Let $X$ be a discrete-time Markov process in $S$ with invariant distribution $\nu$. Show that for any measurable set $B\subset S$ such that $$P_{\nu}\{X_n \in B\, \textrm{i.o.} \}\geq \nu B.$$
I'm honestly also unsure what $P_{\nu}$ means here. Is that supposed to be the initial distribution (e.g. the distribution of $X_0$)? If that's the case, I guess I can prove this using a series argument for the LHS.
The measure $P_\nu$ specifies that the initial distribution is $\nu$. Therefore $$ P_\nu [A] = \sum_{i \in S} \nu(i) P_i[A] = \sum_{i \in S} \nu(i) P[A \mid X_0 = i] $$ for any measurable set $A$.
Notice that if $\nu$ is the invariant distribution for the chain then $$ P_\nu[ X_n \in B \text{ i.o.}] = P_\nu[ \limsup \{X_n \in B \}] \geq \limsup P_\nu[X_n \in B] = \sum_{i \in S} \nu(i) \sum_{j \in B} K_{ij}^n $$ where $K$ is the kernel for the chain. Since $\nu$ is a left evector for $K$ with evalue $1$ (this is what it means to be invariant) the last term above is $\nu(B)$.