ProbabilityTheory.hasCondDistrib_measurableEquiv_comp_right_iff
Lemma
From the authors
HasCondDistrib.comp_right is an equivalence for measurable equivalences.
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 -
e : β ≃ᵐ γEquivalences between measurable spaces. -
κ : Kernel γ ΩA kernel from a measurable spaceαto another measurable spaceβis a measurable functionκ : α → Measure β.IsSFiniteKernel κA kernel is s-finite if it can be written as the sum of countably many finite kernels.
Then
HasCondDistrib Y (⇑e ∘ X) κ μ ↔ HasCondDistrib Y X (κ.comap ⇑e ⋯) μMeasurableSpace : Type u_6 → Type u_6A measurable space is a space equipped with a σ-algebra.
MeasureTheory.SFinite : {α : Type u_1} → {m0 : MeasurableSpace α} → MeasureTheory.Measure α → PropA measure is called s-finite if it is a countable sum of finite measures.
MeasureTheory.Measure : (α : Type u_5) → [MeasurableSpace α] → Type u_5A 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`.
MeasurableEquiv : (α : Type u_6) → (β : Type u_7) → [MeasurableSpace α] → [MeasurableSpace β] → Type (max u_6 u_7)Equivalences between measurable spaces. Main application is the simplification of measurability statements along measurable equivalences.
ProbabilityTheory.IsSFiniteKernel : {α : Type u_1} →
{β : Type u_2} → {mα : MeasurableSpace α} → {mβ : MeasurableSpace β} → ProbabilityTheory.Kernel α β → PropA 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)`.
Iff : Prop → Prop → PropIf and only if, or logical bi-implication. `a ↔ b` means that `a` implies `b` and vice versa. By `propext`, this implies that `a` and `b` are equal and hence any expression involving `a` is equivalent to the corresponding expression with `b` instead. Conventions for notations in identifiers: * The recommended spelling of `↔` in identifiers is `iff`. * The recommended spelling of `<->` in identifiers is `iff` (prefer `↔` over `<->`).
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 → PropPredicate 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 → δFunction 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 βPullback 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_measurableEquiv_comp_right_iff (e : β ≃ᵐ γ) {κ : Kernel γ Ω}
[IsSFiniteKernel κ] :
HasCondDistrib Y (e ∘ X) κ μ ↔ HasCondDistrib Y X (κ.comap e e.measurable) μProof
hasCondDistrib_measurableEmbedding_comp_right_iff e.measurableEmbedding
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, 14 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.