Given a stochastic matrix $p(x,y)$ on a finite or countable set $S$ with invariant probability measure $\mu$. Verify that the adjoint of the Markov operator $P:L_2(\mu)\rar L_2(\mu)$ is given by
$$
P^*f(x)=\sum_y f(y)q(x,y),
\quad\mbox{where}\quad
q(x,y)=\frac{p(y,x)\mu(y)}{\mu(x)}
$$
and conclude that $\mu$ is reversible iff
$$
\forall x,y\in S:\quad
\mu(x)p(x,y)=\mu(y)p(y,x)~.
$$
Give an example of a stochastic matrix with multiple reversible measures.
Put $\la f,g\ra\colon=\sum f(x)g(x)\mu(x)$. Assuming $\mu(y) > 0$ for all $y\in S$ we get
\begin{eqnarray*}
\la P^*f,g\ra
&=&\la f,Pg\ra
=\sum_xf(x)Pg(x)\mu(x)\\
&=&\sum_{x,y}f(x)g(y)\frac{p(x,y)\mu(x)}{\mu(y)}\mu(y)
=\sum_y\sum_x f(x)q(y,x)g(y)\mu(y)
\end{eqnarray*}
and therefore: $P^*f(y)=\sum_x f(x)q(y,x)$.