Convergence of $ L^{p} $-integrals implies convergence in $ L^{p} $-norm?

Question

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!!

Answer 1

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.