TITLE: Proof of Krylov-Bogoliubov Theorem
QUESTION [22 upvotes]: Where can I find a proof (in English) of the Krylov-Bogoliubov theorem, which states if $X$ is a compact metric space and $T\colon X \to X$ is continuous, then there is a $T$-invariant Borel probability measure? The only reference I've seen is on the Wikipedia page, but that reference is to a journal that I cannot find.
Of course, feel free to answer this question by providing your own proof.

REPLY [3 votes]: Then the theorem also doesn't hold. Take $X = \mathbb{R}$, and $T\colon x \mapsto x+1$. Then $T$ has no invariant probability measure. (all the measures "escape to infinity").