Using the Baire Category Theorem to prove $\mu$ is trivial

Question

Suppose we have a probability measure space $(X,\mathcal A,\mu,T)$ where $T$ is measure-preserving. Then if for every $A,B\in\mathcal A$ we have $\mu\left(A\cap T^{-n}B\right)=\mu(A)\mu(B)$ for all $n\geq N$ for some $N\in\mathbb N$ (depending on $A,B$, probably), then show that $\mu(A)=0$ or $1$ for every $A\in\mathcal A$. Do it by first fixing $B$ with $0<\mu(B)<1$ and then look for $A$ using Baire Category Theorem.

I cannot figure out how to use Baire Category Theorem here. We are not in a metric space. How can I talk about dense sets and all that? I would very much like a hint. Thanks.

Answer 1

We can endow $\mathcal A$ with a pseudo-metric defining $\rho\left(A,B\right):=\mu\left(A\Delta B\right)$, where $\Delta$ denotes the symmetric difference operator. In this way, $\left(\mathcal A,\rho\right)$ is a complete pseudo-metric space.

For the details, see this thread.