Investigating an abstract Cauchy problem on the space of measures with bounded variation I came up with the following space:

Let $\operatorname{BV}[a,b]$ the space of all functions $f:[0, 1] \to \mathbb C$ with bounded variation, i.e., the supremum of $\sum_{i = 1}^n \lvert f(x_i) - f(x_{i - 1}) \rvert$ over all finite partitions $\lbrace x_0, \dots, x_n\rbrace$, $n \in \mathbb N$, is finite. Now consider the "Sobolev measure space"
$$\mathrm{M}^1[a,b] := \lbrace \mu \in \mathrm{M}[a,b] : \exists \, f \in \operatorname{BV}[a,b] : \mu = f \, \mathrm d x\rbrace. $$
Since each function of bounded variation is differentiable almost everywhere, it is possible, to define the operator 
$$A := \frac{\mathrm d}{\mathrm d x}, \quad A\mu := f', \quad D(A) := \mathrm{M}^1[a,b]. $$
I would like to know if operators in spirit of $A$ and spaces in spirit of $\mathrm{M}^1[a,b]$ are already covered in the existing literature. 

**Remark:** A concept closely related to my questing is the so called *Skohorod differentiability* that can be found for instance in <cite authors="Bogachev, Vladimir I.">_Bogachev, Vladimir I._, Differentiable measures and the Malliavin calculus, [ZBL1247.28001](https://zbmath.org/?q=an:1247.28001).</cite> However, the definition of this kind of differentiability is then defined only for measures on $\mathbb R$ via the duality to $C_b(\mathbb R)$. It is then proven that a measure on $\mathbb R$ is Skohorod differentiable if and only if it has a density of bounded variation, which was my main motiviation to define the space above. But I cannot believe that I am the first to come up with the pretty simple idea to consider this space.