Differentiation of measures

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

A theorem in classical measure theory, used often in Geometric measure theory and due to Besicovitch. The theorem gives an explicit characterization of the Radon-Nykodim decomposition for locally finite Radon measures on the euclidean space.

Theorem 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

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 $\mu$ (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

[AFP]	L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. MR1857292Zbl 0957.49001
[De]	C. De Lellis, "Rectifiable sets, densities and tangent measures" Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2008.
[Fe]	H. Federer, "Geometric measure theory". Volume 153 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag New York Inc., New York, 1969.
[Ma]	P. Mattila, "Geometry of sets and measures in euclidean spaces". Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. MR1333890 Zbl 0911.28005