Definition of $L^0$ space From Wikipedia:

The vector space of (equivalence classes of) measurable functions on $(S, Σ, μ)$ is denoted $L^0(S, Σ, μ)$. 

This doesn't seem connected to the definition of $L^p(S, Σ, μ), \forall p \in (0, \infty)$ as being the set of measurable functions $f$ such that $\int_S |f|^p d\mu <\infty$. So I wonder if I miss any connection, and why use the notation $L^0$ if there is no connection?
Thanks and regards!

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.
