Let $M$ be a compact space (not necessarily Hausdorff), $T:M\rightarrow M$ continuous bijective and $\mu$ an ergodic $T$-invariant probabilistic measure on $M$. A point $x\in M$ is generic if $$\mu=\lim_{n\to \infty}\frac{1}{n}\sum_{k=0}^{n-1} \delta_{T^{k}(x)}$$ (here $\delta_x$ denotes the Dirac measure on $x$). Now, I want to know if there exists a generic point $x\in M$.
I know that this result is true when $M$ is a compact metric space just assuming that $T$ is measurable. Essentially, this is a consequence of Birkhoff's ergodic theorem and Stone–Weierstrass theorem. However, for a compact space I cannot always use the latter theorem.