In measure theory, a field of mathematics, the Hausdorff density measures how concentrated a Radon measure is at some point.
Let μ be a Radon measure and a ∈ 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 and
Θ ∗ s ( μ , a ) = lim inf r → 0 μ ( B r ( a ) ) r s where B r ( a ) is the ball of radius r > 0 centered at a. Clearly, Θ ∗ s ( μ , a ) ≤ Θ ∗ s ( μ , a ) for all a ∈ 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 Θ s ( μ , a ) .
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
R d . Suppose that the
s-density
Θ s ( μ , a ) exists and is positive and finite for
a in a set of positive
μ 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
μ be a Radon measure on
R d . Suppose that
m ≥ 1 is an integer and the
m-density
Θ m ( μ , a ) exists and is positive and finite for
μ almost every
a in the
support of
μ . Then
μ is
m-rectifiable, i.e.
μ ≪ H m (
μ is absolutely continuous with respect to
Hausdorff measure H m ) and the support of
μ is an
m-rectifiable set.
