Let $P = (\Omega, \mathcal{F}, \mathbb{P}, T)$ be a D2839: Probability-preserving system.
Then $P$ is an ergodic probability-preserving system if and only if \begin{equation} \forall \, E \in \mathcal{F} \left( T^{-1} E = E \quad \implies \quad \mathbb{P}(E) \in \{ 0, 1 \} \right) \end{equation}

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

(i)	$\mathcal{S}$ is the D4490: Set of stationary events in $P$

Then $P$ is an ergodic probability-preserving system if and only if \begin{equation} \forall \, E \in \mathcal{S} : \mathbb{P}(E) \in \{ 0, 1 \} \end{equation}