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 $a\in {ℝ}^n$ some point in Euclidean space. The s-dimensional upper and lower Hausdorff densities are defined to be, respectively,

${\mathrm{\Theta }}^{*s}(\mu ,a)=\underset{r\to 0}{\mathrm{lim\; sup}}\frac{\mu (B_r(a))}{r^s}$

and

${\mathrm{\Theta }}_{*}^s(\mu ,a)=\underset{r\to 0}{\mathrm{lim\; inf}}\frac{\mu (B_r(a))}{r^s}$

where $B_r(a)$ is the ball of radius r > 0 centered at a. Clearly, ${\mathrm{\Theta }}_{*}^s(\mu ,a)\le {\mathrm{\Theta }}^{*s}(\mu ,a)$ for all $a\in {ℝ}^n$. In the event that the two are equal, we call their common value the s-density of $\mu$ at a and denote it ${\mathrm{\Theta }}^s(\mu ,a)$.

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 ${ℝ}^d$. Suppose that the s-density ${\mathrm{\Theta }}^s(\mu ,a)$ 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 ${ℝ}^d$. Suppose that m$\ge 1$ is an integer and the m-density ${\mathrm{\Theta }}^m(\mu ,a)$ 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.
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