Hausdorff density

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

:$\backslash Theta^\{*s\}(\backslash mu,a)=\backslash limsup\_\{r\backslash rightarrow\; 0\}\backslash frac\{\backslash mu(B\_\{r\}(a))\}\{r^\{s\}\}$

and

:$\backslash Theta\_\{*\}^\{s\}(\backslash mu,a)=\backslash liminf\_\{r\backslash rightarrow\; 0\}\backslash frac\{\backslash mu(B\_\{r\}(a))\}\{r^\{s\}\}$

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

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

: Marstrand's theorem: Let $\backslash mu$ be a Radon measure on $\backslash mathbb\{R\}^\{d\}$. Suppose that the s-density $\backslash Theta^\{s\}(\backslash mu,a)$ exists and is positive and finite for a in a set of positive $\backslash mu$ measure. Then s is an integer.

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

• [http://www.encyclopediaofmath.org/index.php/Density_of_a_set Density of a set] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
• [http://www.encyclopediaofmath.org/index.php/Rectifiable_set Rectifiable set] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]

• Pertti Mattila, Geometry of sets and measures in Euclidean spaces. Cambridge Press, 1995.
• {{cite journal

| last = Preiss | first = David

| author-link = David Preiss

| title = Geometry of measures in $R^n$: distribution, rectifiability, and densities

| jstor = 1971410

| journal = Ann. Math. | volume = 125

| issue = 3 | pages = 537–643 | year = 1987 | doi=10.2307/1971410

| hdl = 10338.dmlcz/133417

| hdl-access = free}}

Category:Measure theory