Английская Википедия:Frostman lemma
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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 μ on Rn satisfying μ(A) > 0, and such that
- <math>\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>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 Шаблон:Frac = 0. As before, the measure <math>\mu</math> is unsigned.) It follows from Frostman's lemma that for Borel A ⊂ Rn
- <math>\mathrm{dim}_H(A)= \sup\{s\ge 0:C_s(A)>0\}.</math>
Web pages
Further reading
Шаблон:Fractal-stub
Шаблон:Metric-geometry-stub