ProbabilityTheory.Kernel.hasLaw_eval_zero_trajMeasure
Lemma
No docstring.
Given
-
X : ℕ → Type u_2(n : ℕ) → MeasurableSpace (X n)A measurable space is a space equipped with a σ-algebra. -
κ : (n : ℕ) → Kernel ((i : ↥(Finset.Iic n)) → X ↑i) (X (n + 1))A kernel from a measurable spaceαto another measurable spaceβis a measurable functionκ : α → Measure β.∀ (n : ℕ), IsMarkovKernel (κ n)A kernel is a Markov kernel if every measure in its image is a probability measure. -
μ₀ : MeasureTheory.Measure (X 0)A 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.
Then
HasLaw (fun x => x 0) μ₀ (trajMeasure μ₀ κ)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.
Nat : TypeThe natural numbers, starting at zero. This type is special-cased by both the kernel and the compiler, and overridden with an efficient implementation. Both use a fast arbitrary-precision arithmetic library (usually [GMP](https://gmplib.org/)); at runtime, `Nat` values that are sufficiently small are unboxed.
ProbabilityTheory.IsMarkovKernel : {α : Type u_1} →
{β : Type u_2} → {mα : MeasurableSpace α} → {mβ : MeasurableSpace β} → ProbabilityTheory.Kernel α β → PropA kernel is a Markov kernel if every measure in its image is a probability measure.
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)`.
Finset.Iic : {α : Type u_1} → [inst : Preorder α] → [LocallyFiniteOrderBot α] → α → Finset αThe finset $(-∞, b]$ of elements `x` such that `x ≤ b`. Basically `Set.Iic b` as a finset.
HAdd.hAdd : {α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : HAdd α β γ] → α → β → γ`a + b` computes the sum of `a` and `b`. The meaning of this notation is type-dependent. Conventions for notations in identifiers: * The recommended spelling of `+` in identifiers is `add`.
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.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.Kernel.trajMeasure : {X : ℕ → Type u_1} →
[inst : (n : ℕ) → MeasurableSpace (X n)] →
MeasureTheory.Measure (X 0) →
(κ : (n : ℕ) → ProbabilityTheory.Kernel ((i : ↥(Finset.Iic n)) → X ↑i) (X (n + 1))) →
[∀ (n : ℕ), ProbabilityTheory.IsMarkovKernel (κ n)] → MeasureTheory.Measure ((n : ℕ) → X n)Distribution of the trajectory obtained by starting with `μ₀` and iterating the kernels `κ`.
Code
lemma hasLaw_eval_zero_trajMeasure : HasLaw (fun x ↦ x 0) μ₀ (trajMeasure μ₀ κ) where aemeasurable
Proof
(measurable_pi_apply 0).aemeasurable
map_eq := by
have h := trajMeasure_map_frestrictLe (κ := κ) (μ₀ := μ₀) 0
rw [partialTraj_self, Measure.id_comp] at h
have h2 := congrArg (Measure.map (MeasurableEquiv.piUnique (fun i : Iic 0 ↦ X i))) h
rw [Measure.map_map (MeasurableEquiv.measurable _) (by fun_prop)] at h2
exact h2.trans (MeasurableEquiv.map_map_symm _)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, 24 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.