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The Radon–Nikodym theorem states that,

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!

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    @Tim: I'll add the above as a comment, because I haven't thought deeply about it. Concerning your point on R-N, I have to think a little bit about what exactly I wanted to say, I'll get back to it at some point (ping me in case I forget).2011-05-12

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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.

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    @ncmathsadist: Thanks! I was wondering where in Royden's Real Analysis "these functions are integrals of their derivatives and they are, in fact Radon-Nikodym derivatives" is mentioned? I searched to no avail.2012-03-09