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this is a review question for a second semester measure theory class that has me stumped. I know I need the dominated convergence theorem somewhere, and maybe some sort of estimate using an increasing sequence of sets to the whole space, but other than that I am stuck:

Let $(X,A,\mu)$ be a sigma-finite measure space, and $1<p<q<\infty$ such that $1/p+1/q=1$. Let $f_n$ converge to $f$ almost everywhere, with $sup_{n}\|f_n\|_p<\infty$.

Prove that if $g \in L^p$, that $lim_{n \to \infty} \int f_ng$ = $\int fg$.

AND does this extend to the case where $p=1$ and $q=\infty$?

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@ Johnny Apple : Have you tried Hölder inequality ? –  TheBridge Mar 10 at 7:33

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