= Weakly mixing measure-preserving transformation
= Weak mixing
{synonym}
A probability <measure-preserving transformation> $T$ is weakly mixing when, for every pair of measurable sets $A,B$, the average absolute correlation discrepancy tends to zero:
$$
\frac1N\sum_{n=0}^{N-1}|\mu(T^{-n}A\cap B)-\mu(A)\mu(B)|\longrightarrow0.
$$
Equivalently, the correlations converge to their products in <convergence in density of a sequence>, its <Cartesian product> with itself is an <ergodic transformation>, or its product with every <ergodic transformation> is ergodic. The absolute values are essential: signed <Cesaro convergence of a sequence> of the correlations to their products characterizes <ergodic transformations> and does not imply weak mixing. For example, the transformation interchanging two equally weighted points has discrepancy $(-1)^n/4$ for a singleton $A=B$: its signed average tends to zero while its average absolute discrepancy stays $1/4$.
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