Square-correlation proof of weak mixing from product ergodicity
= Square-correlation proof of weak mixing from product ergodicity
{title2=$\frac1N\sum_{n<N}|\langle U^nf,g\rangle|^2\to0$}
For mean-zero $f$, apply the <mean ergodic theorem> on an ergodic product system to $f\otimes\overline f$, paired with $g\otimes\overline g$. Its correlation is $|\langle U^nf,g\rangle|^2$. The <Cauchy-Schwarz inequality> then gives averaged absolute correlation decay, the definition of <weak mixing>.