# Criterium involving ergodicity and weak mixing

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 $V$, $\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1} |m(T^{-j}U\cap V)-m(U)m(V)|=0$.

How to prove that the following properties are equivalent:

1) $T$ is weakly mixing,

2) $T\times T$ is ergodic with respect to $m\times m$,

3) $T\times T$ is weakly mixing with respect to $m\times m$.

3) implies 2), but how to prove the rest?

Any help is welcome. Thanks in advance.

For a bounded sequence $(a_n)$ of real numbers, then $$\lim_n\dfrac{1}{n}\sum_{j=1}^n|a_n|=0 \text{ if and only if } \lim_n\dfrac{1}{n}\sum_{j=1}^n(a_n)^2=0.$$