Weakly mixing measure-preserving transformation

A probability measure-preserving transformation $T$ is weakly mixing when, for every pair of measurable sets $A,B$, the average absolute correlation discrepancy tends to zero:Equivalently, the correlations converge to their products in convergence in density of a sequence, its Cartesian product with itself is an ergodic transformation, or its product with every ergodic transformation is ergodic. The absolute values are essential: signed Cesaro convergence of a sequence of the correlations to their products characterizes ergodic transformations and does not imply weak mixing. For example, the transformation interchanging two equally weighted points has discrepancy $(-1{)}^n/4$ for a singleton $A=B$: its signed average tends to zero while its average absolute discrepancy stays $1/4$.

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.

Every positive power of a weak mixing transformation is weakly mixing: its nonnegative discrepancies are a subsequence whose average is bounded by $r$ times the full average. Products are weakly mixing because correlations of simple tensors factor; the absolute product discrepancy is bounded by the sum of bounded multiples of the single-system discrepancies. Density of finite tensor sums gives the full $L^2$ assertion.

If $T$ is weakly mixing, then $T\times S$ is ergodic exactly when $S$ is ergodic. In particular, ergodicity of $T\times T$ is equivalent to weak mixing of $T$.

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 $|⟨U^{n}f,g⟩{|}^2$. The Cauchy-Schwarz inequality then gives averaged absolute correlation decay, the definition of weak mixing.

A weakly mixing system has no nonconstant Koopman operator eigenfunction. If $U_{T}f=\lambda f$, then $f(x)\overline{f(y)}$ is invariant under $T\times T$, contradicting the product characterization of weak mixing unless $f$ is constant.

A system is weakly mixing exactly when, for every positive-measure $A,B,C$, some $n\ge 1$ makes both $T^{-n}A\cap B$ and $T^{-n}A\cap C$ have positive measure. This is the finite simultaneous-hitting form of the density-one correlation characterization.