Struggling with the intuitive picture behind the Radon-Nikodym derivative

Question

I understand the math and the formal statements behind this theorem (well perhaps not), but I cannot seem to grasp an intuitive picture behind it like how one can intuitively understand an ordinary derivative as a simple rate of change. I have read through a few answers on here but most seem to focus on probability applications. Can anyone provide an intuitive picture of what is going on behind this theorem?

Answer 1

[Following Masacroso's notation]

There is the common notation $\frac{d\nu}{d\mu}$ for the function $f$ that follows the intuition that this function represents a "rate of change of the density $\nu$ with respect to $\mu$." This notation is quite suggestive, as the equation in Masacroso's comment can be written as $\int_A \, d\nu = \int_A \frac{d\nu}{d\mu} \, d\mu$. Although this isn't exactly a derivative in the usual sense (since $\nu$ and $\mu$ are measures, not functions), the Radon-Nikodym derivative does have many similar properties.

In some instances the Radon-Nikodym derivative is literally a derivative of some function. For example, if $\nu$ is some measure on the real line with distribution function $F(x) := \nu((-\infty, x])$ satisfying some condition (absolute continuity as a function), then the Radon-Nikodym derivative with respect to the Lebesgue measure is precisely the derivative of $F$, as $\nu((a,b]) = F(b)-F(a) = \int_a^b F'(x) \, dx$ by the fundamental theorem of calculus.