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 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 1⁄∞ = 0. As before, the measure is unsigned.) It follows from Frostman's lemma that for Borel A ⊂ Rn
- 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, MR 1333890
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