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 $L^p$?).
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}