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Definition D2840
Ergodic measure-preserving system

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

Let $M = (X, \mathcal{F}, \mu, T)$ 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 $$\forall \, E \in \mathcal{S} : \mu(E) = 0 \text{ or } \mu(X \setminus E) = 0$$
Children
 ▶ Ergodic probability-preserving system