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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.

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  • $\begingroup$ Cauchy property. $L_p$ is a complete metric space. $\endgroup$
    – GEdgar
    Jun 5, 2022 at 23:06
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    $\begingroup$ @GEdgar Where does completeness come in? $\endgroup$
    – geetha290krm
    Jun 5, 2022 at 23:49

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

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  • $\begingroup$ This answer is correct imo. However I wouldn't suggest addressing other answer's/comment's problems in one's own answer, so that the thread does not become chaotic. $\endgroup$
    – Snoop
    Jun 5, 2022 at 23:41
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    $\begingroup$ Thank you for your answer and for the counter-example!! Very useful $\endgroup$
    – Aron
    Jun 5, 2022 at 23:43
  • $\begingroup$ 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. $\endgroup$
    – GEdgar
    Jun 6, 2022 at 11:56

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