This question is related to a comment in Rick Durret's book, Probability: Theory and Examples (version 5). In the proof for Theorem 5.5.9, which is about the uniqueness of stationary measure for irreducible and recurrent Markov chains with a countable state space $S$. It's stated that for $a \in S$, $P_\nu(X_j \neq a, 0 \le j <n)$ might not go to zero if $\nu$ is an infinite measure. I don't understand why this is true, and how to show it mathematically.
Note that here $\nu$ is a stationary measure, $P_\nu$ is a probability measure on the path space where the initial measure is $\nu$.