Definition:Hausdorff Measure
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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