Frostman lemma

Short description: Tool for estimating the Hausdorff dimension of sets

Frostman's lemma provides a convenient tool for estimating the Hausdorff dimension of sets in mathematics, and more specifically, in the theory of fractal dimensions.

Lemma

Lemma: Let A be a Borel subset of Rⁿ, and let s > 0. Then the following are equivalent:

• H^s(A) > 0, where H^s denotes the s-dimensional Hausdorff measure.
• There is an (unsigned) Borel measure μ on Rⁿ satisfying μ(A) > 0, and such that

$\mu (B(x,r))\le r^s$

holds for all x ∈ Rⁿ and r>0.

Otto Frostman proved this lemma for closed sets A as part of his PhD dissertation at Lund University in 1935. The generalization to Borel sets is more involved, and requires the theory of Suslin sets.

A useful corollary of Frostman's lemma requires the notions of the s-capacity of a Borel set A ⊂ Rⁿ, which is defined by

$C_s(A):=\sup \left\{\right(\underset{A\times A}{\int }\frac{d\mu (x)d\mu (y)}{|x-y{|}^s}{)}^{-1}:\mu \text{ is a Borel measure and }\mu (A)=1\}.$

(Here, we take inf ∅ = ∞ and ​¹⁄_∞ = 0. As before, the measure $\mu$ is unsigned.) It follows from Frostman's lemma that for Borel A ⊂ Rⁿ

${\mathrm{d}\mathrm{i}\mathrm{m}}_H(A)=\sup \{s\ge 0:C_s(A)>0\}.$

Web pages

• Illustrating Frostman measures

Further reading

• Mattila, Pertti (1995), Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, 44, Cambridge University Press, ISBN 978-0-521-65595-8