Suppose $p>1$ and $q$ is its conjugate exponent. Suppose $f_n \rightarrow f~a.e.$ and $\sup_n\|f_n\|_p < \infty $. prove that if $g \in L^q,$ then $\lim_{n \rightarrow \infty} \int f_ng=\int fg.$ Does this extend to the case where $p=1$ and $q=\infty$? If not, give a counter example.
Progress
I know I need to prove $|\int (f_ng-fg)| < \epsilon$ and $|\int (f_ng-fg)| < |\int (f_n-f)g|$, if $f_n \rightarrow f$ in $ L^p $. I can use Holder's inequality to get the result. My question is how to get $f_n \rightarrow f $ in $L^p $ by the hypothesis $f_n \rightarrow f~a.e.$ and $\sup_n\|f_n\|_p < \infty$ or use other method to get the result. thanks
Best Answer
You have $\int |f|^p = \int \liminf_n |f_n|^p \le \liminf_n \int |f_n|^p < \infty$, so $f \in L^p$.
Counterexample for $p=1$: Let $X=[0,1]$, $f_n=n1_{[0,{1 \over n}]}$, $\|f_n\|_1 = 1$, $f_n(x) \to 0$ ae., and let $g=1$. Then $\int f_n g = 1$ for all $n$, but $\int 0 g = 0$.