I'm trying to prove a criterion whose proof is omitted in Wolfgang Woess' book named "Random Walks on Infinite Graphs and Groups"
Theorem. (a) $(X,P)$ is recurrent if and only if there's an invariant measure $\nu$ such that every positive excessive measure is a multiple of $\nu$.
Definition of recurrency is given by Green's function $G(x,y)= \sum_{n=0}^{\infty} p^{(n)}(x,y)$
Definition(Recurrency). The Markov chain $(X,P)$ is recurrent if $G(x,y)=\infty$ for some $x,y \in X$. Or equivalently $(X,P)$ is is recurrent if $U(x,x) = \sum_{n=0}^{\infty} \mathbb{P}_x[t^x=n]=1 $ for some $x \in X$ where $t^x=min\{n \geq 1 : Z_n=x\}$
Excessive and invariant measures. $P$ acts on non-negative measures on $X$ by $\nu P(y) = \sum_{x} \nu(x) p(x,y)$. We say that $\nu$ is excessive if $\nu P \leq \nu$ pointwise, and invariant if $\nu P=\nu$.
NOTE: I have already proved that if $(X,P)$ is recurrent, there's an invariant measure such that every positive excessive measure is multiple of it. But I don't where to start with the other direction.