ThmDex – An index of mathematical definitions, results, and conjectures.
 ▼ Set of symbols ▼ Alphabet ▼ Deduction system ▼ Theory ▼ Zermelo-Fraenkel set theory ▼ Set ▼ Subset ▼ Power set ▼ Hyperpower set sequence ▼ Hyperpower set ▼ Hypersubset ▼ Subset algebra ▼ Subset structure ▼ Measurable space ▼ Measure space ▼ Measure-preserving endomorphism ▼ Measure-preserving system ▼ Stationary measurable set ▼ Set of stationary measurable sets ▼ Ergodic measure-preserving system
Definition D4492
Ergodic probability-preserving system

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

Let $P = (\Omega, \mathcal{F}, \mathbb{P}, T)$ be a D2839: Probability-preserving system such that
 (i) $\mathcal{S}$ is the D4490: Set of stationary events in $P$
Then $P$ is an ergodic probability-preserving system if and only if $$\forall \, E \in \mathcal{S} : \mathbb{P}(E) \in \{ 0, 1 \}$$