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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 $L^1.$

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    $\begingroup$ Measurability of $f_n$ is an implicit assumption. $\endgroup$ – Giuseppe Negro Mar 10 '13 at 20:14
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Hint: dominated convergence on a subsequence.

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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.

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