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.

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 $R^{n}$: distribution, rectifiability, and densities". Ann. Math. 125 (3): 537–643. doi:10.2307/1971410. hdl:10338.dmlcz/133417. JSTOR 1971410.