Density of a set

2020 Mathematics Subject Classification: Primary: 28A33 Secondary: 49Q15 [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, 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. $x\not \in E$.

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 \[ \theta^{\alpha,*} (\mu, x) := \limsup_{r\downarrow 0} \frac{\mu (B_r (x))}{\omega_\alpha r^\alpha} \qquad \mbox{and}\qquad \theta^\alpha_* (\mu, x) =\liminf_{r\downarrow 0} \frac{\mu (B_r (x))}{\omega_\alpha r^\alpha}\, , \] 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.

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.

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 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 $f(x)$.

The converse is also true 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 $\lambda$-a.e. $x$, then the measure $\mu$ is given by the formula \ref{e:densita} where $f = \theta^n (\mu, \cdot)$.

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 \[ \theta^{\alpha,*} (\mu, x) := \limsup_{r\downarrow 0} \frac{\mathcal{H}^\alpha (E\cap B_r (x))}{\omega_\alpha r^\alpha} \qquad \mbox{and}\qquad \theta^\alpha_* (\mu, x) =\liminf_{r\downarrow 0} \frac{\mathcal{H}^\alpha (E \cap B_r (x))}{\omega_\alpha r^\alpha}\, . \] They correspond, therefore, to the $\alpha$-dimensional densities of the Radon measure $\mu$ given by \[ \mu (A) := \mathcal{H}^\alpha (A\cap E)\, . \] The following is a classical theorem in Geometric measure theory:

Theorem 5 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. $x\in E$.

that is measurable on the real line , at a point

The limit (if it exists) of the ratio

(1)

where is any segment containing and is its length. If one considers an outer measure instead of a measure, one obtains the definition of the outer density of at . Similarly one can introduce the density in -dimensional space. Here the lengths of the segments in are replaced by the volumes of the corresponding -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 it is useful to employ the concept of the right (left) density of a set at a point , which is obtained from the general definition if in it one considers only segments having left (right) ends at . Very often, the concept of density is used when the density of the set is equal to one (see Density point) or zero (see Thinness of a set).

References

[1]	I.P. Natanson, "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M. (1961) (Translated from Russian)
[2]	S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French)

Comments

See [a1] for a nice topological application of these notions.

References

[a1]	F.D. Tall, "The density topology" Pacific J. Math , 62 (1976) pp. 275–284