In measure theory, a field of mathematics, the Hausdorff density measures how concentrated a Radon measure is at some point.
Definition Let μ {\displaystyle \mu } be a Radon measure and a ∈ R n {\displaystyle a\in \mathbb {R} ^{n}} some point in Euclidean space. The s-dimensional upper and lower Hausdorff densities are defined to be, respectively,
Θ ∗ s ( μ , a ) = lim sup r → 0 μ ( B r ( a ) ) r s {\displaystyle \Theta ^{*s}(\mu ,a)=\limsup _{r\rightarrow 0}{\frac {\mu (B_{r}(a))}{r^{s}}}}
and
Θ ∗ s ( μ , a ) = lim inf r → 0 μ ( B r ( a ) ) r s {\displaystyle \Theta _{*}^{s}(\mu ,a)=\liminf _{r\rightarrow 0}{\frac {\mu (B_{r}(a))}{r^{s}}}}
where B r ( a ) {\displaystyle B_{r}(a)} is the ball of radius r > 0 centered at a. Clearly, Θ ∗ s ( μ , a ) ≤ Θ ∗ s ( μ , a ) {\displaystyle \Theta _{*}^{s}(\mu ,a)\leq \Theta ^{*s}(\mu ,a)} for all a ∈ R n {\displaystyle a\in \mathbb {R} ^{n}} . In the event that the two are equal, we call their common value the s-density of μ {\displaystyle \mu } at a and denote it Θ s ( μ , a ) {\displaystyle \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 μ {\displaystyle \mu } be a Radon measure on R d {\displaystyle \mathbb {R} ^{d}} . Suppose that the s-density Θ s ( μ , a ) {\displaystyle \Theta ^{s}(\mu ,a)} exists and is positive and finite for a in a set of positive μ {\displaystyle \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 μ {\displaystyle \mu } be a Radon measure on R d {\displaystyle \mathbb {R} ^{d}} . Suppose that m ≥ 1 {\displaystyle \geq 1} is an integer and the m-density Θ m ( μ , a ) {\displaystyle \Theta ^{m}(\mu ,a)} exists and is positive and finite for μ {\displaystyle \mu } almost every a in the support of μ {\displaystyle \mu } . Then μ {\displaystyle \mu } is m-rectifiable, i.e. μ ≪ H m {\displaystyle \mu \ll H^{m}} ( μ {\displaystyle \mu } is absolutely continuous with respect to Hausdorff measure H m {\displaystyle H^{m}} ) and the support of μ {\displaystyle \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 {\displaystyle R^{n}} : distribution, rectifiability, and densities". Ann. Math. 125 (3): 537–643. doi:10.2307/1971410. hdl:10338.dmlcz/133417. JSTOR 1971410.
