ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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A deterministic dynamical system as a Markov kernel

Statement

Assume Choice. If $T : (E, E) \to (E, E)$ is measurable, then $K (x, A) = 1_{A} (T x)$ is a probability kernel. Every adapted process satisfying $X_{n + 1} = T (X_{n})$ almost surely is a $K$ -chain.

Facts & Assumptions

Given: Choice, the measurable map $T$ , and the stated adapted process.

[F1]

A probability kernel has probability-measure sections and measurable evaluations. (Measure kernel and probability kernel)

[F2]

The Markov condition is $P (X_{n + 1} \in A ∣ F_{n}) = K (X_{n}, A)$ . (Time-homogeneous Markov chain with transition kernel)

Verification

1.1

For fixed $x$ , $K (x, \cdot) = δ_{T x}$ is a probability measure. For fixed [F1] $A$ , $K (x, A) = 1_{T^{- 1} A} (x)$ is measurable. Thus [F1] holds, including empty and full $A$ and a one-point state space.

2.1

For $A \in E$ , [F2, step 1.1] $1_{{X_{n + 1} \in A}} = 1_{A} (T X_{n}) = K (X_{n}, A) a.s.$ The last variable is $F_{n}$ -measurable, so it is its own conditional expectation given $F_{n}$ . By [F2], $X$ is a $K$ -chain. This also covers $n = 0$ , constant maps, fixed points, and deterministic cycles. Choice is used only for the conditional-expectation class in [F2]; the kernel construction is choice-free.

F2step 1.1∎

Depends on

Used by

Identical one-time marginals do not determine a Markov chain Counterexample

Dependency tree · two levels

9 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

Durrett, Probability: Theory and Examples, Section 5.1 (standard reference, not scraped)