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When $f$ is a Lebesgue measurable function, is $\Vert f \Vert_p$ continuous in $p$, when $p$ is in a set such that f belongs to $L_p$? If yes, how to show that? Thanks!

Yes. It is an exercise taken from Rudin's book. I have spent quite a while thinking but got no idea.

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Is this homework? What have you tried? –  Qiaochu Yuan Nov 10 '10 at 10:16
    
Yes. I have tried to use composition of continous functions still being continous, but the integral is there... –  steveO Nov 10 '10 at 10:32

2 Answers 2

up vote 3 down vote accepted

This may not be the answer you are looking for, but:

Using Holder's inequality you can show that the map $p^{-1}\mapsto \|f\|_p$ is log-convex. Now just use the fact that any convex function on an interval is continuous.

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Thanks, Willie! It looks like the right way to go. –  steveO Nov 10 '10 at 10:54

So you're asking about $p \mapsto \left( \int_X |f|^p d\mu \right)^{1/p}$. Isn't this clearly a composition of continuous functions?

First $p \mapsto |f|^p$ (no hints, but this should be pretty easy), then the Lebesgue integral (hint: it's a linear map between the Banach space $L^p(X,\mu)$ and $\mathbb{R}$. Which linear maps are continuous?), and finally $x \mapsto x^{1/p}$.

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Thanks, Kahen. I thought similar way at first but got stuck at the integral. Looks like your way is correct as well... –  steveO Nov 10 '10 at 10:59

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