AEMeasurable.hasLaw_map
No docstring.
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Ω : Type u_1mΩ : MeasurableSpace ΩA measurable space is a space equipped with a σ-algebra. -
𝓧 : Type u_2m𝓧 : MeasurableSpace 𝓧
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P : MeasureTheory.Measure ΩA 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. -
X : Ω → 𝓧
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hX : AEMeasurable X PA function is almost everywhere measurable if it coincides almost everywhere with a measurable function.
ProbabilityTheory.HasLaw X (MeasureTheory.Measure.map X P) PHasLaw X μ P registers the fact that the random variable X has law μ under the measure P, in other words that P.map X = μ.MeasurableSpace : Type u_6 → Type u_6A measurable space is a space equipped with a σ-algebra.
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`.
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.
ProbabilityTheory.HasLaw : {Ω : Type u_1} →
{𝓧 : Type u_2} →
{mΩ : MeasurableSpace Ω} →
{m𝓧 : MeasurableSpace 𝓧} →
(Ω → 𝓧) → MeasureTheory.Measure 𝓧 → autoParam (MeasureTheory.Measure Ω) ProbabilityTheory.HasLaw._auto_1 → PropThe predicate `HasLaw X μ P` registers the fact that the random variable `X` has law `μ` under the measure `P`, in other words that `P.map X = μ`. We also require `X` to be `AEMeasurable`, to allow for nice interactions with operations on the codomain of `X`. See for instance `HasLaw.comp`, `IndepFun.hasLaw_mul` and `IndepFun.hasLaw_add`.
MeasureTheory.Measure.map : {α : Type u_4} →
{β : Type u_5} →
[inst : MeasurableSpace α] →
[inst_1 : MeasurableSpace β] → (α → β) → MeasureTheory.Measure α → MeasureTheory.Measure βThe pushforward of a measure. If `f` is not an almost everywhere measurable function, we define it to be `0` if `μ = 0`, and to be an arbitrary Dirac mass otherwise. That way we always have `map f 0 = 0`, and the push-forward of a probability measure is always a probability measure.
Code
lemma _root_.AEMeasurable.hasLaw_map {X : Ω → 𝓧} (hX : AEMeasurable X P) :
HasLaw X (P.map X) PProof
⟨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, 5 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.