Carathéodory measurability criterion
E506846
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Carathéodory measurability criterion canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5256252 — 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: Carathéodory measurability criterion Context triple: [Carathéodory’s extension theorem, usesConcept, Carathéodory measurability criterion]
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A.
Carathéodory’s extension theorem
Carathéodory’s extension theorem is a fundamental result in measure theory that guarantees a unique extension of a pre-measure defined on an algebra of sets to a complete measure on the generated σ-algebra.
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B.
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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C.
Carathéodory metric
The Carathéodory metric is an intrinsic distance function in complex analysis that measures how far apart points are in a domain based on holomorphic mappings into the unit disk.
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D.
Lebesgue differentiation theorem
The Lebesgue differentiation theorem is a fundamental result in real analysis stating that, for an integrable function, the averages over shrinking neighborhoods converge almost everywhere to the function’s pointwise value.
-
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: Carathéodory measurability criterion Target entity description: 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.
-
A.
Carathéodory’s extension theorem
Carathéodory’s extension theorem is a fundamental result in measure theory that guarantees a unique extension of a pre-measure defined on an algebra of sets to a complete measure on the generated σ-algebra.
-
B.
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.
-
C.
Carathéodory metric
The Carathéodory metric is an intrinsic distance function in complex analysis that measures how far apart points are in a domain based on holomorphic mappings into the unit disk.
-
D.
Lebesgue differentiation theorem
The Lebesgue differentiation theorem is a fundamental result in real analysis stating that, for an integrable function, the averages over shrinking neighborhoods converge almost everywhere to the function’s pointwise value.
-
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 (47)
| Predicate | Object |
|---|---|
| instanceOf |
concept in measure theory
ⓘ
mathematical criterion ⓘ |
| appliesTo | outer measure spaces ⓘ |
| assumes |
monotonicity of the outer measure
ⓘ
null empty set for the outer measure ⓘ subadditivity of the outer measure ⓘ |
| basedOn | Carathéodory construction of measures NERFINISHED ⓘ |
| category | theorem in analysis ⓘ |
| characterizes | measurable sets ⓘ |
| concerns | structure of measurable sets relative to an outer measure ⓘ |
| contrastsWith |
definitions of measurability via Borel σ-algebra
ⓘ
definitions of measurability via open sets ⓘ |
| coreCondition |
Carathéodory condition for measurability
ⓘ
additivity over disjoint unions ⓘ additivity over intersections and complements ⓘ |
| defines | Carathéodory-measurable set ⓘ |
| ensures |
compatibility of measure with set operations
ⓘ
measurable sets form a σ-algebra ⓘ restriction of outer measure to measurable sets is a measure ⓘ |
| expressedIn | set-theoretic language ⓘ |
| field |
measure theory
ⓘ
real analysis ⓘ |
| formalStatement | A set E is measurable if for every set A, μ*(A) = μ*(A ∩ E) + μ*(A \ E) ⓘ |
| guarantees | completeness of the resulting measure ⓘ |
| hasDomain | subsets of a given set X ⓘ |
| holdsIn |
abstract measure spaces
ⓘ
metric measure spaces ⓘ |
| implies |
countable additivity on the σ-algebra of measurable sets
ⓘ
σ-additivity of the induced measure on measurable sets ⓘ |
| introducedBy | Constantin Carathéodory NERFINISHED ⓘ |
| involves | outer measure μ* ⓘ |
| mathematicalSubjectClassification | 28A12 ⓘ |
| namedAfter | Constantin Carathéodory NERFINISHED ⓘ |
| relatedTo |
Carathéodory extension theorem
NERFINISHED
ⓘ
Lebesgue outer measure NERFINISHED ⓘ σ-algebra of measurable sets ⓘ |
| requires | all subsets A of the underlying space ⓘ |
| timePeriod | early 20th century ⓘ |
| typeOf | necessary and sufficient condition for measurability ⓘ |
| usedFor |
constructing measures from outer measures
ⓘ
defining Lebesgue measurable sets ⓘ extending premeasures to complete measures ⓘ |
| usedIn |
construction of Lebesgue measure on ℝ
ⓘ
construction of product measures ⓘ integration theory ⓘ probability theory ⓘ |
| usesConcept | outer measure ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Carathéodory measurability criterion Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.