For a measure-preserving (finite) system $(X,\mathcal{B},\mu,T)$, is it correct that the following are equivalent?
For every $A,B\in\mathcal{B}$ , $\displaystyle\lim_{n\rightarrow\infty}\mu(A\cap T^{-n}B)=\mu(A)\mu(B)$.
For every $A,B\in\mathcal{B}$ of positive measure, there is some $n_0\in\mathbb{N}$ such that for every $n>n_0$, $\mu(A\cap T^{-n}B)>0$.
Clearly 1 implies 2. Is the opposite direction also correct?