= Decay of autocorrelation implies weak convergence of an observable
For a probability <measure-preserving system>, if $\langle U_T^nf,f\rangle\to|\int f\,d\mu|^2$, then $U_T^nf$ converges weakly in $L^2$ to $(\int f\,d\mu)\mathbf1$. Centre $v=f-(\int f)\mathbf1$; its autocorrelations tend to zero. The <isometry> identity gives the same decay against each $U_T^kv$, hence by approximation against their closed forward span. Against its orthogonal complement the correlations are exactly zero. <Orthogonal decomposition by a closed subspace> finishes the proof without invertibility.
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