Across my studies I came across an object which could be thought of as a "derivative" of a function defined on a set of measures and I was looking for references to the matter (if any). Specifically, let $\mathscr{M}$ be the set of (finite) measures on $\mathbb{R}^n$ and let $P:\mathscr{M}\to\mathbb{R}$ be an arbitrary function. For every $\eta\in\mathscr{M}$ and $x\in \mathrm{Supp}(\eta)$ we can consider the limit $$ \lim\limits_{\varepsilon\to 0}\frac{P(\eta_{\varepsilon})-P(\eta)}{\eta(B_{\varepsilon}(x))} $$ where $B_{\varepsilon}(x)$ is the open ball of radius $\varepsilon$ centered in $x$ and $\eta_{\varepsilon}$ is the measure defined by $\eta_{\varepsilon}(A)=\eta(A)+\eta(A\cap B_{\varepsilon}(x))$. I am looking for references on the study of the previous limit (properties, conditions for its existence, etcetera) or another way of looking at it. I understand one could simply re-express the limit through a function $f:[0,\infty)\to \mathbb{R}$ given by $f(\varepsilon)=P(\eta_{\varepsilon})$ and then multiply/divide by the Lebesgue measure of $ B_{\varepsilon}(x)$ to sort of try to see this as a normal derivative multiplied by a Radon-Nikodym derivative, however I am not aware of the details necessary for carrying this out and it seems to me that work like this has already be done.
Thanks for any help you might provide.