Let $\{f_n\}$ be a sequence of integrable functions on $E$ for which $f_n$ converges to $f$ a.e. on $E$, and $f$ is integrable over $E$. Show that if $\lim_{n\rightarrow\infty}\int_E|f_n|=\int_E|f|$ then $\int_E|f-f_n|\rightarrow 0$.

So I feel like this should be really obvious, but I'm having a hard time proving it. I'd appreciate some advice! The details are what get me!

  • $\begingroup$ not obvious, iirc. go through proof of dominated convergence theorem. $\endgroup$ – mathworker21 Nov 6 '19 at 0:42
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    $\begingroup$ You may want to check the answer in the link. $\endgroup$ – Sangchul Lee Nov 6 '19 at 0:53