Definition:Ergodic Measure-Preserving Transformation/Definition 5

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

Also see

• Equivalence of Defintions of Ergodic Measure-Preserving Transformation

Sources

• 2011: Manfred Einsiedler and Thomas Ward: Ergodic Theory: with a view towards Number Theory $2.3$: Ergodicity