Suppose $\displaystyle f_n,f,g \in L^2 (A)$ and $\displaystyle f_n \rightarrow f in L^2$. Is is true if i say $\displaystyle lim_{n\rightarrow \infty} \int_A f_n g dm = \int_A f g dm$?
$\displaystyle \left|\int_Af_ng-\int_Afg\right|\leq \int_A |f-f_n||g|\leq \sqrt{\int_A |f-f_n|^2}\sqrt{\int_A g^2}$, and you can conclude (the hypothesis is the strong convergence, whereas the conclusion is called weak convergence).