Definition:Ergodic Measure-Preserving Transformation/Definition 5
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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 measurable $f: X \to \C$:
- $f \circ T = f$ holds $\mu$-almost everywhere
- $\implies \exists c \in \C:\, f = c$ holds $\mu$-almost everywhere
- $f \circ T = f$ holds $\mu$-almost everywhere
Also see
Sources
- 2011: Manfred Einsiedler and Thomas Ward: Ergodic Theory: with a view towards Number Theory $2.3$: Ergodicity