Pipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

✓ 1 result · all verified · 1 also independently AI-judged

Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs; all 1 also cleared it.

General-state Markov kernels, invariance, and reversibility

1 · Prerequisites

2 · Summary

For a probability measure $π$ and a pointwise probability kernel $P$ on any measurable state space, the joint flow defined in General-state reversibility by symmetric flow measure is a probability measure on the product sigma-algebra. Reversibility means that this measure is symmetric under swapping its coordinates; it does not require a density or assume invariance in advance. Integrating its nonnegative indicator functions over the two coordinate marginals proves that reversible kernels preserve $π$ (General-state reversibility implies invariance). The existing detailed-balance and convergence results on the stationary Markov-chain page remain countable-state statements; this page makes no general-state ergodicity or convergence claim.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6-luna)Open item page →

General-state reversibility by symmetric flow measure

Definition

Let $(X, X)$ be a measurable space, let $P$ be a probability kernel on it, meaning that $P (x, \cdot)$ is a probability measure for each $x$ and $x \mapsto P (x, A)$ is measurable for each $A \in X$ (Measure kernel and probability kernel), and let $π$ be a probability measure, that is a countably additive measure of total mass one (Probability measures and probability spaces). The product sigma-algebra $X \otimes X$ is generated by measurable rectangles (The product sigma-algebra and its finite iterates). The joint flow measure of $(π, P)$ is the probability measure $Λ_{π, P}$ on $(X \times X, X \otimes X)$ given, for each $C \in X \otimes X$ , by

$Λ_{π, P} (C) : = \int_{X} P (x, C_{x}) π (d x), C_{x} : = {y \in X : (x, y) \in C} .$

Existence, countable additivity, and the nonnegative iterated-integral identity for this measure follow from A prior and model kernel define a joint law. Let $τ (x, y) : = (y, x)$ be the coordinate-swap map. It is measurable because the preimage of a measurable rectangle $A \times B$ is $B \times A$ and such rectangles generate the product sigma-algebra. Define the transpose measure by $Λ_{π, P}^{⊤} (C) : = Λ_{π, P} (τ^{- 1} (C))$ .

The kernel $P$ is reversible with respect to $π$ when $Λ_{π, P} = Λ_{π, P}^{⊤} .$ This definition uses equality of measures and requires no density, standard Borel assumption, or prior assumption that $π$ is invariant.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

General-state reversibility implies invariance

Statement

Let $(X, X)$ be a measurable space, $P$ a probability kernel (Measure kernel and probability kernel) on it, and $π$ a probability measure (Probability measures and probability spaces) on it. If $P$ is reversible with respect to $π$ in the sense of General-state reversibility by symmetric flow measure, then $π$ is invariant for $P$ : for every $A \in X$ ,

$\int_{X} P (x, A) π (d x) = π (A) .$

Facts & Assumptions

Given: A measurable space $(X, X)$ , a pointwise probability kernel $P$ , and a probability measure $π$ for which the joint flow measure $Λ_{π, P}$ is symmetric.

[F1]

The joint flow measure is the probability measure on the product sigma-algebra defined by $Λ_{π, P} (C) = \int P (x, C_{x}) π (d x)$ , and reversibility means $Λ_{π, P} = Λ_{π, P}^{⊤}$ under coordinate swap. (General-state reversibility by symmetric flow measure)

[F2]

The joint-law construction also gives, for every nonnegative product- measurable $h$ , $\int h d Λ_{π, P} = \int [\int h (x, y) P (x, d y)] π (d x)$ ; in particular its rectangle formula holds. (A prior and model kernel define a joint law)

[F3]

A probability $π$ is invariant for $P$ exactly when $\int P (x, A) π (d x) = π (A)$ for every measurable $A$ . (Invariant and stationary distribution for a Markov kernel)

[F4]

A probability kernel has probability-measure rows and measurable event evaluations, and a probability measure has total mass one. (Measure kernel and probability kernel, Probability measures and probability spaces)

Proof

Given: The hypotheses in the statement.

Proof technique: Apply the nonnegative kernel-Tonelli identity to the two marginals of the symmetric joint flow measure.

1.1F2F4given

Fix $A \in X$ . By [F2] on $1_{X \times A}$ , whose section at each $x$ is $A$ , $Λ_{π, P} (X \times A) = \int_{X} P (x, A) π (d x)$ , a nonnegative integral bounded by one.

1.2F1given

Symmetry [F1] and $τ^{- 1} (X \times A) = A \times X$ give $Λ_{π, P} (X \times A) = Λ_{π, P}^{⊤} (X \times A) = Λ_{π, P} (A \times X)$ .

1.3F2F4given

The rectangle formula [F2] gives $Λ_{π, P} (A \times X) = \int_{A} P (x, X) π (d x) = \int_{A} 1 π (d x) = π (A)$ , since each kernel row has mass one by [F4].

2.1F3step 1.1step 1.2step 1.3given∎

Steps 1.1, 1.2, and 1.3 give $\int_{X} P (x, A) π (d x) = π (A)$ for arbitrary $A$ , which is invariance by [F3]. The proof uses only nonnegative indicators, with no density, countability, or standard-Borel hypothesis.

5 · Examples, counterexamples and false statements

None yet.