ProbabilityTheory.hasLaw_fst_prod
No docstring.
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α : Type u_1mα : MeasurableSpace αA measurable space is a space equipped with a σ-algebra. -
β : Type u_2mβ : MeasurableSpace β
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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. -
ν : MeasureTheory.Measure βMeasureTheory.IsProbabilityMeasure νA measureμis called a probability measure ifμ univ = 1.
HasLaw Prod.fst μ (μ.prod ν)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.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`.
MeasureTheory.IsProbabilityMeasure : {α : Type u_1} → {m0 : MeasurableSpace α} → MeasureTheory.Measure α → PropA measure `μ` is called a probability measure if `μ univ = 1`.
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`.
Prod.fst : {α : Type u} → {β : Type v} → α × β → αThe first element of a pair.
MeasureTheory.Measure.prod : {α : Type u_4} →
{β : Type u_5} →
[inst : MeasurableSpace α] →
[inst_1 : MeasurableSpace β] → MeasureTheory.Measure α → MeasureTheory.Measure β → MeasureTheory.Measure (α × β)The binary product of measures. They are defined for arbitrary measures, but we basically prove all properties under the assumption that at least one of them is s-finite.
Code
lemma hasLaw_fst_prod (μ : Measure α) (ν : Measure β) [IsProbabilityMeasure ν] :
HasLaw Prod.fst μ (μ.prod ν) where
map_eqProof
Measure.fst_prod
Meaning last changed in v4.35.0-rc2-1-g61e506b (2026-09-22), the 2th 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, 8 external constants
✓ Proved: no sorry anywhere in its closure
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