Let $T$ be ergodic on $(X,\mu)$.
If $f_1,f_2\in L^1(X,\mu)$ and $\int f_2 d\mu\neq 0$, then $\lim_{n\to\infty}\frac{\sum_{i=0}^{n-1} f_1(T^ix)}{\sum_{i=0}^{n-1} f_2(T^ix)}=\frac{\int_X f_1}{\int_X f_2}$?
How to prove this or where can I read the proof?
Any help is welcome. Thanks.