Differentiation of measures

2020 Mathematics Subject Classification: Primary: 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

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. MR2388959 Zbl 1183.28006
[EG]	L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800
[Fe]	H. Federer, "Geometric measure theory". Volume 153 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag New York Inc., New York, 1969. MR0257325 Zbl 0874.49001
[Ha]	P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Zbl 0040.16802
[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