`ProbabilityTheory.condDistrib_prod_self_left`🔗

Lemma

No docstring.

theorem

ProbabilityTheory.condDistrib_prod_self_left.{u_1, u_2, u_3}
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α}
  {μ : MeasureTheory.Measure α} {mβ : MeasurableSpace β}
  {mγ : MeasurableSpace γ} {X : α → β} {T : α → γ}
  [MeasureTheory.IsFiniteMeasure μ] [StandardBorelSpace β] [Nonempty β]
  [StandardBorelSpace γ] [Nonempty γ] (hX : AEMeasurable X μ)
  (hT : AEMeasurable T μ) :
  ⇑𝓛[fun ω => (X ω, T ω) | T; μ] =ᵐ[MeasureTheory.Measure.map T μ]
    ⇑(Kernel.prod 𝓛[X | T; μ] Kernel.id)
ProbabilityTheory.condDistrib_prod_self_left.{u_1,
    u_2, u_3}
  {α : Type u_1} {β : Type u_2}
  {γ : Type u_3} {mα : MeasurableSpace α}
  {μ : MeasureTheory.Measure α}
  {mβ : MeasurableSpace β}
  {mγ : MeasurableSpace γ} {X : α → β}
  {T : α → γ}
  [MeasureTheory.IsFiniteMeasure μ]
  [StandardBorelSpace β] [Nonempty β]
  [StandardBorelSpace γ] [Nonempty γ]
  (hX : AEMeasurable X μ)
  (hT : AEMeasurable T μ) :
  ⇑𝓛[fun ω => (X ω, T ω) | T;
        μ] =ᵐ[MeasureTheory.Measure.map T
      μ]
    ⇑(Kernel.prod 𝓛[X | T; μ] Kernel.id)

Code

lemma condDistrib_prod_self_left [StandardBorelSpace β] [Nonempty β] [StandardBorelSpace γ]
    [Nonempty γ]
    (hX : AEMeasurable X μ) (hT : AEMeasurable T μ) :
    condDistrib (fun ω ↦ (X ω, T ω)) T μ =ᵐ[μ.map T] condDistrib X T μ ×ₖ Kernel.id

Proof

by
  have h_prod := condDistrib_prod_left hX hT hT (μ := μ)
  have h_fst := condDistrib_comp_self (μ := μ) (fun ω ↦ (T ω, X ω)) (f := Prod.fst) (by fun_prop)
  rw [(compProd_map_condDistrib hX).symm] at h_fst
  have h_fst' := (Measure.ae_compProd_iff (Kernel.measurableSet_eq _ _)).mp h_fst
  filter_upwards [h_prod, h_fst'] with z hz1 hz2
  rw [hz1]
  simp only [Kernel.deterministic_apply] at hz2
  change ∀ᵐ y ∂(condDistrib X T μ z), condDistrib T (fun ω ↦ (T ω, X ω)) μ (z, y) = Measure.dirac z
    at hz2
  ext t ht
  rw [Kernel.compProd_apply ht]
  calc ∫⁻ y, condDistrib T (fun ω ↦ (T ω, X ω)) μ (z, y) (Prod.mk y ⁻¹' t) ∂condDistrib X T μ z
  _ = ∫⁻ y, (Measure.dirac z) (Prod.mk y ⁻¹' t) ∂condDistrib X T μ z :=
    lintegral_congr_ae (hz2.mono fun y hy ↦ by simp only [hy])
  _ = ∫⁻ y, (Prod.mk y ⁻¹' t).indicator 1 z ∂condDistrib X T μ z :=
    lintegral_congr fun y ↦ Measure.dirac_apply' _ (ht.preimage (by fun_prop))
  _ = (condDistrib X T μ z) ((fun y ↦ (y, z)) ⁻¹' t) := by
    rw [← lintegral_indicator_one (ht.preimage (by fun_prop : Measurable fun y ↦ (y, z)))]
    exact lintegral_congr fun _ ↦ rfl
  _ = ((condDistrib X T μ ×ₖ Kernel.id) z) t := by
    rw [Kernel.prod_apply, Kernel.id_apply, Measure.prod_apply_symm ht, lintegral_dirac]

Actions: Source · Open Issue

Meaning last changed in v4.34.0-rc2-1-g439785b (2026-08-23), the 4th 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, 22 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.

ProbabilityTheory.condDistrib_prod_self_left🔗

`ProbabilityTheory.condDistrib_prod_self_left`🔗