Hausdorff Capacity

Abstract

Given a number d ∈ (o, n], the d-dimensional Hausdorff Capacity of a set $E \subset \mathbb{R}^{n}$ is given by

$$\displaystyle{ \Lambda ^{d}(E) =\inf \bigg\{\sum _{ j}r_{j}^{d}: E \subset \cup _{ j}B(x_{j},r_{j}),\;j = 1,2,3,\ldots.\bigg\} }$$

(3.1)

i.e., E is covered by balls B(x _j , r _j ), centered at some x _j and of radius r _j  > 0, and then the infimum is taken over all corresponding sums.