View source for Density of a set

{{MSC|28A05|28A15,49Q15}}

[[Category:Classical measure theory]]


{{TEX|done}}

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

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}}).

'''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 {{Cite|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 {{Cite|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 {{Cite|De}}).

===References===
{|
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|-
|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}}
|-
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|-
|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}}
|-
|}