Definition:Weakly Mixing 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 weakly mixing if and only if:

$T \times T$ is ergodic with respect to $\mu \times \mu$

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

Sources

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