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 $f: X \to \C$:
 * $f \circ T = f$ 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