Definition:Weakly Mixing Measure-Preserving Transformation/Definition 4
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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