Let $T$ be a measure-preserving transformation on a probability space $(\Omega,\mathcal B,\mu)$. Assume that for any pair of measurable sets $A,B\in\mathcal B$ with $\mu(A), \mu(B)>0$, one can find $N$ such that $T^{-n}(A)\cap B\neq\emptyset$ for all $n\geq N$. Is $T$ mixing with respect to $\mu$? This is likely to be a simple exercise. So, apologies, and thanks in advance.



Silly question about mixing
There is a condition known to be intermediate between the one you mention and mixing. A transformation is **lightly mixing** if $\liminf\_{n\to\infty} \mu(T^{-n}(A)\cap B) > 0$ for all $A$ and $B$ of positive measure. For a transformation which is lightly mixing but not mixing, see for example Friedman and King's paper "Rank One Lightly Mixing".


