Some cases are simple enough. For example if \(\displaystyle \mu (X)<\infty\) and \(\displaystyle 1\leq p \leq q \leq \infty\) then \(\displaystyle L^q \subset L^p\) (this is not difficult to prove, just use Hölder's inequality).Intuitively, if on the contrary, \(\displaystyle X\) had no finite measure subset the inclusions would be reversed (I don't yet have a proof for this), and the general case would (?!) follow by decomposing your space in this (or some similar) fashion.