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!
 A: 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]
A: I do think that $L^0$ is nice usage. As is well-known $\lim_{p \to \infty} \|\cdot \|_{L^p} = \|\cdot\|_\infty$ for certain spaces or functions. The case for $L^0$ is not that pretty, but at least still nice.
Recall the distribution function $\mu$ of $f$ given by,
$$\mu(\alpha) := \mu_f(\alpha) := \mu\{|f|>\alpha\}.$$
Fubini gives that,
$$\|f\|_{L^p}^p = p \int_0^\infty \mu_f(\alpha) \alpha^p \frac{\mathrm{d}\alpha}{\alpha}.$$
We can define the Lorentz spaces in a similar way. And indeed, for a finite measure space, we have if $p < q$ that
$$L^q \subseteq L^p.$$
Hence, it is natural to define $L^0$ as,
$$L^0 = \bigcup_{p > 0} L^p.$$
We would like to have that $L^0$ is also complete as a metric space, otherwise the notation would be quite deceiving indeed. For this we need a notation of convergence. On $L^p$ for $0 < p < 1$ it is not the norm that induces the metric, but it is $\|\cdot\|_p^p$.
So, for $0 < p < 1$ we have,
  $$d_p(f, g) = p \int_0^\infty \mu\{|f - g|>\alpha\} \alpha^p \frac{\mathrm{d}\alpha}{\alpha}.$$
$\varepsilon$-neighborhoods $N^p_\varepsilon$ of $f$ in $L^p$ are then given by
$$N^p_\varepsilon(f) = \Biggl\{g : p \int_0^\infty \mu\{|f - g|>\alpha\} \alpha^p \frac{\mathrm{d}\alpha}{\alpha} < \varepsilon \Biggr\}.$$
Too be continued, I wanted to give a brief remark, but I have decided otherwise in due progress.
A: $L^0$ 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. 
A: 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.
