1. ## proof this proposition

this prop. in royden page 270
the proof must be by Fatous lema" applying fatoues lemma to fn+gn>0 and gn-fn>0 "

3. correct me if i'm wrong
from $|f_{n}| \leq g_{n}$, we have $-g_{n} \leq f_{n} \leq g_{n}$.

i) consider $g_{n} - f_{n} \geq 0$
because $g_{n} \to g$ a.e and $f_{n} \to f$ a.e, then $g_{n} - f_{n} \to g - f$ a.e.
by fatou's lemma, we have

$\int g - f \leq \lim \inf \int (g_{n} - f_{n})$

using the linearity of integral, then we get

$\int g - \int f \leq \lim \inf \int g_{n} - \lim \sup \int f_{n}$

because $\int g = \lim \int g_{n} = \lim \sup \int g_{n} = \lim \inf \int g_{n}$, then

$\int f \geq \lim \sup \int f_{n}$

ii) consider $f_{n} + g_{n} \geq 0$
by fatou's lemma, we get

$\int f \leq \lim \sup \int f_{n}$

from (i) and (ii), we conclude that

$\int f = \lim \int f_{n}$