Given two non-negative Borel measures $\mu$, $\nu$ on $\mathbb{R}^n$, that are finite on compact sets, such that $\nu\ll\mu$, it is well known that


$$\frac{d\nu}{d\mu}(x)= \lim\_{\epsilon\to 0} \frac{\nu(B\_\epsilon(x))}{\mu(B\_\epsilon(x))}$$


holds $\mu$ a.e., where $B\_\epsilon(x)$ is the open ball of radius $\epsilon$ centered at $x\in\mathbb{R}$. Can this be generalized to measures on arbitrary metric spaces, or even Banach spaces?


I am particularly interested in the case of probability measures (for which the quantities in the above fraction are finite). Additionally, if necessary, these could also be Radon.



Is the ball ratio theorem for Radon–Nikodým derivative known for general metric spaces?
Arbitrary metric spaces, no.


A classic text discussing such "derivation" is:


*Hayes, C. A.; Pauc, C. Y.*, Derivation and martingales, Ergebnisse der Mathematik und ihrer Grenzgebiete. 49. Berlin-Heidelberg-New York: Springer-Verlag. VII, 203 p. (1970). [ZBL0192.40604](https://zbmath.org/?q=an:0192.40604).


I think you get trouble even in $\mathbb R^2$ if you define your metric so that the $r$-ball centered at $(x,y)$ is something like
$$
[x-r,x+r] \times [y-e^{-1/r},y+e^{-1/r}]
$$




---


*plug*  

This is covered in Chapter 7 of my text with Sucheston:


*Edgar, G. A.; Sucheston, Louis*, [**Stopping times and directed processes.**](http://dx.doi.org/10.1017/CBO9780511574740), Encyclopedia of Mathematics and Its Applications 47. Cambridge: Cambridge University Press (ISBN 978-0-521-13508-5/pbk). xii, 428 p. (2010). [ZBL1189.60074](https://zbmath.org/?q=an:1189.60074).


