Lebesgue measurable set
E898511
A Lebesgue measurable set is a subset of Euclidean space for which a consistent, translation-invariant notion of "size" (Lebesgue measure) can be assigned, forming the foundation of modern measure theory and integration.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Lebesgue measurable functions | 1 |
| Lebesgue measurable set canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10992088 — 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: Lebesgue measurable set Context triple: [Henri Lebesgue, notableConcept, Lebesgue measurable set]
-
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.
Carathéodory measurability criterion
The Carathéodory measurability criterion is a fundamental condition in measure theory that characterizes measurable sets via an outer measure by requiring additivity over intersections and complements.
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C.
Borel set
A Borel set is any set that can be formed from open (or equivalently closed) sets of a topological space through countable unions, intersections, and complements, forming the smallest σ-algebra containing all open sets.
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D.
Lebesgue integration
Lebesgue integration is a foundational measure-theoretic framework for defining and analyzing integrals, particularly suited to handling limits, convergence, and more general functions than those allowed by Riemann integration.
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E.
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.
- 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: Lebesgue measurable set Target entity description: A Lebesgue measurable set is a subset of Euclidean space for which a consistent, translation-invariant notion of "size" (Lebesgue measure) can be assigned, forming the foundation of modern measure theory and integration.
-
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.
Carathéodory measurability criterion
The Carathéodory measurability criterion is a fundamental condition in measure theory that characterizes measurable sets via an outer measure by requiring additivity over intersections and complements.
-
C.
Borel set
A Borel set is any set that can be formed from open (or equivalently closed) sets of a topological space through countable unions, intersections, and complements, forming the smallest σ-algebra containing all open sets.
-
D.
Lebesgue integration
Lebesgue integration is a foundational measure-theoretic framework for defining and analyzing integrals, particularly suited to handling limits, convergence, and more general functions than those allowed by Riemann integration.
-
E.
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.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
measurable set ⓘ subset of Euclidean space ⓘ |
| associatedWith | Lebesgue measure ⓘ |
| characterizedBy |
Carathéodory measurability condition
ⓘ
equality of outer measure of set and outer measure of its intersection with any other set plus that of its complement intersection ⓘ |
| contrastedWith | non-measurable set ⓘ |
| definedOn |
Euclidean space
ⓘ
ℝ ⓘ ℝⁿ ⓘ |
| exampleOf | measurable subset of a measure space ⓘ |
| fieldOfStudy |
integration theory
ⓘ
measure theory ⓘ real analysis ⓘ |
| foundationFor |
Lᵖ spaces
ⓘ
linked to:
Lebesgue spaces
modern probability theory ⓘ modern real analysis ⓘ |
| generalizationOf | Jordan measurable set ⓘ |
| hasCardinalityProperty | collection has cardinality 2^{continuum} ⓘ |
| hasLimitation | not every subset of ℝ is Lebesgue measurable ⓘ |
| hasProperty |
almost-everywhere equality defined via Lebesgue measurable sets
ⓘ
closed under complementation ⓘ closed under countable intersections ⓘ closed under countable unions ⓘ completeness under Lebesgue measure ⓘ countable additivity of measure ⓘ every Borel set is Lebesgue measurable ⓘ forms a σ-algebra ⓘ inner regularity with respect to closed sets ⓘ measure zero sets are Lebesgue measurable ⓘ outer regularity with respect to open sets ⓘ translation invariance of measure ⓘ |
| namedAfter | Henri Lebesgue ⓘ |
| relatedConcept |
Borel set
ⓘ
Lebesgue integral ⓘ
linked to:
Lebesgue integration
completion of a measure space ⓘ measurable function ⓘ null set ⓘ outer measure ⓘ σ-algebra ⓘ |
| subsetOf |
power set of ℝ
ⓘ
σ-algebra of Lebesgue measurable subsets of ℝ ⓘ |
| usedFor |
definition of Lebesgue integral
ⓘ
definition of measurable functions ⓘ formulation of convergence theorems in integration ⓘ probability spaces on ℝⁿ ⓘ |
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: Lebesgue measurable set Description of subject: A Lebesgue measurable set is a subset of Euclidean space for which a consistent, translation-invariant notion of "size" (Lebesgue measure) can be assigned, forming the foundation of modern measure theory and integration.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Lebesgue measurable set