Definition of $L^0$ space

Note that when we restrict ourselves to the probability measures, then this terminology makes sense: $L^p$ is the space of those (equivalence classes of) measurable functions $f$ satisfying $$\int |f|^p<\infty.$$ Therefore $L^0$ should be the space of those (equivalence classes of) measurable functions $f$ satisfying $$\int |f|^0=\int 1=1<\infty,$$ that is the space of all (equivalence classes of) measurable functions $f$. And it is indeed the case.

If the measure of $S$ is finite, the $L^p$ spaces are nested: $L^{p}\subset L^q$ whenever $p\ge q$. The smaller the exponent, the larger the space. Since the space of measurable functions contains all of the $L^p$ for $p>0$, one may be tempted to denote it by $L^0$.

This temptation should be resisted and the notation $L^0$ banished from usage. [/rant]