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
Definition D2840
Ergodic measure-preserving system
Formulation 0
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 \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}
Formulation 1
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 \begin{equation} \forall \, E \in \mathcal{S} : \mu(E) = 0 \text{ or } \mu(X \setminus E) = 0 \end{equation}
Children
D4492: Ergodic probability-preserving system