View source for Differentiation of measures

{{MSC|18A15|49Q15}}

[[Category:Classical measure theory]]


{{TEX|done}}

Some authors use this name for the outcome of the [[Radon-Nikodym theorem]] or for the density of the [[Absolutely continuous measures|Radon-Nykodim decomposition]] (see for instance Section 32 of {{Cite|Ha}}).

Other authors use the name for the following theorem which gives an explicit characterization of the [[Absolutely continuous  measures|Radon-Nykodim decomposition]] for locally finite [[Radon  measure|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 {{Cite|Ma}} and Theorem 2 in Section 1.6 of {{Cite|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 measure|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====
{|
|-
|valign="top"|{{Ref|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.      {{MR|1857292}}{{ZBL|0957.49001}} 
|-
|valign="top"|{{Ref|De}}||  C. De Lellis,  "Rectifiable sets, densities and tangent measures"  Zurich Lectures in  Advanced Mathematics. European Mathematical Society  (EMS), Zürich, 2008. {{MR|2388959}} {{ZBL|1183.28006}}
|-
|valign="top"|{{Ref|EG}}|| L.C. Evans, R.F. Gariepy, "Measure theory   and fine properties of functions" Studies in Advanced Mathematics. CRC   Press, Boca Raton, FL,  1992. {{MR|1158660}} {{ZBL|0804.2800}} 
|-
|valign="top"|{{Ref|Fe}}||   H. Federer, "Geometric measure theory". Volume 153 of Die Grundlehren   der mathematischen Wissenschaften. Springer-Verlag New York Inc., New   York, 1969. {{MR|0257325}} {{ZBL|0874.49001}}
|-
|valign="top"|{{Ref|Ha}}|| P.R. Halmos,   "Measure theory" , v. Nostrand (1950) {{MR|0033869}} {{ZBL|0040.16802}}
|-
|valign="top"|{{Ref|Ma}}||  P. Mattila,  "Geometry of sets and measures in euclidean spaces".     Cambridge  Studies in Advanced Mathematics, 44. Cambridge University     Press,  Cambridge,  1995. {{MR|1333890}} {{ZBL|0911.28005}}
|}