Simultanious convergence in $L^p$ and $L^q$

Question

I have a question and unfortunately now idea yet, how to answer it. Suppose, $p$ and $q$ are numbers $>1$. Suppose $f_n$ are all in $L^p$ and $L^q$ (over some general measure) and $f_n$ converges to $f$ in $L^p$ and to $g$ in $L^q$. Is it true that $f=g$ almost everywhere? From intuition I would say: only if the measure of the space is finite, but I'm not sure. Thanks for advices. :)

Answer 1

Yes, this is true. There is a subsequence $(f_{n_k})$ which converges to $f$ almost everywhere. Since $(f_{n_k})$ converges to $g$ in $L^q$ you can find a subsequence $(f_{n_{k_l}})$ which converges to $g$ almost everywhere. Since the latter subsequence will still converge to $f$ almost everywhere you have $f=g$ almost everywhere.