Let $(X, T, \mathcal F, \mu)$ be a nonatomic standard probability space equipped with a measure preserving transformation $T$. We say $T$ is *uniformly weak mixing* if for every $\varepsilon > 0$, there exists some $N > 0$ such that for all measurable sets $A, B \in \mathcal F$


$$| \frac{1}{n} \sum\_{k=1}^n \mu(T^{-k} A \cap B) - \mu(A) \mu(B)| < \varepsilon$$


for all $n > N$.


**Question:** If $T$ is uniformly weak mixing, then does it hold that for all $f \in L^\infty (X)$, there exists some measurable set $E$ of full measure such that


$$\frac{1}{n} \sum\_{k=1}^n T^k f \to \int f d\mu$$


uniformly on $E$?



Uniformly weak mixing transformations
@John Griesmer answered this question: "I don't think there is a system that satisfies your definition of uniformly weak mixing. Such a system must be ergodic, and therefore admit Rohlin towers. Given $>0$ and a Rohlin tower $\{,,…,^{−1}\}$ with $=()$, you can set $==\cup \cup \ldots \cup ^{/2}$ and find that $\frac{1}{+1}\sum^{+1}\_{=1}\mu(^{−}\cap )\approx \mu()\approx 1/2$. In other words, given an initial segment of integers, every ergodic MPS on a nonatomic probability space admits a subset which is approximately invariant under that segment."


