Frostman lemma

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 $R^n$, 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 $mu$ satisfying $mu(A)>0$ such that $mu(B(x,r))le\; r^s$ holds for all $xinR^n$ 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.

A useful corollary of Frostman's lemma requires the notions of the $s$-capacity of a Borel set $AsubsetR^n$, which is defined by:$C\_s(A):=supBigl\{Bigl(int\_\{A\; imes\; A\}\; |x-y|^\{-s\},dmu(x),dmu(y)Bigr)^\{-1\}:mu\; ext\{\; is\; a\; Borel\; measure\; and\; \}mu(A)=1Bigr\}.$(Here, we take $infemptyset=infty$ and $1/infty=0$. As before, the measure $mu$ is unsigned.) It follows from Frostman's lemma that for Borel $Asubset\; R^n$

:$mathrm\{dim\}\_H(A)=\; sup\{sge\; 0:C\_s(A)>0\}.$

References

* Citation
last1=Mattila
first1=Pertti | title=Geometry of sets and measures in Euclidean spaces | publisher=Cambridge University Press
isbn=978-0-521-65595-8 | year=1995