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

Question

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

Answer 1

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 $$ \int |f_n - f|d\mu \underset{n \to \infty}{\longrightarrow}0 $$

Answer 2

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 $p=\infty$!