- 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.
Contents
Definition
Let μ be a Radon measure and some point in Euclidean space. The s-dimensional upper and lower Hausdorff densities are defined to be, respectively,
and
where Br(a) is the ball of radius r > 0 centered at a. Clearly, for all . 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 . 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 . Suppose that m 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 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.
Categories:- Measure theory
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