($p\ge 1$)$f_n\overset{\mu}{\to}f,\|f_n\|_p\to\|f\|_p$,then:
$\|f_n-f\|_p\to 0$
By Scheffe's lemma,we can get $$\int||f_n|^p-|f|^p|\to 0$$how to get rid of $p$ in the integrand?
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Suppose $\|f_n - f\|$ does not converge to 0, then $\exists \epsilon >0$ and a subsequence of $(f_n)$, denoted $(f_{n_k})$ such that $\|f_{n_k} - f\| \geq \epsilon$
Since $f_{n_k}$ converges in measure to $f$, we can subtract a sub-subsequence which converges almsot everywhere to $f$.
Then apply Scheffe lemma on this sub-subsequence to get a contradiction
Petite Etincelle
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Assume $f_{n_k}$ itself converge to $f$ a.e. and we want to show "$|f_{n_k}-f|_p\to 0$" to get a contradiction.BUT Scheffe's lemma describes the $L_1$ convergence,here is "$L_p$" ,furthermore $|.|_1\le|.|_p$ – Lookout Sep 05 '14 at 10:29
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@paradox The statement in Scheffe's lemma is also true for $p \geq 1$. This can be proven by applying Fatou's lemma on $c(|f|^p + |f_n|^p) - |f- f_n|^p$, where $c$ satisfies the inequality $|x+y|^p \leq c(|x|^p + |y|^p)$ – Petite Etincelle Sep 05 '14 at 10:32