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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Haar measure canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10388988 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Haar measure Context triple: [L’intégration dans les groupes topologiques et ses applications, topic, Haar measure]
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A.
Lebesgue measure
Lebesgue measure is the standard way of assigning a consistent notion of "length," "area," or "volume" to subsets of Euclidean space, forming the foundation of modern measure theory and integration.
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B.
Hausdorff measure
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.
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C.
Wiener measure
Wiener measure is the canonical probability measure on the space of continuous paths that models standard Brownian motion in probability theory and stochastic analysis.
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D.
measure theory
Measure theory is a branch of mathematical analysis that rigorously formalizes the concepts of length, area, volume, and integration for very general sets and functions.
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E.
Introduction to Abstract Harmonic Analysis
Introduction to Abstract Harmonic Analysis is a foundational graduate-level textbook that systematically develops the theory of harmonic analysis on topological groups and related abstract structures.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Haar measure Target entity description: 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.
-
A.
Lebesgue measure
Lebesgue measure is the standard way of assigning a consistent notion of "length," "area," or "volume" to subsets of Euclidean space, forming the foundation of modern measure theory and integration.
-
B.
Hausdorff measure
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.
-
C.
Wiener measure
Wiener measure is the canonical probability measure on the space of continuous paths that models standard Brownian motion in probability theory and stochastic analysis.
-
D.
measure theory
Measure theory is a branch of mathematical analysis that rigorously formalizes the concepts of length, area, volume, and integration for very general sets and functions.
-
E.
Introduction to Abstract Harmonic Analysis
Introduction to Abstract Harmonic Analysis is a foundational graduate-level textbook that systematically develops the theory of harmonic analysis on topological groups and related abstract structures.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Haar measure Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.