I learned Radon-Nikodym theorem in class and I know what exactly it is. But I am not sure about how to compute Radon-Nikodym derivative... Any reference does not explicitly say about how to compute Radon-Nikodym derivative..
Can anybody help me about how to compute it or provide some useful thm regarding it?
If $d\mu = f \, dm$, where $m$ is the Lebesgue measure on $\mathbb{R}^n$, then there is a concrete way of realizing the differentiation of measures; in particular, for almost every $x \in \mathbb{R}^n$, $$ \lim_{r \rightarrow 0} \frac{\mu(B(r,x))}{m(B(r,x))} = f(x)$$
In principle, a similar result holds if $d\mu = f \, d\nu$, but the issue is that then we don't want to use the sets $B(r,x)$ because we don't know how those behave under the measure $\nu$; so ultimately you have to know a lot about the measures explicitly if you want to do any computation.