`ProbabilityTheory.HasCondDistrib.of_measurableEmbedding_comp_right`🔗

Lemma

From the authors

Converse of HasCondDistrib.comp_right for a measurable embedding.

Types

α : Type u_1mα : MeasurableSpace αA measurable space is a space equipped with a σ-algebra.
β : Type u_2mβ : MeasurableSpace β
γ : Type u_3mγ : MeasurableSpace γ
Ω : Type u_4mΩ : MeasurableSpace Ω

Given

μ : MeasureTheory.Measure αA measure is defined to be an outer measure that is countably additive on measurable sets, with the additional assumption that the outer measure is the canonical extension of the restricted measure.MeasureTheory.SFinite μA measure is called s-finite if it is a countable sum of finite measures.
X : α → β
Y : α → Ω
f : β → γ
κ : Kernel γ ΩA kernel from a measurable space α to another measurable space β is a measurable function κ : α → Measure β.IsSFiniteKernel κA kernel is s-finite if it can be written as the sum of countably many finite kernels.

Assuming

hf : MeasurableEmbedding fA map f : α → β is called a *measurable embedding* if it is injective, measurable, and sends measurable sets to measurable sets.
h : HasCondDistrib Y (f ∘ X) κ μPredicate stating that the conditional distribution of Y given X under the measure P is equal to the kernel κ.

Then

HasCondDistrib Y X (κ.comap f ⋯) μ

MeasurableSpace : Type u_6 → Type u_6

A measurable space is a space equipped with a σ-algebra.

MeasureTheory.SFinite : {α : Type u_1} → {m0 : MeasurableSpace α} → MeasureTheory.Measure α → Prop

A measure is called s-finite if it is a countable sum of finite measures.

MeasureTheory.Measure : (α : Type u_5) → [MeasurableSpace α] → Type u_5

A measure is defined to be an outer measure that is countably additive on
measurable sets, with the additional assumption that the outer measure is the canonical
extension of the restricted measure.

The measure of a set `s`, denoted `μ s`, is an extended nonnegative real. The real-valued version
is written `μ.real s`.

ProbabilityTheory.IsSFiniteKernel : {α : Type u_1} →
  {β : Type u_2} → {mα : MeasurableSpace α} → {mβ : MeasurableSpace β} → ProbabilityTheory.Kernel α β → Prop

A kernel is s-finite if it can be written as the sum of countably many finite kernels.

ProbabilityTheory.Kernel : (α : Type u_1) → (β : Type u_2) → [MeasurableSpace α] → [MeasurableSpace β] → Type (max u_1 u_2)

A kernel from a measurable space `α` to another measurable space `β` is a measurable function
`κ : α → Measure β`. The measurable space structure on `MeasureTheory.Measure β` is given by
`MeasureTheory.Measure.instMeasurableSpace`. A map `κ : α → MeasureTheory.Measure β` is measurable
iff `∀ s : Set β, MeasurableSet s → Measurable (fun a ↦ κ a s)`.

MeasurableEmbedding : {α : Type u_1} → {β : Type u_2} → [MeasurableSpace α] → [MeasurableSpace β] → (α → β) → Prop

A map `f : α → β` is called a *measurable embedding* if it is injective, measurable, and sends
measurable sets to measurable sets. The latter assumption can be replaced with “`f` has measurable
inverse `g : Set.range f → α`”, see `MeasurableEmbedding.measurable_rangeSplitting`,
`MeasurableEmbedding.of_measurable_inverse_range`, and
`MeasurableEmbedding.of_measurable_inverse`.

One more interpretation: `f` is a measurable embedding if it defines a measurable equivalence to its
range and the range is a measurable set. One implication is formalized as
`MeasurableEmbedding.equivRange`; the other one follows from
`MeasurableEquiv.measurableEmbedding`, `MeasurableEmbedding.subtype_coe`, and
`MeasurableEmbedding.comp`.

ProbabilityTheory.HasCondDistrib : {Ω : Type u_1} →
  {𝓧 : Type u_2} →
    {𝓨 : Type u_3} →
      {mΩ : MeasurableSpace Ω} →
        {m𝓧 : MeasurableSpace 𝓧} →
          {m𝓨 : MeasurableSpace 𝓨} → (Ω → 𝓨) → (Ω → 𝓧) → ProbabilityTheory.Kernel 𝓧 𝓨 → MeasureTheory.Measure Ω → Prop

Predicate stating that the conditional distribution of `Y` given `X` under the measure `P`
is equal to the kernel `κ`.

Function.comp : {α : Sort u} → {β : Sort v} → {δ : Sort w} → (β → δ) → (α → β) → α → δ

Function composition, usually written with the infix operator `∘`. A new function is created from
two existing functions, where one function's output is used as input to the other.

Examples:
 * `Function.comp List.reverse (List.drop 2) [3, 2, 4, 1] = [1, 4]`
 * `(List.reverse ∘ List.drop 2) [3, 2, 4, 1] = [1, 4]`


Conventions for notations in identifiers:

 * The recommended spelling of `∘` in identifiers is `comp`.

ProbabilityTheory.Kernel.comap : {α : Type u_1} →
  {β : Type u_2} →
    {mα : MeasurableSpace α} →
      {mβ : MeasurableSpace β} →
        {γ : Type u_4} →
          {mγ : MeasurableSpace γ} →
            ProbabilityTheory.Kernel α β → (g : γ → α) → Measurable g → ProbabilityTheory.Kernel γ β

Pullback of a kernel, such that for each set s `comap κ g hg c s = κ (g c) s`.
We include measurability in the assumptions instead of using junk values
to make sure that typeclass inference can infer that the `comap` of a Markov kernel
is again a Markov kernel.

Code

lemma HasCondDistrib.of_measurableEmbedding_comp_right {f : β → γ} (hf : MeasurableEmbedding f)
    {κ : Kernel γ Ω} [IsSFiniteKernel κ] (h : HasCondDistrib Y (f ∘ X) κ μ) :
    HasCondDistrib Y X (κ.comap f hf.measurable) μ

Proof

by
  have hX : AEMeasurable X μ := hf.aemeasurable_comp_iff.mp h.aemeasurable_fst
  have hY : AEMeasurable Y μ := h.aemeasurable_snd
  have hfm : Measurable (Prod.map f (id : Ω → Ω)) := hf.measurable.prodMap measurable_id
  refine ⟨hX.prodMk hY, (hf.prodMap MeasurableEmbedding.id).map_injective ?_⟩
  rw [AEMeasurable.map_map_of_aemeasurable hfm.aemeasurable (by fun_prop)]
  calc μ.map (Prod.map f id ∘ fun ω ↦ (X ω, Y ω))
  _ = μ.map (f ∘ X) ⊗ₘ κ := h.map_eq
  _ = (μ.map X).map f ⊗ₘ κ := by
    rw [AEMeasurable.map_map_of_aemeasurable hf.measurable.aemeasurable hX]
  _ = (μ.map X ⊗ₘ κ.comap f hf.measurable).map (Prod.map f id) := by
    symm
    ext s hs
    rw [Measure.map_apply hfm hs, Measure.compProd_apply (hs.preimage hfm),
      Measure.compProd_apply hs,
      lintegral_map (Kernel.measurable_kernel_prodMk_left hs) hf.measurable]
    rfl

Meaning last changed in v4.34.0-rc2-76-g565f652 (2026-09-10).

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, 10 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.HasCondDistrib.of_measurableEmbedding_comp_right🔗

`ProbabilityTheory.HasCondDistrib.of_measurableEmbedding_comp_right`🔗