I have been reading through Folland, and I am having a hard time answering the following question. Any help will be much appreciated.
Suppose $\lvert f_n \rvert \leq g \in L^1$ and $f_n \rightarrow f$ in measure.
Prove the following:
(a) $\int f = \lim\int f_n.$
(b) $f_n \rightarrow f$ in $L^1.$