As seen on wikipedia, given a measure space $(X,\Sigma,\mu)$ with $\mu(X) < \infty$ and a measure preserving transformation $T: X \mapsto X$. Let $A \subset X$ be a set of positive measure. Define $k_i$ as the power of $T$ such that $T^{k_i}x \in A$ for the $i$th time: that is to say $k_i$ is the "$i$'th return time to $A$". The difference between recurrence times is $R_i = k_i - k_{i-1}$ (assume for simplicity that $k_0 = 0$, that is $x \in A$)

I would like know how to prove the following:

$$ \lim\limits_{n\mapsto\infty} \frac{R_1 + \cdots + R_n}{n} = \frac{\mu(X)}{\mu(A)}$$

The wikipedia article indicates that this is a consequence of the ergodic theorem.

Note that my definition of the $k_i$ above differs slightly from that of wikipedia, in as much as I have omitted to say that the $k_i$s are sorted in increasing order.