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Definition D3059
Ergodic measure

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 $$\forall \, E \in \mathcal{F} \left( T^{-1} E = E \quad \implies \quad \mu(E) = 0 \text{ or } \mu(X \setminus E) = 0 \right)$$

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 $$\forall \, E \in \mathcal{F} \, (T^{-1} E = E \quad \implies \quad \mu(E) = 0 \text{ or } \mu(X \setminus E) = 0)$$
Children
 ▶ D4491: Ergodic probability measure