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{{MSC|28A33|49Q15}}
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{{MSC|28A05|28A15,49Q15}}
  
 
[[Category:Classical measure theory]]
 
[[Category:Classical measure theory]]
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\lim_{r\downarrow 0} \frac{\lambda (B_r (x)\cap  E)}{\omega_n r^n}\, ,
 
\lim_{r\downarrow 0} \frac{\lambda (B_r (x)\cap  E)}{\omega_n r^n}\, ,
 
\]
 
\]
the resulting number is called the density of $E$ at $x$. The following is a classical result in measure theory, due to Lebesgue in the case $n=1$:
+
the resulting number is called the density of $E$ at $x$. The following is a classical result in measure theory (see for instance Corollary 3 in Section 1.7 of {{Cite|EG}}), due to Lebesgue in the case $n=1$:
  
 
'''Theorem 1'''
 
'''Theorem 1'''
The density of a Lebesgue measurable set $E\subset \mathbb R^n$ is $1$ at  $\lambda$--a.e. $x\in E$ and $0$ at $\lambda$--a.e. $x\not \in E$.
+
The density of a Lebesgue measurable set $E\subset \mathbb R^n$ is $1$ at  $\lambda$-a.e. $x\in E$ and $0$ at $\lambda$-a.e. $x\not \in E$.
 +
 
 +
The points of the first type are also called {{Anchor|Density points}} ''density points'' of $E$, whereas the second points are called points of dispersions. The density points and
 +
the points of dispersion are sometimes defined also for non-measurable sets $E$: in that case one uses the Lebesgue outer measure, cp. with Sections 2.9.11 and 2.9.12 of {{Cite|Fe}} (see [[Lebesgue measure]]).
  
 
===Density of a measure===
 
===Density of a measure===
The concept above has been generalized in geometric measure theory to measures, starting from the work of Besicovitch. Consider a (locally finite) [[Radon measure]] $\mu$ in the Euclidean space $\mathbb R^n$, a point $x\in \mathbb R^n$ and a nonnegative real number $\alpha$. The $\alpha$-dimensional upper and lower densities of $\mu$ at $x$ are defined as  
+
The concept above has been generalized in geometric measure theory to measures, starting from the work of Besicovitch. Consider a (locally finite) [[Radon measure]] $\mu$ in the Euclidean space $\mathbb R^n$, a point $x\in \mathbb R^n$ and a nonnegative real number $\alpha$ (see for instance Definition 2.14 of {{Cite|De}} or Definition 6.8 of {{Cite|Ma}}). The $\alpha$-dimensional upper and lower densities of $\mu$ at $x$ are defined as  
 
\[
 
\[
 
\theta^{\alpha,*} (\mu, x) := \limsup_{r\downarrow 0} \frac{\mu (B_r (x))}{\omega_\alpha r^\alpha}
 
\theta^{\alpha,*} (\mu, x) := \limsup_{r\downarrow 0} \frac{\mu (B_r (x))}{\omega_\alpha r^\alpha}
Line 32: Line 35:
 
where the normalizing factor $\omega_\alpha$ is the $\alpha$-dimensional volume of the unit ball in $\mathbb R^\alpha$ when $\alpha$
 
where the normalizing factor $\omega_\alpha$ is the $\alpha$-dimensional volume of the unit ball in $\mathbb R^\alpha$ when $\alpha$
 
is a positive integer and in general $\omega_\alpha = \pi^{\alpha/2} \Gamma (1+\alpha/2)$.  
 
is a positive integer and in general $\omega_\alpha = \pi^{\alpha/2} \Gamma (1+\alpha/2)$.  
If the two numbers coincide, the resulting value is called the $\alpha$-dimensional density of $\mu$ at  $x$. The following Theorem by Marstrand shows that the density might exist and be nontrivial if and only if $\alpha$ is an integer.
+
If the two numbers coincide, the resulting value is called the $\alpha$-dimensional density of $\mu$ at  $x$. The following Theorem by Marstrand shows that the density might exist and be nontrivial if and only if $\alpha$ is an integer (we refer to Chapter 3 of {{Cite|De}} for its proof).
  
