TITLE: Convergence of $ L^{p} $-integrals implies convergence in $ L^{p} $-norm?
QUESTION [6 upvotes]: Let $E$ be a measurable set, 
$\{ f_n \}$ and $f$ are in $L^p(E)$ such that $f_n \to f$ pointwise a.e. 
If $\lim \|f_n \|_p = \| f \|_p$, 
is it true that $\lim \| f_n - f \|_p = 0$?
I have tried using Generalised Lebesgue Dominated Convergence Theorem, 
for all $n$, 
$|f_n-f|^p \leq g_n:=(|f_n|+|f|)^p$, 
then $g_n \to g:=2^p|f|^p$ pointwise a.e. 
But how to show $\lim \int g_n = \int g$? 
Thank you!!

REPLY [5 votes]: Hint: First use Hölder’s Inequality to get $ 2^{p - 1} (|f_{n}|^{p} + |f|^{p}) - |f_{n} - f|^{p} \ge 0 $, then write:
\begin{align}
  & ~ \limsup_{n \to \infty} \int_{E} |f_{n} - f|^{p} ~ \mathrm{d}{\mu} \\
= & ~ \lim_{n \to \infty}
      \int_{E} 2^{p - 1} (|f_{n}|^{p} + |f|^{p}) ~ \mathrm{d}{\mu} -
      \liminf_{n \to \infty}
      \int_{E} \left[ 2^{p - 1} (|f_{n}|^{p} + |f|^{p}) - |f_{n} - f|^{p} \right]
      \mathrm{d}{\mu}.
\end{align}
Finally, use Fatou’s Lemma.