Definition:Ergodic Measure-Preserving Transformation/Definition 4

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 $A,B \in \BB$:
 * $\map \mu A \map \mu B > 0 \implies \exists n \geqslant 1 : \, \map \mu {T^{-n} \sqbrk A \cap B} > 0$

Also see

 * Equivalence of Defintions of Ergodic Measure-Preserving Transformation