= Arithmetic-progression multiple averages under weak mixing
{title2=$\frac1N\sum_{n<N}\prod_{j=1}^kU^{jn}f_j\to\prod_{j=1}^k\int f_j\,d\mu$}
For bounded observables of a probability <weak mixing> system, the displayed convergence holds in $L^2$. Induct on $k$: center the final observable, compute the lagged Hilbert-space correlations, apply the induction hypothesis to the $k-1$ remaining factors, and use <weak mixing> of $T^k$ with the <Van der Corput lemma (Hilbert space sequences)> to obtain zero. Adding back the final mean completes the induction.
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