$\sum f_n$ converges in $L^p(\mu)$ implies $\|f_n\|_p \to 0$ as $n\to\infty$.

Question

I need some help on a measure theory question. If a function series $\sum f_n$ converges in $L^p(\mu)$ this implies $\|f_n\|p \to 0$ as $n\to+\infty$.

How can I show this? I thought about using dominate convergence theorem since, $\sum f_n <+\infty$ almost anywhere, then $f_n(x)\to0$ almost anywhere. But I can't find a dominating function, unless $|f_n| \leq |\sum f_n|$. I kindly thank anyone who would like to help.

Answer 1

If $S_n$ is the $n-$th partial sum then $f_n=S_n-S_{n-1}$ so $||f_n||_p =\|S_n-S_{n-1}||_p \leq \|S_n-S||_p+\|S-S_{n-1}||_p \to 0$ (where $S=\sum f_n$). Completeness is not involved and it is it not true that $\sum \|f_n\|_p <\infty$ [Convergence does not imply absolute convergence even for series of constants.].

Explicit counter-example to $\sum \|f_n\|_p <\infty$: $f_n =\chi_{(0,1)} \frac {(-1)^{n}} n$.