Frostman lemma

Short description: Tool for estimating the Hausdorff dimension of sets

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

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

[math]\displaystyle{ \mu(B(x,r))\le r^s }[/math]

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

[math]\displaystyle{ C_s(A):=\sup\Bigl\{\Bigl(\int_{A\times A} \frac{d\mu(x)\,d\mu(y)}{|x-y|^{s}}\Bigr)^{-1}:\mu\text{ is a Borel measure and }\mu(A)=1\Bigr\}. }[/math]

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

[math]\displaystyle{ \mathrm{dim}_H(A)= \sup\{s\ge 0:C_s(A)\gt 0\}. }[/math]

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