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