# Thread: Sequence in Lebesgue Space

1. ## Sequence in Lebesgue Space

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

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.

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

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

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