Given a measure-preserving system $(X,\mathcal{B},\mu,T)$ with the property that for any $A,B\in\mathcal{B}$, there exists an $N$ such that for all $n>N$:
$$\mu(A\cap T^{-n}B)=\mu(A)\mu(B) $$
Then $\mu$ is trivial in the sense that for each $A\in\mathcal{B}$, $\mu(A)=0$ or $1$
I'm having trouble showing this is true. I know that if $(X,T)$ has such a property then it is ergodic, but I'm not able to show that the measure must be trivial.