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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!

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up vote 2 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.

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