Let $(X,\mathcal A ,\mu)$ be a measure space and let $f$ and $f_1 ,f_2 ,....$ be non-negative functions that belong to $\mathcal L^1(X,\mathcal A,\mu,\mathbb R)$ and satisfy-
(i) $\{f_n\}$ converges to $f$ almost everywhere.
(ii) $\int f\ d\mu=\lim_n \int f_n\ d\mu$.
Show that $\lim_n \int |f_n-f|\ d\mu=0$ .

  • $\begingroup$ What does it mean to say "$(X,\mathcal A,\mu,\mathbb R)$" is a measure space? $\endgroup$ Jul 7 '15 at 17:47
  • $\begingroup$ I tried to consider the case by dropping the "Almost everywhere" condition and replacing it by "Everywhere". But have no clue how to proceed further. $\endgroup$
    – NewB
    Jul 7 '15 at 17:49
  • $\begingroup$ Yes its a measure space , where $\mathcal A$ is the sigma algebra and $\mu$ is the measure. Oops that $\mathbb R$ is unceccesary. Sorry $\endgroup$
    – NewB
    Jul 7 '15 at 17:51
  • $\begingroup$ This question has been already asked for several times... $\endgroup$
    – saz
    Jul 7 '15 at 17:58

First note that since $$ |f_n -f| \leq |f_n| + |f| $$ then $|f_n -f| \in L^1$. Also by (i) we have that $ |f_n| + |f| \to 2|f|$ almost everywhere. Hence hypothesis (ii) gives that $$ \int |f_n| + |f| d\mu \underset{n \to \infty}{\longrightarrow} \int 2|f| d\mu $$ Again (i) gives that $|f-f_n| \to 0 $ almost everywhere, thus by a generalized version of the dominated convergence theorem we have in fact that $$ \int |f_n - f|d\mu \underset{n \to \infty}{\longrightarrow}0 $$

  • $\begingroup$ Thank you so much. This is what I was looking for . $\endgroup$
    – NewB
    Jul 7 '15 at 19:20
  • $\begingroup$ You are welcome @LavKumar ! I am glad I could help ! $\endgroup$ Jul 7 '15 at 19:27

This is a particular version of one of the lemmas of Riesz:

Suppose that $1\leqslant p <\infty$ and that $f_n\to f$ a.e. Then the following are equivalent:

$(1)$ Convergence of norms: $\lVert f_n\rVert_p\to \lVert f\rVert_p$

$(2)$ $L^p$-convergence: $\lVert f-f_n\rVert_p\to 0$.

Proof One direction follows from the triangle inequality. For the converse, recall that $g_n=2^{p-1}(|f|^p+|f_n|^p)-|f-f_n|^p\geqslant 0$, so by the lemma of Fatou $$2^p \int |f|^p= \int \liminf\; g_n\leqslant \liminf\; 2^{p-1}\int (|f|^p+|f_n|^p)-\limsup\; \lVert f-f_n\rVert^p$$

Which gives $\limsup \lVert f-f_n\rVert \leqslant 0$, as desired.

Note The lemma fails for $p=\infty$!


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