LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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A kernel has a countably generated stable sub-sigma-field containing a given observable

Statement

Let $P$ be a probability kernel on an arbitrary measurable space $(X, F)$ , and let $f : X \to R$ be measurable. There is a countably generated sub-sigma-field $G \subseteq F$ such that $f$ is $G$ -measurable and $P$ , restricted to events in $G$ , is a probability kernel $P_{G}$ on $(X, G)$ . For every $n \geq 0$ , $x \in X$ , and $B \in G$ , $P_{G}^{n} (x, B) = P^{n} (x, B) .$ If $π$ is invariant for $P$ , then $π ∣_{G}$ is invariant for $P_{G}$ .

Facts & Assumptions

Given: The measurable space, probability kernel, and real measurable function in the Statement.

[F1]

Measure kernel and probability kernel: $P (x, \cdot)$ is a probability and $x \mapsto P (x, B)$ is measurable for every $B \in F$ .

[F2]

Dynkin's pi-lambda theorem: a lambda-system containing a pi-system contains the sigma-field it generates.

[F3]

Monotone convergence for the integral: increasing nonnegative integrands have increasing integrals with the expected limit.

[F4]

Iterated transition kernels: iterated transitions are defined by kernel integration, with the identity kernel at time zero.

[F5]

Invariant and stationary distribution for a Markov kernel: invariance means $\int P (x, B) π (d x) = π (B)$ for every measurable $B$ .

[F6]

A measurable function between measurable spaces: a real function is measurable when inverse images of Borel sets are measurable.

[F7]

Every nonnegative measurable function admits an explicit increasing sequence of simple approximations: nonnegative measurable functions have increasing nonnegative simple approximations.

Proof

Given: The hypotheses in the Statement.

1.1F1F6constructgiven

Let $A_{0}$ be the algebra generated by ${{f < q} : q \in Q}$ . Inductively, let $A_{j + 1}$ be the algebra generated by $A_{j}$ and all sets ${x : P (x, A) > q}$ with $A \in A_{j}$ and $q \in Q$ . These are subsets in $F$ by [F1] and [F6]. Each algebra is countable: enumerate its generators and then all finite Boolean expressions in them. This also explicitly enumerates $A = ⋃_{j \geq 0} A_{j}$ . Put $G = σ (A)$ . Since rational half-lines generate the real Borel sigma-field, $f$ is $G$ -measurable.

2.1F1F2step 1.1

For $A \in A$ , all rational superlevel sets of $P (\cdot, A)$ belong to $G$ by construction. The class $D$ of $B \in G$ for which $P (\cdot, B)$ is $G$ -measurable contains $A$ and $X$ . It is closed under complements because $P (x, B^{c}) = 1 - P (x, B)$ and under countable disjoint unions because $P (x, ⋃_{l} B_{l}) = \sum_{l} P (x, B_{l})$ is a measurable increasing limit. Thus $D$ is a lambda-system, and [F2] gives $D = G$ . The restriction $P_{G}$ therefore is a probability kernel.

3.1F3F4F7step 2.1

For every nonnegative $G$ -measurable $g$ , integration against $P_{G} (x, \cdot)$ agrees with integration against $P (x, \cdot)$ : this holds for indicators by restriction, for nonnegative simple functions by finite additivity of the integral, and for general $g$ by [F7] and [F3]. Both integrals are allowed to be infinite. Induction in [F4], using this integration identity and step 2.1, gives $P_{G}^{n} (x, B) = P^{n} (x, B)$ for every $B \in G$ , including $n = 0$ .

4.1F5step 3.1step 1.1∎

If $π$ is invariant, the same restriction identity for integrals gives $\int P_{G} (x, B) (π ∣_{G}) (d x) = \int P (x, B) π (d x) = π (B)$ for $B \in G$ . This is invariance of $π ∣_{G}$ by [F5]. All generating enumerations in step 1.1 were explicit, so this construction introduces no choice assumption.

Depends on

Used by

Dependency tree · two levels

31 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Meyn and Tweedie, Markov Chains and Stochastic Stability, measurable kernels and geometric-ergodicity reductions (standard reference, not scraped)