Hausdorff density

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In measure theory, a field of mathematics, the Hausdorff density measures how concentrated a Radon measure is at some point.

Definition[edit]

Let \mu be a Radon measure and a\in\mathbb{R}^{n} some point in Euclidean space. The s-dimensional upper and lower Hausdorff densities are defined to be, respectively,

 \Theta^{*s}(\mu,a)=\limsup_{r\rightarrow 0}\frac{\mu(B_{r}(a))}{r^{s}}

and

 \Theta_{*}^{s}(\mu,a)=\liminf_{r\rightarrow 0}\frac{\mu(B_{r}(a))}{r^{s}}

where  B_{r}(a) is the ball of radius r > 0 centered at a. Clearly, \Theta_{*}^{s}(\mu,a)\leq \Theta^{*s}(\mu,a) for all a\in\mathbb{R}^{n}. In the event that the two are equal, we call their common value the s-density of \mu at a and denote it \Theta^{s}(\mu,a).

Marstrand's theorem[edit]

The following theorem states that the times when the s-density exists are rather seldom.

Marstrand's theorem: Let \mu be a Radon measure on \mathbb{R}^{d}. Suppose that the s-density \Theta^{s}(\mu,a) exists and is positive and finite for a in a set of positive \mu measure. Then s is an integer.

Preiss' theorem[edit]

In 1987 Preiss proved a stronger version of Marstrand's theorem. One consequence is that that sets with positive and finite density are rectifiable sets.

Preiss' theorem: Let \mu be a Radon measure on \mathbb{R}^{d}. Suppose that m\geq 1 is an integer and the m-density \Theta^{m}(\mu,a) exists and is positive and finite for \mu almost every a in the support of \mu. Then \mu is m-rectifiable, i.e. \mu\ll H^{m} (\mu is absolutely continuous with respect to Hausdorff measure H^m) and the support of \mu is an m-rectifiable set.

External links[edit]

References[edit]

  • Pertti Mattila, Geometry of sets and measures in Euclidean spaces. Cambridge Press, 1995.
  • Preiss, David (1987). "Geometry of measures in R^n: distribution, rectifiability, and densities". Ann. Math. 125 (3): 537–643. doi:10.2307/1971410. JSTOR 1971410.