ProbabilityTheory.HasLaw.hasCondDistrib_const
From the authors
Alias of the reverse direction of ProbabilityTheory.hasCondDistrib_const_iff.
**Alias** of the reverse direction of `ProbabilityTheory.hasCondDistrib_const_iff`. --- Conditioning on a constant is the same as having law `κ b`.
Conditioning on a constant is the same as having law κ b.
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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 -
Ω : Type u_4mΩ : MeasurableSpace Ω
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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.IsProbabilityMeasure μA measureμis called a probability measure ifμ univ = 1. -
Y : α → Ω -
κ : Kernel β 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. -
b : β
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HasLaw Y (κ b) μThe predicateHasLaw X μ Pregisters the fact that the random variableXhas lawμunder the measureP, in other words thatP.map X = μ.
HasCondDistrib Y (fun x => b) κ μY given X under the measure P is equal to the kernel κ.MeasurableSpace : Type u_6 → Type u_6A measurable space is a space equipped with a σ-algebra.
MeasureTheory.IsProbabilityMeasure : {α : Type u_1} → {m0 : MeasurableSpace α} → MeasureTheory.Measure α → PropA measure `μ` is called a probability measure if `μ univ = 1`.
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`.
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)`.
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`.
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`.
Code
alias ⟨HasCondDistrib.hasLaw_of_const', HasLaw.hasCondDistrib_const⟩
Proof
hasCondDistrib_const_iff
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
Audit surface: 1 project declarations, 11 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.