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.


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


where 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 at a and denote it .

Marstrand's theorem[edit]

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

Marstrand's theorem: Let be a Radon measure on . Suppose that the s-density exists and is positive and finite for a in a set of positive measure. Then s is an integer.

Preiss' theorem[edit]

In 1987 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 be a Radon measure on . Suppose that m is an integer and the m-density exists and is positive and finite for almost every a in the support of . Then is m-rectifiable, i.e. ( is absolutely continuous with respect to Hausdorff measure ) and the support of is an m-rectifiable set.

External links[edit]


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