Hausdorff density

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

[edit] Definition

Let μ 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 Br(a) is the ball of raduis 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 μ at a and denote it Θs(μ,a).

[edit] Marstrand's theorem

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

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

[edit] References

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