# A.e Convergent Problem.

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• Dec 9th 2011, 08:46 PM
younhock
A.e Convergent Problem.
Suppose i have $\displaystyle \sum_{n=1}^{\infty} ||f_n||_2 < \infty$. How to show that the $\displaystyle \sum_{n=1}^{\infty} f_n$ converges absoutely almost everywhere ,
f=$\displaystyle \sum_{n=1}^{\infty} f_n \in L^2$ , and $\displaystyle ||f||_2 <= \sum_{n=1}^{\infty}||f_n||_2$ .
• Dec 9th 2011, 09:40 PM
Jose27
Re: A.e Convergent Problem.
Quote:

Originally Posted by younhock
Suppose i have $\displaystyle \sum_{n=1}^{\infty} ||f_n||_2 < \infty$. How to show that the $\displaystyle \sum_{n=1}^{\infty} f_n$ converges absoutely almost everywhere ,
f=$\displaystyle \sum_{n=1}^{\infty} f_n \in L^2$ , and $\displaystyle ||f||_2 <= \sum_{n=1}^{\infty}||f_n||_2$ .

All of this is a consequence of Minkowski's integral inequality, we have ($\displaystyle dn$ is the counting measure on $\displaystyle \mathbb{N}$):

$\displaystyle \sum \| f_n \|_2 = \int_{\mathbb{N}} \left( \int_X |f(n,x)|^2dx \right)^{1/2} dn \geq \left( \int_X \left( \int_{\mathbb{N}} |f(n,x)|dn \right)^2 dx \right)^{1/2} = \left( \int_X \left( \sum |f_n(x)| \right)^2 dx \right)^{1/2}\geq \| f\|_2$

Since the left hand side is finite, so is the right side, but an integrable function can only be infinite on a null set so we get that $\displaystyle \left( \sum |f_n| \right)^2$ is finite a.e. The rest follows directly from the inequality above as well.