Let $f$ be a continuous map defined on a compact metric space $X$. Suppose that $f$ preserves the Borel probability measure $\mu$ and that, for every positive-measure set $A\subseteq X$, we have $$\lim_{n\to\infty} \mu(f^n(A))=1.$$ How to prove, as simply as possible, that $$\lim_{n\to \infty} \mu(A\cap f^{-n}(B))=\mu(A)\mu(B),$$ for every $A,B$ measurable sets?

(This question was asked on Math Stack Exchange, with no answers: https://math.stackexchange.com/questions/4883667/simplest-proof-that-exactness-implies-mixing)