Consider the convex cone of quantile functions of random variables on the real line (with finite second moment), that is $$ C := \{ Q_{\mu}: \mu \in \mathcal P_{(2)}(\mathbb R) \}, $$ where $Q_{\mu}(p) := \min\{ x: \mu((-\infty, x]) \ge p \}$, $p \in (0, 1)$ is the quantile function of $\mu$. We interpret $C$ as a subset of the square-integrable functions $\mathcal L^2((0, 1))$ and consequently endow it with the $L^2$-norm.
For $f \colon (0, 1) \to \mathbb R$ we have $f \in C$ if and only if $f$ is non-decreasing and left-continuous (see e.g. section 1.1. of Random variables, monotone relations, and convex analysis by Rockafellar and Royset).
The smallest vector space containing $C$ is $\text{span}(C) = C - C$, the space of differences of quantile functions.
I've read that the space of differences of non-decreasing bounded functions on a compact interval is equal to the space of functions of bounded variation, $\text{BV}$. Is there a similar characterization of $C - C$?
The difference here is that $Q_{\mu}$ is defined on a open interval and might be unbounded, e.g. if $\mu$ is a normal distribution.
