What space is the set of all CDFs? I'm relatively new to functional analysis and am trying to make comparisons across different CDFs (cumulative density functions), i.e. right-continuous, weakly increasing functions $F:\Sigma\subseteq\mathbb{R}\to[0,1]$.
I cannot fit this into the vector space framework since I don't know what a scalar multiplication would mean in such a case. But does there exist some norm over this set, or some category that this space belongs to?
Question Background: Maybe it's useful to explain why I'm asking. In fact, this is related to this question: Is "almost all function" a well defined concept?
Essentially, I'm trying to show that certain CDFs are very rare (a "measure-zero" type of argument). Somebody on mathoverflow recommended to look into the notion of shy sets, but this applies to topological vector spaces. And of course CDFs are already "very rare" among functions, so I currently only see how I could say that CDFs are shy sets among all functions. But that's not what I need, of course. I want to say that those with my property are shy among the space of CDFs. Hence I need some structure over the set of CDFs so I can look into applicable notions.
 A: You can view CDFs as a subset of the real functions $X$ of finite total variation on $\mathbb{R}$ that vanish at $-\infty$ with variation norm:
$$     
                 \|f\|=\lim_{R\rightarrow\infty}V_{R}^{R}(f).
$$
This is a Banach space. It's related to the dual of continuous functions. I won't say exactly because $\infty$ is always an issue. However, on a finite interval, the space $C[a,b]$ of continuous functions on $[a,b]$ with $\max$ norm $\|f\|=\max_{x\in[a,b]}|f(x)|$ has dual $BV[a,b]$ consisting of all complex functions of bounded variation on $[a,b]$ that vanish at $a$ (just for normalization.) The dual correspondence is through the Riemann-Stieltjes integral:
$$
                        F_{\mu}(f) = \int_{a}^{b}f(t)d\mu(t).
$$
In fact, the norm of the linear function is the variation of the representing function:
$$
                       \|F_{\mu}\|_{C[a,b]^{\star}}=V_{a}^{b}(\mu)
$$
Your functions all have norm $1$, and define positive linear functionals. I'm not sure if any of this helps, but it turns it into a classical-looking Functional Analysis formulation.
