Given: $ (X,\mathcal{A},\mu), \mu(X)<\infty, f_n \in L^p, 1<p<\infty, f_n \rightarrow f \ a.e., ||f_n||_p \leq 1 \ \forall \ n, g \in L^q $
Show: $ \int_X f_ng \rightarrow \int_X fg $
Proof: By Fatou, $ ||f||_p < 1 $. Given $ \epsilon > 0 $ we may fix $ \delta>0 $ so small that $ \mu(A)<\delta \Rightarrow ||g\chi_A||_q < \epsilon $, which we are afforded by the absolute continuity of the indefinite integral of $ |g|^q $. By Egorov, we may select closed $ F \subset X, \mu(X-F) < \delta $, such that $ f_n \rightarrow f $ uniformly on F.
We have
$ |\int_X (f-f_n)g| \leq ||(f-f_n)g|| = ||(f-f_n)g\chi_F|| + ||(f-f_n)g\chi_{F^c}|| $
$ ||(f-f_n)g\chi_F|| \leq ||(f-f_n)\chi_F||_p||g\chi_F||_q \rightarrow 0 $ by Holder and the Uniform Convergence Theorem. $ ||(f-f_n)g\chi_{F^c}|| \leq ||(f-f_n)\chi_{F^c}||_p||g\chi_{F^c}||_q < 2\epsilon $, by Holder and the fact that $ ||f-f_n||_p \leq ||f||_p + ||f_n||_p $, since the p-norm is a metric for $ p\geq1 $. $ \square $
-pw
