Differentiation of measures

2020 Mathematics Subject Classification: Primary: 28A15 Secondary: 49Q15 [MSN][ZBL]

Some authors use this name for the outcome of the Radon-Nikodym theorem or for the density of the Radon-Nykodim decomposition (see for instance Section 32 of [Ha]).

Other authors use the name for the following theorem which gives an explicit characterization of the Radon-Nykodim decomposition for locally finite Radon measures on the euclidean space. This theorem is used often in Geometric measure theory and credited to Besicovitch.

Theorem (cp. with Theorem 2.12 of [Ma] and Theorem 2 in Section 1.6 of [EG]) Let $\mu$ and $\nu$ be two locally finite Radon measures on $\mathbb R^n$. Then,

• the limit

\[ f(x) := \lim_{r\downarrow 0} \frac{\nu (B_r (x))}{\mu (B_r (x))} \] exists at $\mu$-a.e. $x$ and defines a $\mu$-measurable map;

• the set

\begin{equation}\label{e:singular} S:= \left\{ x: \lim_{r\downarrow 0} \frac{\nu (B_r (x))}{\mu (B_r (x))} = \infty\right\} \end{equation} is $\nu$-measurable and a $\mu$-null set;

• $\nu$ can be decomposed as $\nu_a + \nu_s$, where

\[ \nu_a (E) = \int_E f\, d\mu \] and \[ \nu_s (E) = \nu (S\cap E)\, . \] Moreover, for $\mu$-a.e. $x$ we have: \begin{equation}\label{e:Lebesgue} \lim_{r\downarrow 0} \frac{1}{\mu (B_r (x))} \int_{B_r (x)} |f(y)-f(x)|\, d\mu (y) = 0\qquad \mbox{and}\qquad \lim_{r\downarrow 0} \frac{\nu_s (B_r (x))}{\mu (B_r (x))}= 0\, . \end{equation}

Comments[edit]

The first identity in \ref{e:Lebesgue} relates to the concept of Lebesgue point.

The theorem can be generalized to signed measures $\nu$ and measures taking values in a finite-dimensional Banach space $V$. In that case:

• $\|\nu (B_r (x))\|_V$ substitutes $\nu (B_r (x))$ in \ref{e:singular};
• $\|f (y)-f(x)\|_V$ substitutes the integrand $|f(y)-f(x)|$ in \ref{e:Lebesgue};
• $|\nu| (B_r (x))$ substitutes $\nu (B_r (x))$ in \ref{e:Lebesgue}, where $|\nu|$ denotes the total variation of $\nu$ (see Signed measure for the relevant definition).

The theorem does not hold in general metric spaces. It holds provided the metric space satisfies some properties about covering of sets with balls, see Covering theorems (measure theory).

References[edit]