LeanMachineLearning

ProbabilityTheory.CondIndepFun.prod_right🔗

Lemma

No docstring.

🔗theorem
ProbabilityTheory.CondIndepFun.prod_right.{u_1, u_2, u_3, u_4}
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4}
  {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α}
  {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ}
  {mδ : MeasurableSpace δ} [MeasureTheory.IsFiniteMeasure μ]
  [StandardBorelSpace α] [StandardBorelSpace β] [Nonempty β]
  [StandardBorelSpace γ] [Nonempty γ] [StandardBorelSpace δ]
  [Nonempty δ] {X : α → β} {Y : α → γ} {Z : α → δ} (hX : Measurable X)
  (hY : Measurable Y) (hZ : Measurable Z)
  (h : CondIndepFun (MeasurableSpace.comap Z inferInstance) ⋯ X Y μ) :
  CondIndepFun (MeasurableSpace.comap Z inferInstance) ⋯ X
    (fun ω => (Y ω, Z ω)) μ
ProbabilityTheory.CondIndepFun.prod_right.{u_1,
    u_2, u_3, u_4}
  {α : Type u_1} {β : Type u_2}
  {γ : Type u_3} {δ : Type u_4}
  {mα : MeasurableSpace α}
  {μ : MeasureTheory.Measure α}
  {mβ : MeasurableSpace β}
  {mγ : MeasurableSpace γ}
  {mδ : MeasurableSpace δ}
  [MeasureTheory.IsFiniteMeasure μ]
  [StandardBorelSpace α]
  [StandardBorelSpace β] [Nonempty β]
  [StandardBorelSpace γ] [Nonempty γ]
  [StandardBorelSpace δ] [Nonempty δ]
  {X : α → β} {Y : α → γ} {Z : α → δ}
  (hX : Measurable X) (hY : Measurable Y)
  (hZ : Measurable Z)
  (h :
    CondIndepFun
      (MeasurableSpace.comap Z
        inferInstance)
      ⋯ X Y μ) :
  CondIndepFun
    (MeasurableSpace.comap Z
      inferInstance)
    ⋯ X (fun ω => (Y ω, Z ω)) μ

Code

lemma CondIndepFun.prod_right [StandardBorelSpace α] [StandardBorelSpace β] [Nonempty β]
    [StandardBorelSpace γ] [Nonempty γ] [StandardBorelSpace δ] [Nonempty δ]
    {X : α → β} {Y : α → γ} {Z : α → δ}
    (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)
    (h : X ⟂ᵢ[Z, hZ; μ] Y) :
    X ⟂ᵢ[Z, hZ; μ] (fun ω ↦ (Y ω, Z ω))
Proof
by
  rw [condIndepFun_iff_condDistrib_prod_ae_eq_prodMkRight hY hX hZ,
    condDistrib_ae_eq_iff_measure_eq_compProd _ (by fun_prop)] at h
  rw [condIndepFun_iff_condDistrib_prod_ae_eq_prodMkRight (by fun_prop) hX hZ,
    condDistrib_ae_eq_iff_measure_eq_compProd _ (by fun_prop)]
  -- Key: condDistrib (Y, Z) Z μ z = (condDistrib Y Z μ z).map (y ↦ (y, z))
  have h_cond : condDistrib (fun ω ↦ (Y ω, Z ω)) Z μ =ᵐ[μ.map Z]
      fun z ↦ (condDistrib Y Z μ z).map (fun y ↦ (y, z)) := by
    suffices condDistrib (fun ω ↦ (Y ω, Z ω)) Z μ =ᵐ[μ.map Z]
        (condDistrib Y Z μ) ×ₖ Kernel.id by
      refine this.trans (ae_of_all _ fun x ↦ ?_)
      simp only
      rw [Kernel.prod_apply, Kernel.id_apply]
      ext s hs
      rw [Measure.map_apply (by fun_prop) hs, Measure.prod_apply_symm hs, lintegral_dirac]
    exact condDistrib_prod_self_left hY.aemeasurable hZ.aemeasurable
  -- Main calculation
  calc μ.map (fun x ↦ ((Z x, X x), (Y x, Z x)))
  _ = (μ.map (fun x ↦ ((Z x, X x), Y x))).map (fun p ↦ (p.1, (p.2, p.1.1))) := by
      rw [Measure.map_map (by fun_prop) (by fun_prop)]; rfl
  _ = (μ.map (fun ω ↦ (Z ω, X ω)) ⊗ₘ (condDistrib Y Z μ).prodMkRight β).map
        (fun p ↦ (p.1, (p.2, p.1.1))) := by rw [h]
  _ = μ.map (fun ω ↦ (Z ω, X ω)) ⊗ₘ (condDistrib (fun ω ↦ (Y ω, Z ω)) Z μ).prodMkRight β := by
    ext s hs
    rw [Measure.map_apply (by fun_prop) hs,
      Measure.compProd_apply (hs.preimage (by fun_prop)), Measure.compProd_apply hs]
    have h_cond' : ∀ᵐ p ∂(μ.map (fun ω ↦ (Z ω, X ω))),
        condDistrib (fun ω ↦ (Y ω, Z ω)) Z μ p.1 =
          (condDistrib Y Z μ p.1).map (fun y ↦ (y, p.1)) := by
      have h_fst : (μ.map (fun ω ↦ (Z ω, X ω))).map Prod.fst = μ.map Z := by
        rw [Measure.map_map (by fun_prop) (by fun_prop)]; rfl
      rw [← h_fst] at h_cond
      exact mem_ae_of_mem_ae_map (by fun_prop) h_cond
    refine lintegral_congr_ae (h_cond'.mono fun ⟨z, x⟩ hzx ↦ ?_)
    simp only [Kernel.prodMkRight_apply, hzx,
      Measure.map_apply (by fun_prop : Measurable fun y ↦ (y, z))
        (hs.preimage (by fun_prop : Measurable (Prod.mk (z, x))))]
    congr 1

Actions: Source · Open Issue

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, 13 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.