Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I earlier asked this question but I have not had a general classification in the posted answers there. So here is a new question.

I am looking now for some special cases as suggested in one of the comments. Consider the space $L^p (\Omega, \mathcal{F}, \mu)$ where $(\Omega, \mathcal{F}, \mu)$ is some measure space satisfying $\Omega$ being uncountable, $\mu$ being a Radon measure and $p \in [1, \infty]$. Is it true that $L^p \subseteq L^q$ for $q \leqslant p$ and if not, what are some known counter-examples?

I would be quite satisfied with just references to articles or sections in books!

share|improve this question

1 Answer 1

up vote 1 down vote accepted

Take $\Omega = \mathbb{N}$ with $\mu$ counting measure. Consider $f(n) = n^\alpha$. Exercise: For which $p$ (in terms of $\alpha$) is $f \in L^p$?

This holds if $\mu$ is a finite measure, and otherwise it usually does not.

share|improve this answer

Your Answer

 
discard

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.