Hausdorff density

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

## Contents

* 1 Definition
* 2 Marstrand's theorem
* 3 Preiss' theorem
* 4 External links
* 5 References

## Definition

Let μ be a Radon measure and a∈ℝn some point in Euclidean space. The s-dimensional upper and lower Hausdorff densities are defined to be, respectively,

Θ*s(μ,a)=lim supr->0μ(Br(a))rs

and

Θ*s(μ,a)=lim infr->0μ(Br(a))rs

where Br(a) is the ball of radius r > 0 centered at a. Clearly, Θ*s(μ,a)≤Θ*s(μ,a) for all a∈ℝn. 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 ℝd. 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 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 μ be a Radon measure on ℝd. Suppose that m≥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. μ≪Hm (μ is absolutely continuous with respect to Hausdorff measure Hm) and the support of μ is an m-rectifiable set.

## External links

* Density of a set at Encyclopedia of Mathematics
* Rectifiable set at Encyclopedia of Mathematics

## 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.

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