 
'''Theorem 2'''
 
'''Theorem 2'''
Let  $\mu$ be a locally finite Radon measure on $\mathbb R^n$ and $\alpha$ a  nonnegative real number such that the $\alpha$-dimensional density of  $\mu$ exists and is positive on a set of positibe $\mu$-measure. Then  $\alpha$ is necessarily an integer.
+
Let  $\mu$ be a locally finite Radon measure on $\mathbb R^n$ and $\alpha$ a  nonnegative real number such that the $\alpha$-dimensional density of  $\mu$ exists and is positive on a set of positive $\mu$-measure. Then  $\alpha$ is necessarily an integer.
  
 
====Lebesgue theorem====
 
====Lebesgue theorem====
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theorem corresponds to the fact that, given a summable function $f$, $\lambda$-a.e. point $x$ is a [[Lebesgue point]] for $f$:
 
theorem corresponds to the fact that, given a summable function $f$, $\lambda$-a.e. point $x$ is a [[Lebesgue point]] for $f$:
  
'''Theorem 3'''
+
'''Theorem 3''' (Theorem 1 in Section 1.7 of {{Cite|EG}})
 
Let $f\in L^1_{loc} (\mathbb R^n)$ and consider the measure  
 
Let $f\in L^1_{loc} (\mathbb R^n)$ and consider the measure  
 
\begin{equation}\label{e:densita}
 
\begin{equation}\label{e:densita}
Line 48: Line 51:
 
Then the $n$--dimensional density of $\mu$ exists at $\lambda$--a.e. $x\in \mathbb R^n$ and coincides with $f(x)$.
 
Then the $n$--dimensional density of $\mu$ exists at $\lambda$--a.e. $x\in \mathbb R^n$ and coincides with $f(x)$.
  
The converse is also true and is a particular case of a more general result on the [[Differentiation of measures]]:
+
A similar result in the opposite direction holds and is a particular case of a more general result on the [[Differentiation of measures]]:
  
 
'''Theorem 4'''
 
'''Theorem 4'''
Let $\mu$ be a locally finite Radon measure on $\mathbb R^n$. If the $n$-dimensional density $\theta^n (\mu, x)$ exists for $\lambda$-a.e. $x$, then the measure $\mu$ is given by the formula \ref{e:densita} where $f = \theta^n (\mu, \cdot)$.
+
Let $\mu$ be a locally finite Radon measure on $\mathbb R^n$. If the $n$-dimensional density $\theta^n (\mu, x)$ exists for $\mu$-a.e. $x$, then the measure $\mu$ is given by the formula \ref{e:densita} where $f = \theta^n (\mu, \cdot)$.
 +
 
 +
The latter theorem can be generalized to Hausdorff $\alpha$-dimensional measures $\mathcal{H}^\alpha$ (cp. with Theorem 6.9 of {{Cite|Ma}}).
 +
 
 +
'''Theorem 5'''
 +
Let $\mu$ be a locally finite Radon measure on $\mathbb R^n$. If the upper $\alpha$-dimensional density exists and it is positive and finite at $\mu$-a.e. $x\in \mathbb R^n$, then there is a Borel function $f$ and a Borel set $E$ with locally finite $\alpha$-dimensional Hausdorff measure such that
 +
\[
 +
\mu (A) = \int_{A\cap E} f\, d\mathcal{H}^\alpha\, .
 +
\]
 +
 
 +
A generalization of Theorem 3 is also possible, but much more subtle (see below).
  
