Let $P = (\Omega, \mathcal{F}, \mathbb{P}, T)$ be a D2839: Probability-preserving system.

Then $P$ is an D4492: Ergodic probability-preserving system if and only if \begin{equation} \forall \, E, F \in \mathcal{F} : \lim_{N \to \infty} \frac{1}{N} \sum_{n = 0}^{N - 1} \mathbb{P}(E \cap T^{-n} F) = \mathbb{P}(E) \mathbb{P}(F) \end{equation}