Sufficient condition for strong mixing. Given a measure-preserving transformation $T:X\to X,$ with $(X,\mathcal{A},\mu)$ a probability space, we say $T$ is strongly mixing if $$\mu(T^{-n}(A)\cap B)\to \mu(A)\mu(B)$$ when $n\to\infty$ for every $A,B\in \mathcal{A}.$ 
I know this definition to be equivalent to that for every $f,g\in L^{\infty}(X,\mathcal{A},\mu)$ we have $$(*) \int_X (f\circ T^n)g d\mu \to \int_X f d\mu \int_X g d\mu$$ when $n\to\infty.$
Now, I was asked to prove (Brin/Stuck exercise 4.3.6) that if $\mu$ is a Borel measure, $X$ is a compact topological space and for every $f,g$ continuous with zero integrals we have  $$\int_X (f\circ T^n)g d\mu \to 0,$$ then $T$ is strongly mixing.
Certainly, every continuous function in $X$ is bounded, and I presume I have to get to (*), but I still haven't come with an idea of how to prove it for every function with nonzero integral. Any idea or suggestion will be appreciated.

Any continuous function can be decomposed as $f = ( f - \int f ) + \int f $ where the first term has $0$ intergral, and the second is constant. You should be able to use your hypothesis and the linearity of integration to get ($*$) for all continuous functions.
Now that you have it for continuous functions, use the Stone-Weierstrass Approximation Theorem and X compact to finish the proof.
