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Let $(X,\mathcal{A}, \mu)$ be an arbitrary measure space and $u_1, \dots, u_n$ be $L^p$ functions, $1\le p <\infty$. Then $w=\max\{|u_1|, |u_2|, \dots, |u_n|\}$ is also in $L^p$.

I know how to show this in the case of $p=1$, but for $p>1$, I am stuck on how to show this. I would greatly appreciate any help.

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    Do you already know that sums of $L^p$ functions are $L^p$? Because the maximum is a sum of a certain collection of $L^p$ functions... – 2017-02-03
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    @Ian yes I do know the first fact, but is there a way to describe the maximum of $n>2$ functions like the case for $n=2$? – 2017-02-03
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    No it's not but you can use induction argument. $u_1,u_2 \in L^p$ so is $a_1 = \max\{|u_1|,|u_2|\}$, then $a_2 = \max\{|u_1|,|u_2|,|u_3|\} = \max\{a_1,|u_3|\}$ is also in $L^p$, etc… – 2017-02-03
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    You can: $\max \{ u_1,\dots,u_n \}=\sum u_k 1_{A_k}$ where $A_k$ is the set where $u_k$ is the biggest. – 2017-02-03

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