Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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
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.

share|cite|improve this answer
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 $x \mapsto x^{1/p}$.

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.