ProbabilityTheory.hasCondDistrib_deterministic_iff
Lemma
No docstring.
Types
-
α : Type u_1mα : MeasurableSpace αA measurable space is a space equipped with a σ-algebra. -
β : Type u_2mβ : MeasurableSpace β -
Ω : Type u_4mΩ : MeasurableSpace ΩMeasurableEq ΩTypeclass for a measurable spaceαfor which the diagonal ofα × αis measurable.
Given
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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 → Ω
Assuming
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hf : Measurable fA functionfbetween measurable spaces is measurable if the preimage of every measurable set is measurable. -
hX : AEMeasurable X μA function is almost everywhere measurable if it coincides almost everywhere with a measurable function. -
hY : AEMeasurable Y μ
Then
HasCondDistrib Y X (Kernel.deterministic f hf) μ ↔ Y =ᵐ[μ] f ∘ XMeasurableSpace : Type u_6 → Type u_6A measurable space is a space equipped with a σ-algebra.
MeasurableEq : (α : Type u_1) → [MeasurableSpace α] → PropTypeclass for a measurable space `α` for which the diagonal of `α × α` is measurable.
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`.
Measurable : {α : Type u_1} → {β : Type u_2} → [MeasurableSpace α] → [MeasurableSpace β] → (α → β) → PropA function `f` between measurable spaces is measurable if the preimage of every measurable set is measurable.
AEMeasurable : {α : Type u_1} →
{β : Type u_2} →
[MeasurableSpace β] →
{_m : MeasurableSpace α} → (α → β) → autoParam (MeasureTheory.Measure α) AEMeasurable._auto_1 → PropA function is almost everywhere measurable if it coincides almost everywhere with a measurable function. A similar notion is `MeasureTheory.NullMeasurable`. That notion is equivalent to `AEMeasurable` if the σ-algebra on the codomain is countably generated, but weaker in general.
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`.
ProbabilityTheory.Kernel.deterministic : {α : Type u_1} →
{β : Type u_2} →
{mα : MeasurableSpace α} → {mβ : MeasurableSpace β} → (f : α → β) → Measurable f → ProbabilityTheory.Kernel α βKernel which to `a` associates the dirac measure at `f a`. This is a Markov kernel.
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`.
Code
lemma hasCondDistrib_deterministic_iff [MeasurableEq Ω] [SFinite μ] {f : β → Ω}
(hf : Measurable f) (hX : AEMeasurable X μ) (hY : AEMeasurable Y μ) :
HasCondDistrib Y X (Kernel.deterministic f hf) μ ↔ Y =ᵐ[μ] f ∘ XProof
by refine ⟨ae_eq_of_hasCondDistrib_deterministic hf hX hY, fun h ↦ ?_⟩ refine HasCondDistrib.congr ?_ ?_ h (X := X) · exact hasCondDistrib_comp_self hf hX · 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, 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.