# Criterium involving ergodicity and weak mixing

A transformation TT on XX is ergodic iff for any two measurable sets
UU and VV holds: limn→∞1n∑n−1j=0m(T−jU∩V)=m(U)m(V)\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 TT is called weakly mixing if for any two measurable sets
UU and VV, \lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1} |m(T^{-j}U\cap V)-m(U)m(V)|=0\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) TT is weakly mixing,

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

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

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

Any help is welcome. Thanks in advance.

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