Prove that convergence of the infinite series of integral of absolue values of a sequence of functions implies convergence
Let $(X,\Sigma,\mu)$ be a measure space and let $f_{n}:X\rightarrow \mathbb{R} $ is an integrable function such that $\sum_{n=1}^{\infty } \int |f_{n}|du$ is convergent.
Prove that $\sum_{n=1}^{\infty } f_{n}$ converges almost everywhere to an integrable function and that
$$\int \sum_{n=1}^{\infty}f_{n} du=\sum_{n=1}^{\infty}\int f_{n}du.$$
Noah Taylor
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Daniel90
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The answer is accepted.
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