I have been wondering for a bit what is the actual utility of a Radon-Nikodym derivative. One thing I've observed is that it provides a general notion of derivative given a measure space (under further assumptions).
As application of this is a classic one is the derivative of standard functions of real variable (so they generalize the riemann integral on the real line) but another one was derivitives w.r.t. counting measures.
To some extent, and correct me if I am wrong, they unify for example the notion of derivative of real function of real variables with finite differences for example.
I wonder if there's another application in graph theory for example, given a graph $G = (V,E)$ we can define a function $f : V \to \mathbb{C}$ as $f(v_i) = x_i$ I am familiar with the notion of Combinatorial Laplacian however I was wondering if it is possible to associate to a graph $G$ some measure space so that the definition of derivative would follow quite naturally from the use of Radon-Nikodym derivative.
I am no expert in graph theory, but this sounds one of those things that someone surely must have tried or thought about, but I don't even know what keywords should I look for.