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

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.

• Cauchy property. $L_p$ is a complete metric space. Jun 5, 2022 at 23:06
• @GEdgar Where does completeness come in? Jun 5, 2022 at 23:49

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 $$\sum \|f_n\|_p <\infty$$: $$f_n =\chi_{(0,1)} \frac {(-1)^{n}} n$$.
• My thought was: if $\sum f_n$ converges then $S_n$ is Cauchy, and then $f_n = S_{n}-S_ {n-1} \to 0$. But (as pointed out here) this direction of [converge iff Cauchy] does not need completeness. Jun 6, 2022 at 11:56