Let $(X,d)$ be a compact [metric space](/page/Metric%20Space), let $\mathcal{B}(X)$ denote its Borel $\sigma$-algebra, let $\mu$ be a Borel probability measure on $X$, and let $T: X \to X$ be a Borel measurable map such that
for every $A \in \mathcal{B}(X)$. Assume that the measure-preserving system $(X,\mathcal{B}(X),\mu,T)$ is ergodic. Then for $\mu$-almost every $x \in X$, the orbit $(T^n x)_{n \geq 0}$ is equidistributed with respect to $\mu$, meaning that for every [continuous function](/page/Continuous%20Function) $f: X \to \mathbb{R}$,