Frostman lemma

{{rewritten|date=February 2025}} {{short description|Tool for estimating the Hausdorff dimension of sets}}

'''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.<ref>{{cite arXiv|last=Dobronravov |first=Nikita P. |title=Frostman lemma revisited |date=2022-04-22 |class=math.CA |eprint=2204.10441 }}</ref> ==Lemma==

'''Lemma:''' Let ''A'' be a Borel subset of '''R'''''n'', and let ''s'' > 0. Then the following are equivalent: *''H''''s''(''A'') > 0, where ''H''''s'' denotes the ''s''-dimensional Hausdorff measure. *There is an (unsigned) Borel measure ''μ'' on '''R'''''n'' satisfying ''μ''(''A'') > 0, and such that ::<math>\mu(B(x,r))\le r^s</math> :holds for all ''x'' ∈ '''R'''''n'' 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.<ref>{{Cite journal |last=Nozaki |first=Yasuo |date=1958 |title=On generalization of Frostman's lemma and its applications |url=https://www.jstage.jst.go.jp/article/kodaimath1949/10/3/10_3_113/_article/-char/ja/ |journal=Kodai Mathematical Seminar Reports |volume=10 |issue=3 |pages=113–126 |doi=10.2996/kmj/1138844025}}</ref>

A useful corollary of Frostman's lemma requires the notions of the ''s''-capacity of a Borel set ''A'' ⊂ '''R'''''n'', 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 {{Frac|1|∞}} = 0. As before, the measure <math>\mu</math> is unsigned.) It follows from Frostman's lemma that for Borel ''A'' ⊂ '''R'''''n''

:<math>\mathrm{dim}_H(A)= \sup\{s\ge 0:C_s(A)>0\}.</math>

==Web pages==

*[https://web.archive.org/web/20210917223752/maths.ed.ac.uk/~jazzam/posts/2020-01-14-frostmann/ Illustrating Frostman measures]

==References== {{reflist}}

==Further reading== * {{Citation | last1=Mattila | first1=Pertti | author1-link = Pertti Mattila | title=Geometry of sets and measures in Euclidean spaces | publisher=Cambridge University Press | isbn=978-0-521-65595-8 | year=1995 | mr = 1333890 |series= Cambridge Studies in Advanced Mathematics | volume = 44}}

Category:Dimension theory Category:Fractals Category:Metric geometry