The evolved-measure form of finite-step invariance

AutoSamplingTheory.TechnicalLemmas.Probability.KernelInvariance.bind_pow_eq · theorem · Teaching coverage

Statement

On an arbitrary measurable space A, let K:A→𝓜(A) be a kernel, μ a measure, and suppose μK=μ. For each natural n, binding μ to Kⁿ gives exactly μ. This is the same finite-step invariance result written as an equality of measures instead of the predicate Invariant; it introduces no additional mathematical conclusion.

All objects and hypotheses

• α : Type* with [MeasurableSpace α].
• κ : Kernel α α and μ : Measure α; neither finiteness nor Markovness is assumed.
• hκ : κ.Invariant μ.
• n : ℕ. No omitted section variables or extra hypotheses.

Notation and interpretation

Measurable space and kernel

A measurable space is a set with a specified sigma-algebra of events. A kernel K from A to B assigns to each input a a measure K(a,·) on B, and a↦K(a,E) is measurable whenever E is measurable. A kernel need not assign probability measures unless Markovness is separately assumed.

\[K:A\to\mathcal M(B),\qquad a\mapsto K(a,E)\text{ measurable for every measurable }E\subseteq B.\]

Measure evolved by a kernel; invariance

Write μK for the measure obtained by starting with μ and applying K. Lean writes μ.bind K or K ∘ₘ μ. Invariance says that this evolved measure is exactly μ. If μ is a probability law this means a stationary initial distribution, but the definitions and these theorems also allow arbitrary measures.

\[(\mu K)(E)=\int_A K(a,E)\,\mu(da),\qquad \mu K=\mu.\]

Composition and powers

Lean's L ∘ₖ K applies K first and L second. The multiplicative identity is the identity kernel a↦δ_a, and powers are repeated composition. All integrals here are nonnegative extended-real (lower Lebesgue) integrals; no signed or Bochner integrability is silently assumed.

\[(L\circ_{\!k}K)(a,E)=\int L(b,E)\,K(a,db),\quad K^0(a,\cdot)=\delta_a,\quad K^{n+1}=K^n\circ_{\!k}K.\]

Markov and s-finite kernels

Markov means every output measure has total mass one. An s-finite kernel is a countable sum of finite kernels, where each finite kernel has a finite uniform bound on its total mass across inputs. Markovness supplies this s-finiteness automatically. S-finite is not the same hypothesis as sigma-finite, and is not merely a pointwise finiteness assertion.

\[K(a,B)=1\ \forall a\quad\text{(Markov)};\qquad K=\sum_{n=0}^{\infty}K_n,\quad \forall n\ \exists C_n<\infty\ \forall a,\ K_n(a,B)\le C_n\quad\text{(s-finite)}.\]

Nonnegative weights

ℝ≥0 consists of finite nonnegative real numbers; ℝ≥0∞ also allows infinity and is the codomain of measure values. The coercion wᵢ↦cᵢ puts finite weights into extended nonnegative arithmetic. Zero weights are permitted, including when component masses are infinite; this arithmetic uses 0·∞=0.

\[w_i\in[0,\infty),\qquad c_i=\iota(w_i)\in[0,\infty].\]

Measurable equivalence and pushforward

A measurable equivalence e is a bijection whose forward and inverse maps are both measurable. Its pushforward measure records the original mass of inverse images. This is transport of measures, not a density transformation with an implicit Jacobian.

\[(e_*\mu)(E)=\mu(e^{-1}(E)),\qquad (e^{-1})_*(e_*\mu)=\mu.\]

Mathematical proof

theorem bind_pow_eq {α : Type*} [MeasurableSpace α]
    {κ : Kernel α α} {μ : Measure α}
    (hκ : κ.Invariant μ) (n : ℕ) :
    μ.bind (κ ^ n) = μ

theorem bind_pow_eq {α : Type*} [MeasurableSpace α]
    {κ : Kernel α α} {μ : Measure α}
    (hκ : κ.Invariant μ) (n : ℕ) :
    μ.bind (κ ^ n) = μ :=
  (invariant_pow hκ n).def

end KernelInvariance
end Probability
end TechnicalLemmas
end AutoSamplingTheory

Scope and omitted-condition boundaries

• Same mathematical boundary as invariant_pow; count this as a reuse/notation wrapper, not a new stationary-distribution or mixing theorem.
• A downstream reversible-kernel use still needs the Markov assumption required by Mathlib IsReversible.invariant before this wrapper can be applied.