Definition:Ergodic Measure-Preserving Transformation/Definition 2

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 $A \in \BB$:

$\map \mu {T^{-1} \sqbrk A \symdif A} = 0 \implies \map \mu A \in \set {0, 1}$

where $\symdif$ denotes the symmetric difference.

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