ProbabilityTheory.hasCondDistrib_prodMk_left_unique_iff
Lemma
From the authors
Conditioning on a pair whose first component takes values in a type with a unique element is the same as conditioning on the second component.
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 Ω -
δ : Type u_6mδ : MeasurableSpace δUnique δUnique αexpresses thatαis a type with a unique termdefault.
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 -
U : α → δ -
η : Kernel (δ × β) ΩA kernel from a measurable spaceαto another measurable spaceβis a measurable functionκ : α → Measure β.IsSFiniteKernel ηA kernel is s-finite if it can be written as the sum of countably many finite kernels.
Then
HasCondDistrib Y (fun ω => (U ω, X ω)) η μ ↔ HasCondDistrib Y X (η.sectR default) μMeasurableSpace : Type u_6 → Type u_6A measurable space is a space equipped with a σ-algebra.
Unique : Sort u → Sort (max 1 u)`Unique α` expresses that `α` is a type with a unique term `default`. This is implemented as a type, rather than a `Prop`-valued predicate, for good definitional properties of the default term.
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`.
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)`.
Prod : Type u → Type v → Type (max u v)The product type, usually written `α × β`. Product types are also called pair or tuple types. Elements of this type are pairs in which the first element is an `α` and the second element is a `β`. Products nest to the right, so `(x, y, z) : α × β × γ` is equivalent to `(x, (y, z)) : α × (β × γ)`. Conventions for notations in identifiers: * The recommended spelling of `×` in identifiers is `Prod`.
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`.
Prod.mk : {α : Type u} → {β : Type v} → α → β → α × βConstructs a pair. This is usually written `(x, y)` instead of `Prod.mk x y`. Conventions for notations in identifiers: * The recommended spelling of `(a, b)` in identifiers is `mk`.
ProbabilityTheory.Kernel.sectR : {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) γ → α → ProbabilityTheory.Kernel β γDefine a `Kernel β γ` from a `Kernel (α × β) γ` by taking the comap of `fun b ↦ (a, b)` for a given `a : α`.
Inhabited.default : {α : Sort u} → [self : Inhabited α] → α`default` is a function that produces a "default" element of any `Inhabited` type. This element does not have any particular specified properties, but it is often an all-zeroes value.
Code
lemma hasCondDistrib_prodMk_left_unique_iff {U : α → δ} {η : Kernel (δ × β) Ω}
[IsSFiniteKernel η] :
HasCondDistrib Y (fun ω ↦ (U ω, X ω)) η μ ↔ HasCondDistrib Y X (η.sectR default) μProof
by have hU : U = fun _ ↦ default := funext fun _ ↦ Unique.eq_default _ subst hU exact hasCondDistrib_measurableEmbedding_comp_right_iff (measurableEmbedding_prodMk_left default)
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.