Hausdorff density
In measure theory, a field of mathematics, the Hausdorff density measures how concentrated a Radon measure is at some point.
Definition
Let [math]\displaystyle{ \mu }[/math] be a Radon measure and [math]\displaystyle{ a\in\mathbb{R}^{n} }[/math] some point in Euclidean space. The s-dimensional upper and lower Hausdorff densities are defined to be, respectively,
- [math]\displaystyle{ \Theta^{*s}(\mu,a)=\limsup_{r\rightarrow 0}\frac{\mu(B_{r}(a))}{r^{s}} }[/math]
and
- [math]\displaystyle{ \Theta_{*}^{s}(\mu,a)=\liminf_{r\rightarrow 0}\frac{\mu(B_{r}(a))}{r^{s}} }[/math]
where [math]\displaystyle{ B_{r}(a) }[/math] is the ball of radius r > 0 centered at a. Clearly, [math]\displaystyle{ \Theta_{*}^{s}(\mu,a)\leq \Theta^{*s}(\mu,a) }[/math] for all [math]\displaystyle{ a\in\mathbb{R}^{n} }[/math]. In the event that the two are equal, we call their common value the s-density of [math]\displaystyle{ \mu }[/math] at a and denote it [math]\displaystyle{ \Theta^{s}(\mu,a) }[/math].
Marstrand's theorem
The following theorem states that the times when the s-density exists are rather seldom.
- Marstrand's theorem: Let [math]\displaystyle{ \mu }[/math] be a Radon measure on [math]\displaystyle{ \mathbb{R}^{d} }[/math]. Suppose that the s-density [math]\displaystyle{ \Theta^{s}(\mu,a) }[/math] exists and is positive and finite for a in a set of positive [math]\displaystyle{ \mu }[/math] measure. Then s is an integer.
Preiss' theorem
In 1987 David Preiss proved a stronger version of Marstrand's theorem. One consequence is that sets with positive and finite density are rectifiable sets.
- Preiss' theorem: Let [math]\displaystyle{ \mu }[/math] be a Radon measure on [math]\displaystyle{ \mathbb{R}^{d} }[/math]. Suppose that m[math]\displaystyle{ \geq 1 }[/math] is an integer and the m-density [math]\displaystyle{ \Theta^{m}(\mu,a) }[/math] exists and is positive and finite for [math]\displaystyle{ \mu }[/math] almost every a in the support of [math]\displaystyle{ \mu }[/math]. Then [math]\displaystyle{ \mu }[/math] is m-rectifiable, i.e. [math]\displaystyle{ \mu\ll H^{m} }[/math] ([math]\displaystyle{ \mu }[/math] is absolutely continuous with respect to Hausdorff measure [math]\displaystyle{ H^m }[/math]) and the support of [math]\displaystyle{ \mu }[/math] is an m-rectifiable set.
External links
References
- Pertti Mattila, Geometry of sets and measures in Euclidean spaces. Cambridge Press, 1995.
- Preiss, David (1987). "Geometry of measures in [math]\displaystyle{ R^n }[/math]: distribution, rectifiability, and densities". Ann. Math. 125 (3): 537–643. doi:10.2307/1971410.
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