Diagonal set-correlation criterion for mixing
= Diagonal set-correlation criterion for mixing
A probability <measure-preserving transformation> is <strong mixing> exactly when $\mu(T^{-n}A\cap A)\to\mu(A)^2$ for every measurable $A$. The converse follows by applying <decay of autocorrelation implies weak convergence of an observable> to $f=\mathbf1_A$ and then testing against $\mathbf1_B$. Diagonal polarization alone controls a sum of the two directed cross-correlations and does not suffice as a proof.