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Given $f_n \rightarrow f$ a.e. Does $||f_n||_p \rightarrow ||f||_p$ imply $f_n\rightarrow f$ in $L^p$?

Clearly this does not hold for $p = \infty$, since given functions with same hight, pointwise convergence does not imply uniform convergence. And when $p=1$, it becomes Scheffe's lemma.

For $ 1< p <\infty$, we have

$$\int |f_n - f|^p dm \leq 2^{p-1}\bigg(\int |f_n|^p + |f|^p dm\bigg). $$ Using General Lebesgue Dominated Convergent theorem, we have $$\lim_n \int |f_n - f|^p dm = \int \lim_n |f_n - f|^p dm,$$ if and only if $$\lim_n \int |f_n|^p + |f|^p dm = \int \lim_n |f_n|^p + |f|^p dm$$ which holds from the assumption that $||f_n||_p \rightarrow ||f||_p$.

Is my argument correct? Thanks!

Xiao
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  • Oops, I didn't see you were asking for a proof verification. – David Mitra Jul 29 '14 at 13:43
  • The first iequality is not true (there is a constant missing). I can't see why the second equality is true? Even if it were true, I don't see how you can finish the proof? – Tomás Jul 29 '14 at 13:53
  • @Tomás I fixed the first one, forgot the constant from the convexity argument. The second one is from general dominated convergent theorem, give $f_n,g_n \rightarrow f,g$ a.e. and $|f_n|\leq g_n$, then $$\lim\int f_n = \int f \text{ only if } \lim \int g_n =\int g.$$ And wouldn't $$\lim \int|f_n - f|^p dm = 0$$ imply $||f_n - f||_p \rightarrow 0$. – Xiao Jul 29 '14 at 13:57
  • Ok, now I undertood, however I fail to see how would you conclude that $\lim\int |f_n-f|^p=0$? – Tomás Jul 29 '14 at 14:18
  • @Tomas Thanks for the reply. Using general dominated convergence theorem, $$\lim\int |f_n - f|^p = \int\lim |f_n - f|^p = 0$$ since $f_n\rightarrow f$ a.e. – Xiao Jul 29 '14 at 14:22

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