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: Hs(A) > 0, where Hs denotes the s-dimensional Hausdorff measure. There is an (unsigned) Borel measure μ on Rn satisfying μ(A) > 0, and such that
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
(Here, we take inf ∅ = ∞ and = 0. As before, the measure is unsigned.) It follows from Frostman's lemma that for Borel A ⊂ Rn
Web pages
Illustrating Frostman measures
References
Further reading