I am looking for a proof (or a reference to a proof) of the following theorem:


Let $X$ be a compact metric space with metric $d$, endow $X$ with the Borel $\sigma$-algebra and a probability measure $\mu$. Let $T\colon X\to X$ be a continuous map which is $\mu$-preserving. Then for $\mu$-almost every $x\in X$ there is a sequence $n\_k\to\infty$ in $\mathbb{N}$ such that $T^{n\_k}x\to x$ as $k\to\infty$.


I tried already for some time and looked for references, unfortunately unsuccessful. It seems that one needs to find the right formulation to be able to use the Poincare recurrence theorem. Any ideas?



Poincare recurrence theorem and convergence on compact metric spaces
Let $\mathcal P\_1,\mathcal P\_2,\ldots$ be a sequence of refining partitions of decreasing diameter (converging to 0) of $X$. Let $\delta\_n$ be the diameter of $\mathcal P\_n$. Let $\mathcal P\_n(x)$ be the element of $\mathcal P\_n$ containing $x$.
For $\mu$-a.e. $x$, $\mu(\mathcal P\_n(x))\ne 0$. 


Now the Poincaré recurrence theorem applies to each element (with positive measure) of $\mathcal P\_n$, saying that for almost every $z\in B\in\mathcal P\_n$, $z$ returns to $B$ infinitely often. Hence there exist infinitely many $k^{(n)}\_i$ such that $d(T^{k^{(n)}\_i}z,z)\le \delta\_n$. Let the union of the exceptional sets for each $B$ in $\mathcal P\_n$ be $E\_n$. 
Let $E$ be the union of the $E\_n$. Then $E$ has measure 0.


Now just put it all together. Let $E'$ be the set of $x'$ such that $\mu(\mathcal P\_n(x))=0$ for some $n$. This is another set of measure 0. Now if $x\in X\setminus(E\cup E')$, we're in business:


$\mu(\mathcal P\_n(x))>0$ for all $n$ and $x$ returns to $\mathcal P\_n(x)$ infinitely often. A diagonal argument finishes things off.


There may be a reference for this in Furstenberg's book.