 
===Lower-dimensional densities of a set===
 
===Lower-dimensional densities of a set===
Assume $E\subset \mathbb R^n$ is a Borel set  with finite $\alpha$-dimensional [[Hausdorff measure]]. The $\alpha$-dimensional upper and lower $\alpha$-dimensional densities $\theta^{\alpha, *} (E, x)$ and $\theta^\alpha_* (E,x)$ of $E$ at $x$ are defined as  
+
Assume $E\subset \mathbb R^n$ is a [[Borel set]] with finite $\alpha$-dimensional [[Hausdorff measure]]. The $\alpha$-dimensional upper and lower densities $\theta^{\alpha, *} (E, x)$ and $\theta^\alpha_* (E,x)$ of $E$ at $x$ are defined as
 
\[
 
\[
\theta^{\alpha,*} (\mu, x) := \limsup_{r\downarrow 0} \frac{\mathcal{H}^\alpha (E\cap B_r (x))}{\omega_\alpha r^\alpha}
+
\theta^{\alpha,*} (E, x) := \limsup_{r\downarrow 0} \frac{\mathcal{H}^\alpha (E\cap B_r (x))}{\omega_\alpha r^\alpha}
\qquad \mbox{and}\qquad \theta^\alpha_* (\mu, x) =\liminf_{r\downarrow 0} \frac{\mathcal{H}^\alpha (E \cap B_r (x))}{\omega_\alpha r^\alpha}\, .
+
\qquad \mbox{and}\qquad \theta^\alpha_* (E, x) =\liminf_{r\downarrow 0} \frac{\mathcal{H}^\alpha (E \cap B_r (x))}{\omega_\alpha r^\alpha}\,  
 
\]
 
\]
They correspond, therefore, to the $\alpha$-dimensional densities of the Radon measure $\mu$ given by  
+
(cp. with Definition 6.1 of {{Cite|Ma}})
 +
They correspond, therefore, to the $\alpha$-dimensional (upper and lower) densities of the Radon measure $\mu$ given by  
 
\[
 
\[
\mu (A) := \mathcal{H}^\alpha (A\cap E)\, .
+
\mu (A) := \mathcal{H}^\alpha (A\cap E)\, \qquad\mbox{for any Borel set } A\subset\mathbb R\, .
 
\]
 
\]
The following is a classical theorem in Geometric measure theory:
+
The following is a classical theorem in Geometric measure theory (cp. with Theorem 6.2 of {{Cite|Ma}}):
  
'''Theorem 5'''
+
'''Theorem 6'''
If $E\subset \mathbb R^n$ is a Borel set  with finite $\alpha$-dimensional [[Hausdorff measure]], then  
+
If $E\subset \mathbb R^n$ is a Borel set  with finite $\alpha$-dimensional Hausdorff measure, then  
 
* $\theta^{\alpha,*} (E, x) =0$ for $\mathcal{H}^\alpha$-a.e. $x\not\in E$.
 
* $\theta^{\alpha,*} (E, x) =0$ for $\mathcal{H}^\alpha$-a.e. $x\not\in E$.
* $1\geq \theta^\alpha_* (E,x) \geq 2^{-\alpha}$ for $\mathcal{H}^\alpha$--a.e. $x\in E$.
+
* $1\geq \theta^\alpha_* (E,x) \geq 2^{-\alpha}$ for $\mathcal{H}^\alpha$-a.e. $x\in E$.
  
 +
====Besicovitch-Preiss theorem and rectifiability====
 +
However the existence of the density fails in general: as a consequence of Marstrand's Theorem 2 the existence of a nontrivial $\alpha$-dimensional
 +
density implies that $\alpha$ is an integer. But even in the case when $\alpha$ is an integer, it was discovered by Besicovitch that the density does
 +
not necessarily exist. Indeed the following generalization of Besicovitch's theorem was achieved by Preiss in the mid eighties (see Chapters 6,7,8 and 9 of {{Cite|De}} for an exposition of Preiss' proof):
  
''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031130/d0311301.png" /> that is measurable on the real line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031130/d0311302.png" />, at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031130/d0311303.png" />''
+
'''Theorem 7'''
 +
Let $E\subset \mathbb R^n$ be a Borel set with positive and finite $k$-dimensional Hausdorff measure, where $k\in \mathbb N$. The $k$-dimensional density
 +
exists at $\mathcal{H}^k$-a.e. $x\in E$ if and only if the set $E$ is [[Rectifiable set|rectifiable]], i.e. if there are countably many $C^1$ $k$-dimensional submanifolds
 +
of $\mathbb R^n$ which cover $\mathcal{H}^k$-almost all $E$.
  
