Definition:Weakly Mixing 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 weakly mixing if and only if:

for every ergodic measure-preserving system $\struct {Y, \BB_Y, \nu, S}$, the system:

$\struct {X \times Y, \BB \otimes \BB_Y, \mu \times \nu, T \times S}$

is ergodic

where $\mu \times \nu$ denotes the product measure on $\struct {X \times Y, \BB \otimes \BB_Y}$

Sources

• 2011: Manfred Einsiedler and Thomas Ward: Ergodic Theory: with a view towards Number Theory $2.7$: Strong-Mixing and Weak-Mixing