A transformation $T$ on $X$ is ergodic iff for any two measurable sets $U$ and $V$ holds: $\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1} m(T^{-j}U\cap V)=m(U)m(V)$,
(or equivalently iff every invariant measurable function is constant almost everywhere, or iff every T-invariant set has full measure or measure 0).
A transformation $T$ is called weakly mixing if for any two measurable sets $U$ and $V$, $\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1} |m(T^{-j}U\cap V)-m(U)m(V)|=0$.
$T$ is weakly mixing if and only if $T\times T$ is ergodic.
It is known that weakly mixing implies that ergodicty, but the converse is not true, in general(consider for example irrational rotation of circle).
What conditions imply that the converse holds?
Please help me to know it.