I look at a reversible Markov chain on a countable set $G$, i.e. if $p_{xy}$ is the transition probability from $x$ to $y$, there is a positive function $\pi$ such that $$ \pi(x) p_{xy} = \pi(y) p_{yx} $$
Assume that there is a finite invariant $\nu$. I want to show that then $$ \sum_{x,y} \pi(x) p_{xy} < \infty $$
What I tried: I looked at the function $$ f(x) = \frac{\nu(x)}{\pi(x)} $$ and checked that it is harmonic. If I knew this function is bounded, I could deduce that it must be constant and then the conclusion follows.
But I can't see why this function needs to be bounded.
Thanks.
Might as well answer my own question, just in case.
First, note that the result is false if we do not assume irreducibility. So let us assume irreducibility!
Existence of the finite invariant measure implies that the Markov chain is recurrent.
Recurrence and irreducibility imply that any positive harmonic function has to be constant.
Hence the conclusion.