Is the ball ratio theorem for Radon–Nikodým derivative known for general metric spaces?

Question

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.

Answer 1

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.

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 $$ [x-r,x+r] \times [y-e^{-1/r},y+e^{-1/r}] $$

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

Edgar, G. A.; Sucheston, Louis, Stopping times and directed processes., Encyclopedia of Mathematics and Its Applications 47. Cambridge: Cambridge University Press (ISBN 978-0-521-13508-5/pbk). xii, 428 p. (2010). ZBL1189.60074.