I am looking for 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.
EDIT (after martini's answer): $\mu$ has to be $T$-invariant.
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?
This is wrong. Let $X = \{x,y\}$ with $x\ne y$ and the discrete metric, let $\mu = \delta_x$ and $T$ the constant map to $y$. Then the only elment $z$ for which $(T^n z) = (z, y,y,y,\ldots)$ has a subsequence converging to $z$ is $z = y$, but $\delta_x(\{y\}) = 0 \ne 1$.