Arithmetic-progression multiple averages under weak mixing ($\frac{1}{N}{\sum }_{n<N}{\prod }_{j=1}^k{U}^{jn}{f}_j\to {\prod }_{j=1}^k\int f_{j}d\mu$)

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 to obtain zero. Adding back the final mean completes the induction.

For real bounded observables, the Cesaro limit of the correlation comes from arithmetic-progression multiple averages under weak mixing. Apply the same result on the weakly mixing product system to $f_j\otimes f_j$ to obtain its second moment. The averaged squared error tends to zero, so the mean-square criterion for convergence in density gives the displayed density limit.