# Thread: A.e Convergent Problem.

1. ## A.e Convergent Problem.

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

2. ## Re: A.e Convergent Problem.

Originally Posted by younhock
Suppose i have $\sum_{n=1}^{\infty} ||f_n||_2 < \infty$. How to show that the $\sum_{n=1}^{\infty} f_n$ converges absoutely almost everywhere ,
f= $\sum_{n=1}^{\infty} f_n \in L^2$ , and $||f||_2 <= \sum_{n=1}^{\infty}||f_n||_2$ .
All of this is a consequence of Minkowski's integral inequality, we have ( $dn$ is the counting measure on $\mathbb{N}$):

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

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 $\left( \sum |f_n| \right)^2$ is finite a.e. The rest follows directly from the inequality above as well.