Then $P$ is a

**weakly mixing 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} \left| \mathbb{P}(E \cap T^{-n} F) - \mathbb{P}(E) \mathbb{P}(F) \right| = 0 \end{equation}