Definition:Weakly Mixing Measure-Preserving Transformation/Definition 1

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:

$\ds \forall A, B \in \BB : \lim_{N \mathop\to \infty} \frac 1 N \sum _{n \mathop = n}^{N-1} \size {\map \mu {A \cap T^{-n} \sqbrk B} - \map \mu A \map \mu B} = 0$

where $T^{-n} \sqbrk B$ denotes the preimage of $B$ under the power $T^n$.

Sources

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