In mathematics, and more specifically, in the theory of fractal dimensions, Frostman's lemma provides a convenient tool for estimating the

Hausdorff dimension In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of ...

of sets. 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 In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that as ...

. *There is an (unsigned) Borel measure ''μ'' on R^''n'' satisfying ''μ''(''A'') > 0, and such that ::

\mu(B(x,r))\le r^s

:holds for all ''x'' ∈ R^''n'' and ''r''>0.

Otto Frostman Otto Albin Frostman (3 January 1907 – 29 December 1977) was a Swedish mathematician, known for his work in potential theory and complex analysis. Frostman earned his Ph.D. in 1935 at Lund University under the Hungarian-born mathematician ...

proved this lemma for closed sets ''A'' as part of his PhD dissertation at

Lund University , motto = Ad utrumque , mottoeng = Prepared for both , established = , type = Public research university , budget = SEK 9 billion Suslin sets. 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 :

C_s(A):=\sup\Bigl\.

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

\mu

is unsigned.) It follows from Frostman's lemma that for Borel ''A'' ⊂ R^''n'' :

\mathrm_H(A)= \sup\.

Web pages

Illustrating Frostman measures

Web pages

Further reading