Let $X, \mathcal{A}, \mu$ be a probability space and $T: X \rightarrow X$ a measure preserving measurable map (i.e. $\mu (T^{-1}(A)) = \mu (A)$ for all $A \in \mathcal{A}$). We say $T$ is mixing for sets $A, B \in \mathcal{A}$ if $$\lim_{n\rightarrow \infty}\mu \{T^{-n}(A) \cap B \} = \mu (A) \mu (B)$$
I'm trying to show that it suffices to check the mixing property on an algebra generating $\mathcal{A}$. In other words:
if $\mathcal{A}_0$ is an algebra that generates $\mathcal {A}$, then $T$ is mixing for all $A, B \in \mathcal {A}$ if $T$ is mixing for all $A, B \in \mathcal {A}_0$.
I figured the easiest thing to do would be start by showing that, for fixed $B$, the set
$$\Gamma _B := \{A : T \text{ is mixing for the pair } A,B\}$$
is a "monotone class" (i.e. closed under nested countable unions and intersections). This would show $\mathcal {A} \subset \Gamma _B$. Then try to make the same argument with $A$ and $B$ flipped. However, when I do this I run into a double limit that doesn't seem to commute.
I'm pretty sure that I can show (using the dominated convergence theorem) that $\Gamma _B$ is closed under countable disjoint unions as well as compliments. Therefore it would also suffice to show its closed under finite intersection. But I'm not sure how to do that either.
