Frostman lemma

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:
 * Hs(A) > 0, where Hs denotes the s-dimensional Hausdorff measure.
 * There is an (unsigned) Borel measure &mu; on Rn satisfying &mu;(A) > 0, and such that
 * $$\mu(B(x,r))\le r^s$$
 * holds for all x &isin; 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 &sub; Rn, which is defined by


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

(Here, we take inf &empty; = &infin; and $1/&infin;$ = 0. As before, the measure $$\mu$$ is unsigned.) It follows from Frostman's lemma that for Borel A &sub; Rn


 * $$\mathrm{dim}_H(A)= \sup\{s\ge 0:C_s(A)>0\}.$$

Web pages

 * Illustrating Frostman measures