Suppose we have a Markov Chain $X_n$ on $\mathbb{R}$, and let $f:\mathbb{R} \longrightarrow [0,1]$ be a measurable function.
Does a law of large numbers hold, i.e. $\frac{f(X_1) +...+f(X_n)}{n} \longrightarrow p\in [0,1]$ in probability or a.s.?
Without the assumption that $f$ is bounded this is clearly not true (since we can take $f(x)=x$ and $X_n$ to be some deterministic increasing Markov Chain), but is this assumption enough?