Consider the following measure-theoretic analogy to the concept of derivative.
Let $(\Omega, m)$ be a metric space ($\Omega\neq\emptyset$). Denote the topology on $\Omega$ induced by $m$ as $\tau$. Let $\mu, \nu$ be measures on $\sigma(\tau)$, and let $\omega^*\in\Omega$ belong to $\mu$'s support. Suppose the following holds. There is a number $d \in \mathbb{R}$ such that, for every sequence $(B_1, B_2, \dots)$ in $\sigma(\tau)$ consisting of $\tau$-neighborhoods of $\omega^*$, whose $m$-diameters converge to $0$, $$ d = \lim_{n\rightarrow\infty} \frac{\nu(B_n)}{\mu(B_n)}. $$ Define $D^\nu_\mu(\omega^*):= d$.
Has this concept been studied? Does it have a name?
How this concept comes about
This concept arises naturally in the settings of conditional probability. For instance, in his textbook Basic Probability Theory, Robert Ash writes: "A reasonable approach to the conditional probability $P(R_2 \in B\ |\ R_1 = x_0)$ is to look at $P(R_2 \in B\ |\ x_0-h< R_1 < x_0+h)$ and let $h\rightarrow 0$." (p. 136)
The limit $$ \lim_{h\rightarrow\infty} P(R_2 \in B\ |\ x_0-h< R_1 < x_0+h) = \lim_{h\rightarrow\infty}\frac{P(R_2 \in B, x_0-h< R_1 < x_0+h)}{P(x_0-h< R_1 < x_0+h)} $$ can be restated (sort of) in the notation I introduced above as $D^\nu_\mu(x_0)$, where $\Omega = \mathbb{R}$, $m$ is the Euclidean metric, $\mu = P_{R_1}$ (i.e. $\mu$ is $R_1$'s distribution), and $\nu$ is the measure that assigns to every Borel set $A$, $\nu(A) := P(R_2 \in B, R_1 \in A)$.
As a further evidence to how natural this intuitive view of conditional probability is, take this question posted yesterday on this forum, where a user who appears to be a beginning student of probability, is suggesting a definition of conditional probability essentially identical to Ash's "definition" above.
In my answer to said question I proved the following result, which I reformulate using the notation introduced above.
Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $(S,m)$ be a metric space. Denote by $\tau$ the topology on $S$ induced by $m$. Let $Y:\Omega\rightarrow S$ be $\mathcal{F}/\sigma(\tau)$-measurable and let $A \in \mathcal{F}$.
Denoting the Euclidean topology on $\mathbb{R}$ by $\mathcal{E}$, suppose that, for some version $f$ of the conditional probability $P(A\ |\ Y=y)$, $f$ is $\tau/\mathcal{E}$-continuous at some $y^*\in S$ that lies in $P_Y$'s support.
Then $$ f(y^*) = D^\nu_\mu(y^*),\hspace{1cm}(*) $$ where $\mu := P_Y$, and $\nu$ is the measure on the Borel field on $\mathbb{R}$ that assigns to every Borel set $B$, $\nu(B):=P(A \cap \{Y\in B\})$. In particular, the expression on the right hand side of $(*)$ is well-defined.