# Sequence in Lebesgue Space

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• Dec 13th 2009, 01:18 PM
pascal4542
Sequence in Lebesgue Space
Show that if $\displaystyle (f_n)_n$ is a sequence in $\displaystyle L^1[0, 1]$ is such that $\displaystyle \sum_{n=1}^{\infty} || f_n ||_1 < \infty$ then $\displaystyle \sum_{n=1}^{\infty} | f_n(s) | < \infty$ for almost every $\displaystyle s \in [0, 1]$.

This looks simple to prove. However, I do not see how to prove this right now. Any hints on how to proceed would be appreciated.
• Dec 13th 2009, 01:48 PM
Laurent
Quote:

Originally Posted by pascal4542
Show that if $\displaystyle (f_n)_n$ is a sequence in $\displaystyle L^1[0, 1]$ is such that $\displaystyle \sum_{n=1}^{\infty} || f_n ||_1 < \infty$ then $\displaystyle \sum_{n=1}^{\infty} | f_n(s) | < \infty$ for almost every $\displaystyle s \in [0, 1]$.

This looks simple to prove. However, I do not see how to prove this right now. Any hints on how to proceed would be appreciated.

You probably know that if $\displaystyle f$ is nonnegative and $\displaystyle \int f<\infty$, then $\displaystyle f(s)<\infty$ for almost every $\displaystyle s$ (this is almost by definition of the integral).

Then apply that to $\displaystyle f=\sum_{n=1}^\infty |f_n(s)|$... (You will need the monotone convergence theorem to justify the computation of $\displaystyle \int f(=\int \lim_N \sum_{n=1}^N |f_n|$)