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I just want to check if there are any other references on the definition of derivative of a regular borel measure beside the one I am reading:

A regular Borel measure on an open set $\Omega \in \mathbb{R}^n$ is an outer measure $\nu$ on $\Omega$ such that

  1. every open set is measurable,
  2. every point has a neighbourhood of finite measure,
  3. every set is contained in a $G_{\delta}$ of the same measure.

The regular Borel measure $\nu$ is differentiable at the point $x$ if the limit

$D\nu(x) = \displaystyle\lim_{|B| \rightarrow 0} \frac{\nu(B)}{|B|}$

exists, where $B$ denotes an open ball containing $x$. $D\nu(x)$ is called the derivative fo $\nu$ at $x$.

So I have looked at this and that, and the only thing comes close to it is Radon-Nikodym derivative. I am not sure if these two definitions are equivalent though.

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This is covered well in Rudin's Real and Complex Analysis. Recall that every $\sigma-$ finite regular Borel measure on $\mathbb{R}^d$ admits a decomposition into two measures, one of which is absolutely continuous with respect to the Lebesgue measure and one of which is singular with respect to the Lebesgue measure (this is the Lebesgue decomposition theorem). What you are computing is a Radon-Nikodym derivative, but it is the Radon-Nikodym derivative of the absolutely continuous part of $\nu$ with respect to the Lebesgue measure (possibly after restricting to a sufficiently small neighborhood to get finiteness).

Once the technical stuff is dealt with, the fact that this is a.e. equal to the density of the absolutely continuous part of $\nu$ follows from the Lebesgue differentiation theorem.

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  • $\begingroup$ Thanks for the reference, greatly appreciated. $\endgroup$ – Lei Lei Apr 3 '13 at 1:19

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