Definition:Ergodic Measure-Preserving Transformation/Definition 3

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 for any $A \in \BB$:
 * $\ds \map \mu A > 0 \implies \map \mu {\bigcup_{n \mathop = 1}^\infty T^{-n} \sqbrk A} = 1$

Also see

 * Equivalence of Defintions of Ergodic Measure-Preserving Transformation