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
μ ( B ( x , r ) ) ≤ r s {\displaystyle \mu (B(x,r))\leq 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 { ( ∫ A × A d μ ( x ) d μ ( y ) | x − y | s ) − 1 : μ is a Borel measure and μ ( A ) = 1 } . {\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 \}}.}
(Here, we take inf ∅ = ∞ and 1⁄∞ = 0. As before, the measure μ {\displaystyle \mu } is unsigned.) It follows from Frostman's lemma that for Borel A ⊂ Rn
d i m H ( A ) = sup { s ≥ 0 : C s ( A ) > 0 } . {\displaystyle \mathrm {dim} _{H}(A)=\sup\{s\geq 0:C_{s}(A)>0\}.}
Web pages Illustrating Frostman measures
References
Further reading Mattila, Pertti (1995), Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, ISBN 978-0-521-65595-8, MR 1333890