Hausdorff density

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

Definition
Let $$\mu$$ 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 $$ B_{r}(a)$$ 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 $$\mu$$ at a and denote it $$\Theta^{s}(\mu,a)$$.

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


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