Hausdorff measure
E608815
Hausdorff measure is a fundamental concept in geometric measure theory that generalizes the notion of length, area, and volume to sets with arbitrary fractal or irregular structure in metric spaces.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
concept in geometric measure theory
ⓘ
measure ⓘ outer measure ⓘ |
| assignsFiniteMeasureTo | many fractal sets ⓘ |
| assignsInfiniteMeasureTo | sets of Hausdorff dimension greater than s for s-dimensional measure ⓘ |
| assignsZeroMeasureTo | sets of Hausdorff dimension less than s for s-dimensional measure ⓘ |
| codomain | extended nonnegative real numbers ⓘ |
| coincidesWith |
Lebesgue measure on R^n up to constant factor
ⓘ
n-dimensional volume on sufficiently regular n-dimensional manifolds ⓘ |
| constructionUses |
countable covers by sets of small diameter
ⓘ
gauge function r^s ⓘ limit as diameter bound tends to zero ⓘ |
| definedOn | metric spaces ⓘ |
| dependsOn | metric ⓘ |
| domain | all subsets of a metric space ⓘ |
| field |
fractal geometry
ⓘ
geometric measure theory ⓘ measure theory ⓘ |
| generalizes |
Lebesgue measure on Euclidean space
ⓘ
notion of area ⓘ notion of length ⓘ notion of volume ⓘ |
| hasGeneralization |
Hausdorff–Carathéodory measure
NERFINISHED
ⓘ
gauge Hausdorff measure ⓘ |
| hasVariant | s-dimensional Hausdorff measure ⓘ |
| introducedIn | early 20th century ⓘ |
| isBorelRegular | true ⓘ |
| isMetricDependent | true ⓘ |
| isOuterMeasure | true ⓘ |
| isSigmaFiniteOn | many natural geometric sets ⓘ |
| namedAfter | Felix Hausdorff NERFINISHED ⓘ |
| notation | H^s ⓘ |
| parameterizedBy | real dimension parameter s ⓘ |
| relatedTo |
Carathéodory construction of measures
NERFINISHED
ⓘ
Minkowski dimension NERFINISHED ⓘ packing measure ⓘ |
| usedIn |
analysis on metric spaces
ⓘ
calculus of variations ⓘ geometric analysis ⓘ minimal surface theory ⓘ potential theory ⓘ study of fractal sets ⓘ theory of rectifiable sets ⓘ |
| usedToCharacterize | rectifiable sets via approximate tangent planes ⓘ |
| usedToDefine |
Hausdorff dimension
ⓘ
perimeter of sets of finite perimeter ⓘ surface measure on boundaries of sets in R^n ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.