# Radon–Nikodym derivative and “normal” derivative

given a measurable space $(X,\Sigma)$, if a $\sigma$-finite measure $\nu$ on $(X,\Sigma)$ is absolutely continuous with respect to a $\sigma$-finite measure $\mu$ on $(X,\Sigma)$, then there is a measurable function $f$ on $X$ and taking values in $[0,\infty)$, such that

$$\nu(A) = \int_A f \, d\mu$$

for any measurable set $A$.

$f$ is called the Radon–Nikodym derivative of $\nu$ wrt $\mu$.

I was wondering in what cases the concept of Radon–Nikodym derivative and the concept of derivative in real analysis can coincide and how?

Thanks and regards!

#### Solutions Collecting From Web of "Radon–Nikodym derivative and “normal” derivative"

Consider absolute continuity of functions as defined in Royden’s Real Analysis. These functions are integrals of their derivatives and they are, in fact Radon-Nikodym derivatives.