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Definition D4491
Ergodic probability measure

Let $S = (\Omega, \mathcal{F}, \mathbb{P}, T)$ be a D2839: Probability-preserving system.
Then $\mathbb{P}$ is an ergodic probability measure on $S$ if and only if $$\forall \, E \in \mathcal{F} \, (T^{-1} E = E \quad \implies \quad \mathbb{P}(E) = 0 \text{ or } \mathbb{P}(\Omega \setminus E) = 0)$$

Let $S = (\Omega, \mathcal{F}, \mathbb{P}, T)$ be a D2839: Probability-preserving system.
Then $\mathbb{P}$ is an ergodic probability measure on $(\Omega, \mathcal{F})$ with respect to $T$ if and only if $$\forall \, E \in \mathcal{F} \, (T^{-1} E = E \quad \implies \quad \mathbb{P}(E) = 0 \text{ or } \mathbb{P}(\Omega \setminus E) = 0)$$