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 .
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.
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.
- Pertti Mattila, Geometry of sets and measures in Euclidean spaces. Cambridge Press, 1995.