Definition:Ergodic Measure-Preserving Transformation/Definition 4
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Definition
Let $\struct {X, \BB, \mu}$ be a probability space.
Let $T: X \to X$ be a measure-preserving transformation.
$T$ is said to be ergodic if and only if:
- for all $A, B \in \BB$:
- $\map \mu A \map \mu B > 0 \implies \exists n \ge 1 : \, \map \mu {T^{-n} \sqbrk A \cap B} > 0$
Also see
Sources
- 2011: Manfred Einsiedler and Thomas Ward: Ergodic Theory: with a view towards Number Theory $2.3$: Ergodicity