Haar measure
E860100
Haar measure is a fundamental concept in harmonic analysis and topological group theory, providing a translation-invariant way to assign measures to subsets of locally compact groups.
Statements (51)
| Predicate | Object |
|---|---|
| instanceOf |
concept in harmonic analysis
ⓘ
concept in topological group theory ⓘ mathematical concept ⓘ measure ⓘ |
| appliesTo |
compact groups
ⓘ
locally compact abelian groups ⓘ non-abelian locally compact groups ⓘ |
| correspondsTo |
Lebesgue measure on R^n
ⓘ
counting measure on finite groups ⓘ normalized arc length on the circle group ⓘ |
| definedOn | locally compact group ⓘ |
| domain | Borel sigma-algebra of a locally compact group ⓘ |
| ensures | existence of invariant integration on locally compact groups ⓘ |
| hasProperty |
inner regularity
ⓘ
left-translation invariance ⓘ local finiteness ⓘ nontriviality ⓘ outer regularity ⓘ regularity ⓘ right-translation invariance ⓘ uniqueness up to scalar multiple ⓘ |
| introducedBy | Alfréd Haar NERFINISHED ⓘ |
| invarianceType |
bi-invariant measure on unimodular groups
ⓘ
left-invariant measure ⓘ right-invariant measure ⓘ |
| mathematicalField |
abstract algebra
ⓘ
analysis ⓘ topology ⓘ |
| normalizationProperty | total measure of compact group can be chosen to be 1 ⓘ |
| relatedConcept |
Fourier transform on locally compact abelian groups
ⓘ
Plancherel measure NERFINISHED ⓘ Pontryagin duality NERFINISHED ⓘ amenability of groups ⓘ invariant integral ⓘ modular function of a locally compact group ⓘ |
| specialCaseOn |
Euclidean space as additive group
NERFINISHED
ⓘ
circle group ⓘ finite groups ⓘ real line as additive group ⓘ |
| uniquenessStatement | any two left Haar measures differ by a positive scalar multiple ⓘ |
| usedIn |
Fourier analysis on groups
ⓘ
abstract harmonic analysis ⓘ automorphic forms ⓘ ergodic theory ⓘ harmonic analysis ⓘ number theory ⓘ probability on groups ⓘ representation theory of Lie groups ⓘ representation theory of locally compact groups ⓘ topological group theory ⓘ |
| yearIntroduced | 1933 ⓘ |
Referenced by (1)
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