The limit (if it exists) of the ratio
+
For non-rectifiable sets $E$ the lower-dimensional density might display a variety of different behavior. Besicovitch proved that for $1$-dimensional non-rectifiable sets
 +
the lower dimensional density cannot be larger than $\frac{3}{4}$ and advanced the following long-standing conjecture (cp. with Conjecture 10.5 of {{Cite|De}}).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031130/d0311304.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
'''Conjecture 8'''
 +
Let $E\subset \mathbb R^2$ be a Borel set with positive and finite $1$-dimensional Hausdorff measure. If $\theta^1_* (E, x)>\frac{1}{2}$ for $\mathcal{H}^1$-a.e.
 +
$x\in E$, then the set $E$ is rectifiable.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031130/d0311305.png" /> is any segment containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031130/d0311306.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031130/d0311307.png" /> is its length. If one considers an outer measure instead of a measure, one obtains the definition of the outer density of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031130/d0311308.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031130/d0311309.png" />. Similarly one can introduce the density in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031130/d03113010.png" />-dimensional space. Here the lengths of the segments in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031130/d03113011.png" /> are replaced by the volumes of the corresponding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031130/d03113012.png" />-dimensional parallelepipeds with faces parallel to the coordinate planes, while the limit is considered as the diameters of the parallelepipeds tend to zero. For sets from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031130/d03113013.png" /> it is useful to employ the concept of the right (left) density of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031130/d03113014.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031130/d03113015.png" />, which is obtained from the general definition if in it one considers only segments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031130/d03113016.png" /> having left (right) ends at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031130/d03113017.png" />. Very often, the concept of density is used when the density of the set is equal to one (see [[Density point|Density point]]) or zero (see [[Thinness of a set|Thinness of a set]]).
+
Besicovitch's $\frac{3}{4}$ treshold has been improved by Preiss and Tiser in {{Cite|PT}}.
  
====References====
+
===Besicovitch-Marstrand-Preiss Theorem===
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.P. Natanson,  "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M.  (1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Saks,  "Theory of the integral" , Hafner  (1952)  (Translated from French)</TD></TR></table>
+
Combining the various theorems exposed so far we reach the following characterization of measures $\mu$ for which densities exist and are nontrivial almost everywhere.
  
 +
'''Theorem 9'''
 +
Let $\mu$ be a locally finite Radon measure and $\alpha$ a nonnegative real number. Then $\theta^\alpha (\mu, x)$ exists, it is finite and positive
 +
at $\mu$-a.e. $x\in \mathbb R^n$ if and only $\alpha$ is an integer $k$ and there are a rectifiable $k$-dimensional Borel set $E$ and a Borel function $f: E\to ]0, \infty[$ such that
 +
\[
 +
\mu (A) = \int_{A\cap E} f\, d\mathcal{H}^k\qquad \mbox{for any Borel } A\subset \mathbb R^n\, .
 +
\]
 +
Moreover, in this case $\theta^k (E,x)=f(x)$ for $\mathcal{H}^k$-a.e. $x\in E$ and $\theta^k (E, x) =0$ for $\mathcal{H}^k$-a.e. $x\not\in E$.
  
 +
====Comments====
 +
See {{Cite|Ta}} for a nice topological application of the classical notion of Lebesgue density.
  
====Comments====
+
The definition of $\alpha$-dimensional density of a Radon measure can be generalized to metric spaces. In general, however, very little is known outside of the Euclidean setting (cp. with Section 10.0.2 of {{Cite|De}}).
See [[#References|[a1]]] for a nice topological application of these notions.
 
