TITLE: Definition of $L^0$ space
QUESTION [22 upvotes]: 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!

REPLY [2 votes]: $L^0$ is just a notation to refer to the weakness of the topology of convergence in measure. It is not locally bounded but is metrizable if the underlying measure space is non-atomic and $\sigma$-finite. The proper terminology is F-norm for complete metric linear spaces(F-spaces) which are not locally bounded. When locally bounded, such spaces are quasi-Banach spaces which $L^0$ is not. Thus, the main difference between a norm(quasi-norm) and an F-norm is the homogeneity. For quasi-norms $||cx||\le |c|||x||$ for all scalars c but for F-spaces $||cx||\le ||x||$ only for $|c|\le 1$. Despite being almost 40 years old, "An F-space Sample" by Kalton, Peck, and Roberts is still one of the best sources for this general view of functional analysis which focusses on topology and not on operator theory for Hilbert spaces.