Multifractals
Hausdorff dimension bounded by measure
If the measure of discs about every point of a set A has a power law scaling with exponent s, then the s-dimensional Hausdorff measure is bounded by the measure of A.
Specifically, let D
e
(x) denote the disc of radius e centered at x and suppose
lim
e -> 0
mu(B
e
(x))/e
s
<= c for all x in A.
then H
s
(A) >= mu(A)/c.