Question from Folland on modes of convergence

Question

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 $L^1.$

Answer 1

Hint: dominated convergence on a subsequence.

Answer 2

Here is a possible way when the space is assumed $\sigma$-finite. It can always be assumed as we can write $S=\bigcup_{n\geqslant 1}\{x, |g(x)|>n^{-1}\}$ and work with $S$ as underlying space.