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
$$\int \sum_{n=1}^{\infty}f_{n} du=\sum_{n=1}^{\infty}\int f_{n}du.$$


Answers can only be viewed under the following conditions:
  1. The questioner was satisfied with and accepted the answer, or
  2. The answer was evaluated as being 100% correct by the judge.
View the answer
The answer is accepted.
Join Matchmaticians Affiliate Marketing Program to earn up to a 50% commission on every question that your affiliated users ask or answer.