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!