Hausdorff Measure
Suppose
and . Then for all , we define where is the diameter. Then the -dimensional Hausdorff measure is defined as
Remark
As
decreases, the class of permissible covers of in is reduced. Therefore, the infimum increases, or at least does not decrease. So this limit exists for any subset of , although the limiting value can be (and usually is) or .
Proposition
The Hausdorff measure is a measure on
.
Hausdorff Dimension
Roughly speaking, the Hausdorff dimension of a set
is the value of at which the -dimensional Hausdorff measure ‘jump’ from to occurs. Formally, we define the Hausdorff dimension of as
Corollary
- If
is a Lipschitz transformation, then
- If
is a bi-Lipschitz transformation, i.e., where , then