Hausdorff Measure - Relation With Hausdorff Dimension

Relation With Hausdorff Dimension

One of several possible equivalent definitions of the Hausdorff dimension is

$\operatorname{dim}_{\mathrm{Haus}}(S)=\inf\{d\ge 0:H^d(S)=0\}=\sup\bigl(\{d\ge 0:H^d(S)=\infty\}\cup\{0\}\bigr),$

where we take

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