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}