I am currently thinking about a problem I work on and I came up with the space:
Let $K \subseteq \mathbb R^d$ be a compact set and $\operatorname{rca}(K)$ the space of regular complex Borel measures on $K$. It is well-known that $\operatorname{rca}(K)$ is a Banach space with the total variation norm. Now one could define some kind of "Sobolev space" like $$ W_{\operatorname{rca}}(K) = \{ \mu \in \operatorname{rca}(K) : \mu' \in \operatorname{rca}(K)\},$$ where the derivate $ \mu'$ is defined in the distributional sense, namely $ \mu'$ is called the derivative of $\mu$ if $$ \int_K \varphi \, \mathrm d\mu' = \int_K \varphi' \, \mathrm d\mu \qquad \text{for all } \ \varphi \in C(K). $$ Now I am very curious if spaces of this form are well understood and if there is some kind of citable literature on these kind of topics. I couldn't find literature about it but would suppose that people in stochastics and variational calculus work with these kinds of spaces.
I recently used VERY deep results of Stanislav Smirnov Decomposition of solenoidal vector charges into elementary solenoids, and the structure of normal one-dimensional flows about more general vector measures $\mu=(\mu_1,\ldots,\mu_n)$ on $\mathbb R^n$ such that the distributional divergence $\sum\limits_{j=1}^n \partial_j \mu_j$ is again a measure. (Just ask Google for the title, the first hit is the pdf of this article.) Smirnov proves a kind of Choquet decomposition of such vector charges into very simple ones induced by Lipschitz curves. Unfortunately, this article is not easy to digest (at least for me).