  
====References====
+
===References===
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> F.D. Tall,  "The density topology"  ''Pacific J. Math'' , '''62'''  (1976)  pp. 275–284</TD></TR></table>
+
{|
 +
|-
 +
|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|Be}}|| S. Besicovitch, "On the fundamental geometrical properties of linearly measurable plane sets of points II", ''Math. Ann.'', '''115''' (1938), pp. 296–329. {{ZBL|64.0193.01}}
 +
|-
 +
|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|Mar}}|| J. M. Marstrand, "The (φ, s) regular subset of n space". ''Trans. Amer. Math. Soc.'', '''113''' (1964), pp. 369–392. {{MR|0166336}} {{ZBL|0144.04902}}
 +
|-
 +
|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}}
 +
|-
 +
|valign="top"|{{Ref|Pr}}|| D. Preiss, "Geometry of measures in $\mathbb R^n$ : distribution, rectifiability, and densities". ''Ann. of Math.'', '''125''' (1987), pp. 537–643. {{MR|0890162}} {{ZBL|0627.28008}}
 +
|-
 +
|valign="top"|{{Ref|PT}}|| D. Preiss, J. Tiser, "On Besicovitch’s $\frac{1}{2}$-problem. ''J. London Math. Soc. (2)'', '''45''' (1992), pp. 279–287. {{MR|1171555}} {{ZBL|0762.28003}}
 +
|-
 +
|valign="top"|{{Ref|Ta}}|| F.D. Tall,  "The density topology"  ''Pacific J. Math'' , '''62'''  (1976)  pp. 275–284 {{MR|0419709}} {{ZBL|0305.54039}}
 +
|-
 +
|}

Latest revision as of 09:58, 16 August 2013

2020 Mathematics Subject Classification: Primary: 28A05 Secondary: 28A1549Q15 [MSN][ZBL]

A concept of classical measure theory generalized further in Geometric measure theory

Lebesgue density of a set

Given a Lebesgue measurable set $E$ in the standard Euclidean space $\mathbb R^n$ and a point $x\in\mathbb R^n$, the upper and lower densities of $E$ at $x$ are defined respectively as \[ \limsup_{r\downarrow 0} \frac{\lambda (B_r (x)\cap E)}{\omega_n r^n} \qquad \mbox{and} \qquad \liminf_{r\downarrow 0} \frac{\lambda (B_r (x)\cap E)}{\omega_n r^n}\, , \] where $\lambda$ denotes the Lebesgue measure and $\omega_n$ the volume of the unit $n$-dimensional ball. If the two numbers coincides, i.e. if the following limit exists, \[ \lim_{r\downarrow 0} \frac{\lambda (B_r (x)\cap E)}{\omega_n r^n}\, , \] the resulting number is called the density of $E$ at $x$. The following is a classical result in measure theory (see for instance Corollary 3 in Section 1.7 of [EG]), due to Lebesgue in the case $n=1$:

Theorem 1 The density of a Lebesgue measurable set $E\subset \mathbb R^n$ is $1$ at $\lambda$-a.e. $x\in E$ and $0$ at $\lambda$-a.e. $x\not \in E$.

The points of the first type are also called density points of $E$, whereas the second points are called points of dispersions. The density points and the points of dispersion are sometimes defined also for non-measurable sets $E$: in that case one uses the Lebesgue outer measure, cp. with Sections 2.9.11 and 2.9.12 of [Fe] (see Lebesgue measure).

Density of a measure

The concept above has been generalized in geometric measure theory to measures, starting from the work of Besicovitch. Consider a (locally finite) Radon measure $\mu$ in the Euclidean space $\mathbb R^n$, a point $x\in \mathbb R^n$ and a nonnegative real number $\alpha$ (see for instance Definition 2.14 of [De] or Definition 6.8 of [Ma]). The $\alpha$-dimensional upper and lower densities of $\mu$ at $x$ are defined as \[ \theta^{\alpha,*} (\mu, x) := \limsup_{r\downarrow 0} \frac{\mu (B_r (x))}{\omega_\alpha r^\alpha} \qquad \mbox{and}\qquad \theta^\alpha_* (\mu, x) =\liminf_{r\downarrow 0} \frac{\mu (B_r (x))}{\omega_\alpha r^\alpha}\, , \] where the normalizing factor $\omega_\alpha$ is the $\alpha$-dimensional volume of the unit ball in $\mathbb R^\alpha$ when $\alpha$ is a positive integer and in general $\omega_\alpha = \pi^{\alpha/2} \Gamma (1+\alpha/2)$. If the two numbers coincide, the resulting value is called the $\alpha$-dimensional density of $\mu$ at $x$. The following Theorem by Marstrand shows that the density might exist and be nontrivial if and only if $\alpha$ is an integer (we refer to Chapter 3 of [De] for its proof).

