Frostman lemma

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. ^[1]

Lemma

Lemma: Let A be a Borel subset of Rⁿ, 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 Rⁿ satisfying μ(A) > 0, and such that

${\displaystyle \mu (B(x,r))\leq r^{s}}$

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. ^[2]

A useful corollary of Frostman's lemma requires the notions of the s-capacity of a Borel set A ⊂ Rⁿ, which is defined by

${\displaystyle C_{s}(A):=\sup {\Bigl \{}{\Bigl (}\int _{A\times A}{\frac {d\mu (x)\,d\mu (y)}{|x-y|^{s}}}{\Bigr )}^{-1}:\mu {\text{ is a Borel measure and }}\mu (A)=1{\Bigr \}}.}$

(Here, we take inf ∅ = ∞ and 1⁄∞ = 0. As before, the measure ${\displaystyle \mu }$ is unsigned.) It follows from Frostman's lemma that for Borel A ⊂ Rⁿ

${\displaystyle \mathrm {dim} _{H}(A)=\sup\{s\geq 0:C_{s}(A)>0\}.}$

Web pages

• Illustrating Frostman measures

References

1. Dobronravov, Nikita P. (2022-04-22), Frostman lemma revisited, arXiv: 2204.10441 , retrieved 2025-02-07
2. Nozaki, Yasuo (1958). "On generalization of Frostman's lemma and its applications". Kodai Mathematical Seminar Reports. 10 (3): 113–126. doi:10.2996/kmj/1138844025.

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