Let $M = (X, \mathcal{F}, \mu, T)$ be a D2827: Measure-preserving system.
Then $M$ is an ergodic measure-preserving system if and only if \begin{equation} \forall \, E \in \mathcal{F} \left( T^{-1} E = E \quad \implies \quad \mu(E) = 0 \text{ or } \mu(X \setminus E) = 0 \right) \end{equation}

Let $M = (X, \mathcal{F}, \mu, T)$ be a D2827: Measure-preserving system such that

(i)	$\mathcal{S}$ is the D2842: Set of stationary measurable sets in $M$

Then $M$ is an ergodic measure-preserving system if and only if \begin{equation} \forall \, E \in \mathcal{S} : \mu(E) = 0 \text{ or } \mu(X \setminus E) = 0 \end{equation}