Frostman lemma - Alchetron, The Free Social Encyclopedia

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:

H^s(A) > 0, where H^s 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

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 } .

(Here, we take inf ∅ = ∞ and ¹⁄_∞ = 0. As before, the measure μ 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 } .

References

Frostman lemma Wikipedia

(Text) CC BY-SA