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).