Frostman lemma explained

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 on Rⁿ satisfying μ(A) > 0, and such that

\mu(B(x,r))\le 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):=\supl\{l(\int} _{A x} (d\mu(x)d\mu(y))/(|x - y| ^{s}) r ) ^{-1} :\mu isaBorelmeasureand \mu(A) =1 r \}.

(Here, we take inf ∅ = ∞ and  = 0. As before, the measure

\mu is unsigned.) It follows from Frostman's lemma that for Borel A ⊂ Rⁿdim _{H(A)=} \sup\{s\ge 0 :C _{s(A)>0\}.}

Web pages

• Illustrating Frostman measures