Reversible measure and detailed balance

Let $p$ be the transition matrix of a countable state space $E$ (Transition matrices and n-step probabilities), so that $p(x,y)\ge 0$ and ${\sum }_{y\in E}p(x,y)=1$ for every $x\in E$. A state measure is a function $\mu :E\to [0,+\mathrm{\infty })$ with $\mu (x)<+\mathrm{\infty }$ for every $x$; it is nonzero when $\mu (x)>0$ for at least one $x$. Such a measure $\mu$ is reversible for $p$, and $\mu$ satisfies detailed balance for $p$, when

$\mu (x)p(x,y)=\mu (y)p(y,x)\text{for all }x,y\in E\mathrm{.}$

If in addition ${\sum }_{x\in E}\mu (x)=1$, then $\mu$ is a reversible probability distribution for $p$.

Because each value $\mu (x)$ is finite and each transition entry lies in $[0,1]$, every product $\mu (x)p(x,y)$ is a well-defined element of $[0,+\mathrm{\infty })$; the identity is between nonnegative numbers and involves no subtraction of infinite quantities. Every entry $p(x,x)$ satisfies the identity trivially, including when $\mu (x)=0$. A state with $\mu (x)=0$ may have $p(x,\cdot )$ arbitrary; the identity then forces $\mu (y)p(y,x)=0$ for all $y$, so no flow from the positive support of $\mu$ enters $x$. The definition imposes no irreducibility, aperiodicity or normalization hypothesis, and a nonzero reversible measure may have infinite total mass; normalization to total mass one is stated separately. The reversal interpretation of detailed balance is given by Time reversal of a stationary Markov chain, and the lemma Detailed balance implies invariance shows that a reversible measure is invariant even when its total mass is infinite.