this prop. in royden page 270
the proof must be by Fatous lema" applying fatoues lemma to fn+gn>0 and gn-fn>0 "
correct me if i'm wrong
from $\displaystyle |f_{n}| \leq g_{n}$, we have $\displaystyle -g_{n} \leq f_{n} \leq g_{n} $.
i) consider $\displaystyle g_{n} - f_{n} \geq 0$
because $\displaystyle g_{n} \to g$ a.e and $\displaystyle f_{n} \to f$ a.e, then $\displaystyle g_{n} - f_{n} \to g - f$ a.e.
by fatou's lemma, we have
$\displaystyle \int g - f \leq \lim \inf \int (g_{n} - f_{n})$
using the linearity of integral, then we get
$\displaystyle \int g - \int f \leq \lim \inf \int g_{n} - \lim \sup \int f_{n}$
because $\displaystyle \int g = \lim \int g_{n} = \lim \sup \int g_{n} = \lim \inf \int g_{n}$, then
$\displaystyle \int f \geq \lim \sup \int f_{n}$
ii) consider $\displaystyle f_{n} + g_{n} \geq 0$
by fatou's lemma, we get
$\displaystyle \int f \leq \lim \sup \int f_{n}$
from (i) and (ii), we conclude that
$\displaystyle \int f = \lim \int f_{n}$