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](https://en.wikipedia.org/wiki/Krylov%E2%80%93Bogolyubov_theorem) 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.



Proof of Krylov-Bogoliubov theorem
First, fix $x \in X$ and let $\mu\_1 := \delta\_x$ be the Dirac measure supported at $x$. Then define a sequence of probability measures $\mu\_n$ such that for any $f \in C^0 (X)$,
$$ \int\_X f(y) \mathrm{d} \mu\_n (y) = \frac{1}{n} \sum\_{k=0}^{n-1} \int\_X f \circ T^k (y) \mathrm{d} \mu\_1 (y). $$
Apply the Banach-Alaouglu Theorem to deduce there exists a subsequence $\mu\_{n\_j}$ which converges in the weak-$\star$ topology. It is then very easy to prove that this limit measure is in fact T-invariant, using the formulation that $\mu$ is T-invariant if and only if $$\int\_X f \circ T \mathrm{d} \mu = \int\_X f \mathrm{d}\mu$$ for all continuous $f$. 


