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Definition 2024

Hausdorff content ‎(plural Hausdorff contents)

the d-dimensional Hausdorff content of S is defined by $C_{H}^{d}(S):=\lim _{\sup _{i}r_{i}\rightarrow 0}\inf {\Bigl \{}\sum _{i}r_{i}^{d}:{\text{ there is a cover of }}S{\text{ by balls with radii }}r_{i}>0{\Bigr \}}.$
- 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.