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 Bogachev, Vladimir I., Differentiable measures and the Malliavin calculus, ZBL1247.28001. 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.
