Density of a set
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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[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 |
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[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 |
[Ta] | F.D. Tall, "The density topology" Pacific J. Math , 62 (1976) pp. 275–284 MR0419709 Zbl 0305.54039 |
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