 Weakly mixing measure-preserving transformation

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: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 $1/4$.