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 :
 * for all measurable $f: X \to \C$:
 * $f \circ T = f \quad \mbox{$\mu$-a.e.} \implies \exists c \in \C:\, f = c \quad \mbox {$\mu$-a.e.}$

where a.e. stands for almost everywhere.

Also see

 * Equivalence of Defintions of Ergodic Measure-Preserving Transformation