Theorem 2 Let $\mu$ be a locally finite Radon measure on $\mathbb R^n$ and $\alpha$ a nonnegative real number such that the $\alpha$-dimensional density of $\mu$ exists and is positive on a set of positive $\mu$-measure. Then $\alpha$ is necessarily an integer.

Lebesgue theorem

Concerning $n$-dimensional densities, the following theorem corresponds to the fact that, given a summable function $f$, $\lambda$-a.e. point $x$ is a Lebesgue point for $f$:

Theorem 3 (Theorem 1 in Section 1.7 of [EG]) Let $f\in L^1_{loc} (\mathbb R^n)$ and consider the measure \begin{equation}\label{e:densita} \mu (A):= \int_A f\, d\lambda\, . \end{equation} Then the $n$--dimensional density of $\mu$ exists at $\lambda$--a.e. $x\in \mathbb R^n$ and coincides with $f(x)$.

A similar result in the opposite direction holds and is a particular case of a more general result on the Differentiation of measures:

Theorem 4 Let $\mu$ be a locally finite Radon measure on $\mathbb R^n$. If the $n$-dimensional density $\theta^n (\mu, x)$ exists for $\mu$-a.e. $x$, then the measure $\mu$ is given by the formula \ref{e:densita} where $f = \theta^n (\mu, \cdot)$.

The latter theorem can be generalized to Hausdorff $\alpha$-dimensional measures $\mathcal{H}^\alpha$ (cp. with Theorem 6.9 of [Ma]).

Theorem 5 Let $\mu$ be a locally finite Radon measure on $\mathbb R^n$. If the upper $\alpha$-dimensional density exists and it is positive and finite at $\mu$-a.e. $x\in \mathbb R^n$, then there is a Borel function $f$ and a Borel set $E$ with locally finite $\alpha$-dimensional Hausdorff measure such that \[ \mu (A) = \int_{A\cap E} f\, d\mathcal{H}^\alpha\, . \]

A generalization of Theorem 3 is also possible, but much more subtle (see below).

Lower-dimensional densities of a set

Assume $E\subset \mathbb R^n$ is a Borel set with finite $\alpha$-dimensional Hausdorff measure. The $\alpha$-dimensional upper and lower densities $\theta^{\alpha, *} (E, x)$ and $\theta^\alpha_* (E,x)$ of $E$ at $x$ are defined as \[ \theta^{\alpha,*} (E, x) := \limsup_{r\downarrow 0} \frac{\mathcal{H}^\alpha (E\cap B_r (x))}{\omega_\alpha r^\alpha} \qquad \mbox{and}\qquad \theta^\alpha_* (E, x) =\liminf_{r\downarrow 0} \frac{\mathcal{H}^\alpha (E \cap B_r (x))}{\omega_\alpha r^\alpha}\, \] (cp. with Definition 6.1 of [Ma]) They correspond, therefore, to the $\alpha$-dimensional (upper and lower) densities of the Radon measure $\mu$ given by \[ \mu (A) := \mathcal{H}^\alpha (A\cap E)\, \qquad\mbox{for any Borel set } A\subset\mathbb R\, . \] The following is a classical theorem in Geometric measure theory (cp. with Theorem 6.2 of [Ma]):

Theorem 6 If $E\subset \mathbb R^n$ is a Borel set with finite $\alpha$-dimensional Hausdorff measure, then

  • $\theta^{\alpha,*} (E, x) =0$ for $\mathcal{H}^\alpha$-a.e. $x\not\in E$.
  • $1\geq \theta^\alpha_* (E,x) \geq 2^{-\alpha}$ for $\mathcal{H}^\alpha$-a.e. $x\in E$.

