General-state reversibility implies invariance

Given: A measurable space $(\mathsf{X},\mathcal{X})$, a pointwise probability kernel $P$, and a probability measure $\pi$ for which the joint flow measure ${\mathrm{\Lambda }}_{\pi ,P}$ is symmetric.

The joint flow measure is the probability measure on the product sigma-algebra defined by ${\mathrm{\Lambda }}_{\pi ,P}(C)=\int P(x,C_x)\pi (dx)$, and reversibility means ${\mathrm{\Lambda }}_{\pi ,P}={\mathrm{\Lambda }}_{\pi ,P}^{\mathrm{\top }}$ under coordinate swap. (General-state reversibility by symmetric flow measure)

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

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

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)