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
$$\nu(A) = \int_A f \, d\mu$$
- Finitely but not countably additive set function
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- Assume that $ f ∈ L()$ and $\int x^nf(x)dx=0$ for $n=0,1,2…$.
- Examples of measurable and non measurable functions
- Convolution of an integrable function of compact support with a bump function.
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!
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.