Radon-Nikodym derivatives as limits of ratios

Question

Let $\mu_1$ and $\mu_2$ be measures with $\mu_1 \ll \mu_2$. Suppose we can characterize (a version of) their Radon-Nikodym derivative this way:

$$\frac{d\mu_1}{d\mu_2}(x) = \lim_{n \to \infty} \frac{\mu_1(B_n)}{\mu_2(B_n)}$$

where $\bigcap_{n \in \mathbb{N}} B_n = \{x\}$.

Q. Under what conditions for $\mu_1$, $\mu_2$ and $(B_n)_{n \in \mathbb{N}}$ does this hold a.e.?

For example, it appears to hold if $\mu_1$ and $\mu_2$ are defined on the Borel sets of $\mathbb{R}$ and have densities w.r.t. Lebesgue measure, and $(B_n)_{n \in \mathbb{N}}$ is a sequence of intervals converging on $\{x\}$.

I'm interested in the most general known conditions under which Radon-Nikodym derivatives can be defined analogously to how derivatives are defined in differential calculus.

EDIT: In real life, I've just been pointed to a fairly general theorem: Bogachev, Measure Theory, vol. 1, Theorem 5.8.8. Restated:

Theorem. Let $\mu_1$ and $\mu_2$ be nonnegative measures on the Borel sets of $\mathbb{R}^n$ that are finite on all balls, such that $\mu_1 \ll \mu_2$. Let $(B_n)_{n \in \mathbb{N}}$ be a descending sequence of balls, each with center $x$, with intersection $\{x\}$. Then the above equation holds $\mu_2$-a.e.

I suspect this can be extended to sequences of measurable sets with a uniformity condition, as in Rudin's generalization of Lebesgue differentiation (Real and Complex Analysis, Theorem 7.10).

Answer 1

This question has been thoroughly investigated in the literature a while back. For example see most of the book

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.,

or chapter 7 in the book

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.

Three results which stand out are that it is enough to assume any one of the following conditions:

1 the underlying space is the real line ($\mu_1,\mu_2$ are arbitrary measures finite on compacts), and $B_n$ are intervals which contain $x$ and whose length goes to zero: see J.L. Doob's book

Doob, Joseph L., Measure theory, Graduate Texts in Mathematics. 143. New York: Springer-Verlag. xii, 210 p. (1994). ZBL0791.28001.

2 the sets $B_n$ have some uniformity in their shape and they contain $x$ in their interior (as you described in the book by Rudin)

3 $f:=\frac{d\mu_1}{d\mu_2}$ is not just in $L^1(\mu_2)$ but rather satisfies stronger integrability properties. For example in $\mathbb{R}^n$ if $\mu_2$ is the Lebesgue measure it is enough to ask that $|f|\log^+(|f|)^{n-1}\in L^1(\mu_2)$; this latter result appears in both books cited above, and also as Theorem 2.2.1 in the book

Khoshnevisan, Davar, Multiparameter processes. An introduction to random fields, Springer Monographs in Mathematics. New York, NY: Springer. xix, 584 p. (2002). ZBL1005.60005.