Hausdorff density

Hausdorff density

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

Contents

Definition

Let μ 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 Br(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 μ at a and denote it Θs(μ,a).

Marstrand's theorem

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

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

Preiss' theorem

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

References

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

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