# Differentiation of measures

2010 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}

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).