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 and only if:

for all $A \in \BB$:

$\ds \map \mu A > 0 \implies \map \mu {\bigcup_{n \mathop = 1}^\infty T^{-n} \sqbrk A} = 1$

Also see

• Equivalence of Defintions of Ergodic Measure-Preserving Transformation

Sources

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