Noun
Hausdorff content (plural Hausdorff contents)
the d-dimensional Hausdorff content of S is defined by
C
H
d
(
S
)
:=
lim
sup
i
r
i
→
0
inf
{
∑
i
r
i
d
:
there is a cover of
S
by balls with radii
r
i
>
0
}
.
{\displaystyle C_{H}^{d}(S):=\lim _{\sup _{i}r_{i}\rightarrow 0}\inf {\Bigl \{}\sum _{i}r_{i}^{d}:{\text{ there is a cover of }}S{\text{ by balls with radii }}r_{i}>0{\Bigr \}}.}
The d-dimensional Hausdorff content of a set is a measure of the size of that set in the d dimension.
The 0-dimensional Hausdorff content of a set is the number of points in that set.