Ergodicity vs Weak Mixing

Question

I have been trying to prove the following:

Let $(\Omega,\mathcal{F})$ be a measurable space endowed with probability measure $\mathbb{P}$. Suppose $\tau : \Omega \to \Omega$ is a measure preserving transform, i.e. $\forall A \in \mathcal{F}$, $\mathbb{P}(\tau^{-1}A) = \mathbb{P}(A)$.

Prove that $\tau$-ergodic iff for any measurable $A,B$ \begin{equation} \frac{1}{n} \sum_{k=0}^{n-1} \mathbb{P}[A \cap \tau^{-k}B] \rightarrow \mathbb{P}[A] \mathbb{P}[B] \end{equation}

Above, $\tau$-ergodic means for every $E$ with property $E = \tau^{-1}E$, $\mathbb{P}[E] = 0 $ or $1$.

I have proved that if the given claim (a.k.a weak mixing) holds, then $\tau$ is indeed ergodic. However, I failed to prove the other direction, I did it when $A$ is a $\tau$-invariant event but could not extend to the general case. I am not %100 sure whether the statement is correct. Any proof or counterexample? Any help is appreciated.

PS: Sorry if something is wrong with my notation, I am very new into Ergodic theory.

Answer 1

First let me note that weak mixing is something different, in fact stronger than ergodicity.

For the other direction note that $$ \begin{split} \mathbb P(A)\mathbb P(B) &=\mathbb P(A)\int_\Omega 1_B\,d\mathbb P\\ &=\int_\Omega\mathbb P(A)1_B\,d\mathbb P\\ &=\int_\Omega\lim_{n\to\infty}\frac1n\sum_{k=0}^{n-1}(1_A\circ \tau^k) 1_B\,d\mathbb P\\ &=\lim_{n\to\infty}\frac1n\sum_{k=0}^{n-1} \int_\Omega 1_{\tau^{-k}A\cap B}\,d\mathbb P, \end{split} $$ using dominated convergence in the last identity. Therefore, $$ \mathbb P(A)\mathbb P(B)=\lim_{n\to\infty}\frac1n\sum_{k=0}^{n-1}\mathbb P(\tau^{-k}A\cap B). $$