Let $P = (\Omega, \mathcal{F}, \mathbb{P}, T)$ be a D2839: Probability-preserving system.
Then the following statements are equivalent
| (1) | $P$ is an D4492: Ergodic probability-preserving system |
| (2) | \begin{equation} \forall \, E \in \mathcal{F} \left( \mathbb{P}(E \triangle T^{-1} E) = 0 \quad \implies \quad \mathbb{P}(E) \in \{ 0, 1 \} \right) \end{equation} |
| (3) | \begin{equation} \forall \, E \in \mathcal{F} \left[ \mathbb{P}(E) > 0 \quad \implies \quad \mathbb{P} \left( \bigcup_{n = 1}^{\infty} T^{-n} E \right) = 1 \right] \end{equation} |
| (4) | \begin{equation} \forall \, E, F \in \mathcal{F} \left( \mathbb{P}(E), \mathbb{P}(F) > 0 \quad \implies \quad \exists \, n \in 1, 2, 3, \ldots : \mathbb{P}(E \cap T^{-n} F) > 0 \right) \end{equation} |