Proof of "strong law of large numbers" in Markov Chains

Question

I have been given a theorem stating an analogue of the strong law of large numbers for Markov chains. It states that if $X=(X_n)_{n\in\mathbb{N}}$ is a Markov chain with transition matrix $p$ and $\pi$ is its invariant probability and $f:E\to\mathbb{R}$ is a function integrable with respect to $\pi$, then setting $Y_n:=\frac{X_1+\dotso+X_n}{n}$ one gets: $$\lim_{n\to\infty}Y_n=\sum_{i\in E}f(i)\pi_i=E_{\pi}[f(X_1)],$$ that is, this limit is the integral of $f(X_1)$ with respect to $\pi$. I looked into a couple of references and was unable to find a proof of this, and looking for it on the web is basically impossible because I don't know how to word this in such a way that the query has any efficiency whatsoever. Can someone point me to a reference (not Google Books please, the probability of me not being able to see the page of the proof is close to 1) or post a proof of the theorem here?

Answer 1

Quite late, but convergence in probability for finite (maybe countable?) state-space is shown in James Norris's book Markov Chains, Section 1.10.

First, the ergodic theorem mentioned by @AlexR is established. This is then extended to the law of large numbers requested by the OP.