- Source: 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 Rn, and let s > 0. Then the following are equivalent:
Hs(A) > 0, where Hs denotes the s-dimensional Hausdorff measure.
There is an (unsigned) Borel measure μ on Rn satisfying μ(A) > 0, and such that
μ
(
B
(
x
,
r
)
)
≤
r
s
{\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.
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
}
.
{\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
μ
{\displaystyle \mu }
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
}
.
{\displaystyle \mathrm {dim} _{H}(A)=\sup\{s\geq 0:C_{s}(A)>0\}.}
Web pages
Illustrating Frostman measures
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