Hausdorff content

Hausdorff content (plural Hausdorff contents)

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

• Hausdorff dimension