_5, u_6}
  {α : Type u_1} {β : Type u_2} {Ω : Type u_5} {Ω' : Type u_6}
  {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α}
  {mβ : MeasurableSpace β} [MeasurableSpace Ω] [StandardBorelSpace Ω]
  [Nonempty Ω] [mΩ' : MeasurableSpace Ω'] [StandardBorelSpace Ω']
  {X : α → β} {Y : α → Ω} {Z : α → Ω'} [StandardBorelSpace α]
  [StandardBorelSpace β] [Nonempty β] [Countable β] [Countable Ω']
  [MeasureTheory.IsZeroOrProbabilityMeasure μ] (hZ : Measurable Z)
  (h : CondIndepFun (MeasurableSpace.comap Z inferInstance) ⋯ Y X μ)
  (hX : Measurable X) (hY : Measurable Y) {b : β} {ω : Ω'}
  (hμ : μ (Z ⁻¹' {ω} ∩ X ⁻¹' {b}) ≠ 0) :
  𝓛[Y | Z ⁻¹' {ω} ∩ X ⁻¹' {b}; μ] = 𝓛[Y | Z in {ω}; μ]
ProbabilityTheory.cond_of_condIndepFun.{u_1,
    u_2, u_5, u_6}
  {α : Type u_1} {β : Type u_2}
  {Ω : Type u_5} {Ω' : Type u_6}
  {mα : MeasurableSpace α}
  {μ : MeasureTheory.Measure α}
  {mβ : MeasurableSpace β}
  [MeasurableSpace Ω]
  [StandardBorelSpace Ω] [Nonempty Ω]
  [mΩ' : MeasurableSpace Ω']
  [StandardBorelSpace Ω'] {X : α → β}
  {Y : α → Ω} {Z : α → Ω'}
  [StandardBorelSpace α]
  [StandardBorelSpace β] [Nonempty β]
  [Countable β] [Countable Ω']
  [MeasureTheory.IsZeroOrProbabilityMeasure
      μ]
  (hZ : Measurable Z)
  (h :
    CondIndepFun
      (MeasurableSpace.comap Z
        inferInstance)
      ⋯ Y X μ)
  (hX : Measurable X) (hY : Measurable Y)
  {b : β} {ω : Ω'}
  (hμ : μ (Z ⁻¹' {ω} ∩ X ⁻¹' {b}) ≠ 0) :
  𝓛[Y | Z ⁻¹' {ω} ∩ X ⁻¹' {b}; μ] =
    𝓛[Y | Z in {ω}; μ]

Code

lemma cond_of_condIndepFun [StandardBorelSpace α] [StandardBorelSpace β] [Nonempty β] [Countable β]
    [Countable Ω']
    [IsZeroOrProbabilityMeasure μ]
    (hZ : Measurable Z)
    (h : CondIndepFun (MeasurableSpace.comap Z inferInstance) hZ.comap_le Y X μ)
    (hX : Measurable X) (hY : Measurable Y) {b : β} {ω : Ω'}
    (hμ : μ (Z ⁻¹' {ω} ∩ X ⁻¹' {b}) ≠ 0) :
    (μ[|Z ⁻¹' {ω} ∩ X ⁻¹' {b}]).map Y = (μ[|Z ⁻¹' {ω}]).map Y
Proof
by
  symm at h
  have h := (condIndepFun_iff_condDistrib_prod_ae_eq_prodMkRight hY hX hZ).mp h
  have h_left := condDistrib_ae_eq_cond (hZ.prodMk hX) hY (μ := μ)
  have h_right := condDistrib_ae_eq_cond hZ hY (μ := μ)
  rw [Filter.EventuallyEq, ae_iff_of_countable] at h h_left h_right
  specialize h (ω, b)
  specialize h_left (ω, b)
  specialize h_right ω
  rw [Measure.map_apply (by fun_prop) (measurableSet_singleton _)] at h h_left h_right
  rw [← Set.singleton_prod_singleton, Set.mk_preimage_prod] at h h_left
  have hZ_ne : μ (Z ⁻¹' {ω}) ≠ 0 := fun h ↦ hμ (measure_mono_null Set.inter_subset_left h)
  rw [← h_right hZ_ne, ← h_left hμ, h hμ]
  simp

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Meaning last changed in v4.34.0-rc2-1-g439785b (2026-08-23), the 3th recorded change.

Self-contained, with its dependencies inlined and proofs replaced by sorry: download the raw file · open it in the Lean web editor.

Dependency graph

Nothing to draw. Its statement rests on no other declaration in this project, and names nothing from a package left unaudited — so the graph is this declaration alone. That is the answer, not a missing picture.

Audit surface: 0 project declarations, 28 external constants

✓ Proved: no sorry anywhere in its closure

This is the tool's own reading of one build's recorded axioms, and it is not robust against an author who wants it to pass. Checking meant to be relied on should go through Comparator, which replays the proof through the kernel from an export against an explicit list of permitted axioms.