Definition:Hausdorff Measure

Definition
Let $\powerset {\R^n}$ be the power set of the real Euclidean space $\R^n$.

Given $U \in \powerset {\R^n}$, let $\size U$ denote the diameter of $U$.

Let $s \in \R_{\ge 0}$.

The $s$-dimensional Hausdorff measure on $\R^n$ is an outer measure:
 * $\HH^s: \powerset {\R^n} \to \overline \R_{\ge 0}$

defined by:
 * $\ds \map {\HH^s} F := \lim_{\delta \to 0^+} \map {\HH^s_\delta} F$

where:
 * $\ds \map {\HH^s_\delta} F := \inf \leftset {\sum \size {U_i}^s : \sequence {U_i} }$ is a $\delta$-cover of $\rightset {F}$

Also see

 * Hausdorff Measure is Outer Measure
 * Restriction of Hausdorff Measure is Borel Measure
 * Higher Dimensional Hausdorff Measure than Euclidean Space is Zero
 * Definition:Hausdorff-Besicovitch Dimension