# I am having trouble solving this problem from the book " Measure Theory" by Donald L.Cohn.

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 $\lim_n \int |f_n-f|\ d\mu=0$ .

• What does it mean to say "$(X,\mathcal A,\mu,\mathbb R)$" is a measure space? Jul 7 '15 at 17:47
• 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.
– NewB
Jul 7 '15 at 17:49
• Yes its a measure space , where $\mathcal A$ is the sigma algebra and $\mu$ is the measure. Oops that $\mathbb R$ is unceccesary. Sorry
– NewB
Jul 7 '15 at 17:51
– 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 $$\int |f_n - f|d\mu \underset{n \to \infty}{\longrightarrow}0$$

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