Besicovitch-Preiss theorem and rectifiability

However the existence of the density fails in general: as a consequence of Marstrand's Theorem 2 the existence of a nontrivial $\alpha$-dimensional density implies that $\alpha$ is an integer. But even in the case when $\alpha$ is an integer, it was discovered by Besicovitch that the density does not necessarily exist. Indeed the following generalization of Besicovitch's theorem was achieved by Preiss in the mid eighties (see Chapters 6,7,8 and 9 of [De] for an exposition of Preiss' proof):

Theorem 7 Let $E\subset \mathbb R^n$ be a Borel set with positive and finite $k$-dimensional Hausdorff measure, where $k\in \mathbb N$. The $k$-dimensional density exists at $\mathcal{H}^k$-a.e. $x\in E$ if and only if the set $E$ is rectifiable, i.e. if there are countably many $C^1$ $k$-dimensional submanifolds of $\mathbb R^n$ which cover $\mathcal{H}^k$-almost all $E$.

For non-rectifiable sets $E$ the lower-dimensional density might display a variety of different behavior. Besicovitch proved that for $1$-dimensional non-rectifiable sets the lower dimensional density cannot be larger than $\frac{3}{4}$ and advanced the following long-standing conjecture (cp. with Conjecture 10.5 of [De]).

Conjecture 8 Let $E\subset \mathbb R^2$ be a Borel set with positive and finite $1$-dimensional Hausdorff measure. If $\theta^1_* (E, x)>\frac{1}{2}$ for $\mathcal{H}^1$-a.e. $x\in E$, then the set $E$ is rectifiable.

Besicovitch's $\frac{3}{4}$ treshold has been improved by Preiss and Tiser in [PT].

Besicovitch-Marstrand-Preiss Theorem

Combining the various theorems exposed so far we reach the following characterization of measures $\mu$ for which densities exist and are nontrivial almost everywhere.

Theorem 9 Let $\mu$ be a locally finite Radon measure and $\alpha$ a nonnegative real number. Then $\theta^\alpha (\mu, x)$ exists, it is finite and positive at $\mu$-a.e. $x\in \mathbb R^n$ if and only $\alpha$ is an integer $k$ and there are a rectifiable $k$-dimensional Borel set $E$ and a Borel function $f: E\to ]0, \infty[$ such that \[ \mu (A) = \int_{A\cap E} f\, d\mathcal{H}^k\qquad \mbox{for any Borel } A\subset \mathbb R^n\, . \] Moreover, in this case $\theta^k (E,x)=f(x)$ for $\mathcal{H}^k$-a.e. $x\in E$ and $\theta^k (E, x) =0$ for $\mathcal{H}^k$-a.e. $x\not\in E$.

Comments

See [Ta] for a nice topological application of the classical notion of Lebesgue density.

The definition of $\alpha$-dimensional density of a Radon measure can be generalized to metric spaces. In general, however, very little is known outside of the Euclidean setting (cp. with Section 10.0.2 of [De]).

References

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[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
[Mar] J. M. Marstrand, "The (φ, s) regular subset of n space". Trans. Amer. Math. Soc., 113 (1964), pp. 369–392. MR0166336 Zbl 0144.04902
[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
[Pr] D. Preiss, "Geometry of measures in $\mathbb R^n$ : distribution, rectifiability, and densities". Ann. of Math., 125 (1987), pp. 537–643. MR0890162 Zbl 0627.28008
[PT] D. Preiss, J. Tiser, "On Besicovitch’s $\frac{1}{2}$-problem. J. London Math. Soc. (2), 45 (1992), pp. 279–287. MR1171555 Zbl 0762.28003
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How to Cite This Entry:
Density of a set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Density_of_a_set&oldid=27340
This article was adapted from an original article by V.A. Skvortsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article