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Is $\int_{\Omega} \bigg( \sum_{n=1}^{\infty} |f_n| \bigg)^p d \mu$ really in $L^p$?

What confuses me that I think that $|f_n|$ should have some power of $p$.

$f_n$ are elements of $L^p$. $\sum_n f_n$ is an absolutely convergent sequence in $L^p$.

Def. of $\| \cdot \|_p$ of $L^p$:

https://en.wikipedia.org/wiki/Lp_space#Lp_spaces

So I think the form I give, doesn't look like that. Yet my notes claim it's that (in order for the series to be in $L^p$?).

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  • $\begingroup$ By absolutely convergent, do you mean $\sum_{n=1}^\infty ||f_n||_p <\infty$ ? $\endgroup$ – Danny Pak-Keung Chan May 9 at 19:37
  • $\begingroup$ @DannyPak-KeungChan Yes. $\endgroup$ – mavavilj May 9 at 19:37
  • $\begingroup$ Are you asking whether $$ \int_\Omega \left( \sum_n |f_n| \right)^p \,\mathrm d\mu < +\infty, $$ i.e. whether the function $$ \sum_n |f_n| $$ is in $L^p$? $\endgroup$ – MisterRiemann May 9 at 19:45
  • $\begingroup$ @MisterRiemann That's part of the proof I'm looking at, but I don't understand how it concludes that $\sum_{n=1}^{\infty} |f_n|$ $\in L^p$. Because I think $|f_n|$ should have some power of $p$. Since the formulation given here doesn't look like the one for $L^p$. $\endgroup$ – mavavilj May 9 at 19:46
  • $\begingroup$ A similar proof is here: "12 proposition". folk.ntnu.no/hanche/notes/lp/lp-a4.pdf $\endgroup$ – mavavilj May 9 at 19:50
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By Minkowski, $$ \left\Vert \sum_{n=1}^N |f_n| \right\Vert_p \leq \sum_{n=1}^N \Vert f_n\Vert_p, $$ and so by Fatou, $$ \left\Vert \sum_{n=1}^\infty |f_n| \right\Vert_p = \left\Vert \lim_N\sum_{n=1}^N |f_n| \right\Vert_p \leq \liminf_N \left\Vert \sum_{n=1}^N |f_n| \right\Vert_p \leq \liminf_N \sum_{n=1}^N \Vert f_n\Vert_p = \sum_{n=1}^\infty \Vert f_n\Vert_p < +\infty. $$ EDIT: By request, the line above (skipping a few steps) could also be written in the form \begin{align} \left(\int_\Omega \left(\sum_{n=1}^\infty |f_n|\right)^p \,\mathrm d\mu\right)^{1/p} \leq \liminf_N \left(\int_\Omega \left(\sum_{n=1}^N |f_n|\right)^p \,\mathrm d\mu\right)^{1/p} \leq \liminf_N \sum_{n=1}^N \left( \int_{\Omega} |f_n|^p \,\mathrm d\mu \right)^{1/p}. \end{align}

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  • $\begingroup$ So are you referring to the $\| \cdot \|_p$ of $L^p$? How does one apply that "integral form" to an infinite series? $\endgroup$ – mavavilj May 9 at 19:53
  • $\begingroup$ Yes, of course. $\endgroup$ – MisterRiemann May 9 at 19:53
  • $\begingroup$ Shouldn't $\sum_{n=1}^{\infty} |f_n|$ have power $p$? Since in the $\| \cdot \|_p$ of $L^p$ there's $|f|^p$. $\endgroup$ – mavavilj May 9 at 19:54
  • $\begingroup$ I don't understand why this would affect the argument. $\endgroup$ – MisterRiemann May 9 at 19:56
  • $\begingroup$ It doesn't, but I'm asking about this notational confusement. See the question and the title. I'm asking why is that form I give in $L^p$, because it doesn't look the same as the $\|\cdot\|_p$ of $L^p$. $\endgroup$ – mavavilj May 9 at 19:56

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