There is a result in the wikipedia article about $L_p$ space embedding:

a. Let $0 ≤ p < q ≤ ∞$. $L_q(S, μ)$ is contained in $L_p(S, μ)$ iff $S$ does not contain sets of arbitrarily large measure;

b. Let $0 ≤ p < q ≤ ∞$. $L_p(S, μ)$ is contained in $L_q(S, μ)$ iff $S$ does not contain sets of arbitrarily small non-zero measure.

I can only find one which is *similar* to the result above: *Another note on the inclusion $L^p(\mu) ⊂ L^q(\mu)$*(by A. Villani, The American Mathematical Monthly, Vol. 92 (1985), No. 7, 485–487):

The following conditions on measure space $(X,\Sigma,\mu)$ are equivalent:

1. $\sup_{E\in{\mathscr A}_{\infty}}\mu(E)<+\infty$,where ${\mathscr A}_\infty=\{E\in\Sigma:\mu(E)<+\infty\}$.

2. $L^p(\mu)\subset L^q(\mu)$ for all $p,q\in(0,\infty)$ with $p>q$.

and

the following conditions on measure space $(X,\Sigma,\mu)$ are equivalent:

3. $\inf_{E\in{\mathscr A}_{0}}\mu(E)>0$,where ${\mathscr A}_0=\{E\in\Sigma:\mu(E)>0\}$.

4. $L^p(\mu)\subset L^q(\mu)$ for all $p,q\in(0,\infty]$ with $p<q$.

Is there a typo in (a) ($0\leq p<q\leq\infty$ should be $0\leq p<q<\infty$?)? Can someone come up with a reference for the results (a) and (b)?