Suppose $f : A \rightarrow B$ is a continuous function between locally compact topological spaces equipped with regular measures, lets call them $\mu_A$ and $\mu_B$. Lets also assume that $\mu_A$ has empty support. Then for each $a \in A$, we can study what $f$ does to the measure of a little neighborhood of $a$. To ensure the image of such a neighbourhood is measurable, lets focus just on the compact neighbourhoods. In particular, we can study the net $f[a] : \mathrm{CompactNbhd}(a) \rightarrow \mathbb{R}$ given by $$f[a](N) = \frac{\mu_B(f(N))}{\mu_A(N)}.$$ The numerator makes sense because $f(N)$ is compact, hence closed, hence Borel. The denominator is non-zero because the $\mu_A$ has non-empty support. If this net converges, presumably this tells us something important about what $f$ is doing to the measures of things near $a$. For example, if $A$ and $B$ are Euclidean spaces of equal dimension, it should be the case that this number converges to $|\mathrm{det} Jf_a|$, where $J$ is the Jacobian. I guess something similar can be probably be said if $A$ and $B$ are Riemannian manifolds.
Question. When it exists, what is this number called?