General-state reversibility by symmetric flow measure

Let $(\mathsf{X},\mathcal{X})$ 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 $A\in \mathcal{X}$ (Measure kernel and probability kernel), and let $\pi$ be a probability measure, that is a countably additive measure of total mass one (Probability measures and probability spaces). The product sigma-algebra $\mathcal{X}\otimes \mathcal{X}$ is generated by measurable rectangles (The product sigma-algebra and its finite iterates). The joint flow measure of $(\pi ,P)$ is the probability measure ${\mathrm{\Lambda }}_{\pi ,P}$ on $(\mathsf{X}\times \mathsf{X},\mathcal{X}\otimes \mathcal{X})$ given, for each $C\in \mathcal{X}\otimes \mathcal{X}$, by

${\mathrm{\Lambda }}_{\pi ,P}(C):={\int }_{\mathsf{X}}P(x,C_x)\pi (dx),C_x:=\{y\in \mathsf{X}:(x,y)\in C\}\mathrm{.}$

Existence, countable additivity, and the nonnegative iterated-integral identity for this measure follow from A prior and model kernel define a joint law. Let $\tau (x,y):=(y,x)$ 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 ${\mathrm{\Lambda }}_{\pi ,P}^{\mathrm{\top }}(C):={\mathrm{\Lambda }}_{\pi ,P}({\tau }^{-1}(C))$.

The kernel $P$ is reversible with respect to $\pi$ when ${\mathrm{\Lambda }}_{\pi ,P}={\mathrm{\Lambda }}_{\pi ,P}^{\mathrm{\top }}\mathrm{.}$ This definition uses equality of measures and requires no density, standard Borel assumption, or prior assumption that $\pi$ is invariant.