Uniformly weak mixing transformations

Question

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 $E$?

Answer 1

@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 $𝑁>0$ and a Rohlin tower $\{𝐸,𝑇𝐸,…,𝑇^{𝑚−1}𝐸\}$ with $𝑁=𝑜(𝑚)$, you can set $𝐴=𝐵=𝐸\cup 𝑇𝐸 \cup \ldots \cup 𝑇^{𝑚/2}𝐸$ and find that $\frac{1}{𝑁+1}\sum^{𝑁+1}_{𝑘=1}\mu(𝑇^{−𝑘}𝐴\cap 𝐴)\approx \mu(𝐴)\approx 1/2$. 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."