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

Source of Name

This entry was named for Felix Hausdorff.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Hausdorff measure

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• 2014: Kenneth Falconer: Fractal Geometry: Mathematical Foundations and Applications (3rd ed.): $3.1$ Hausdorff measure