Carathéodory’s extension theorem

E118705

mathematical theorem result in measure theory

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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All labels observed (5)

Label	Occurrences
Carathéodory extension theorem	3
Carathéodory’s criterion for measurability	1
Carathéodory’s extension theorem canonical	1
Carathéodory’s measure extension theorem	1
Carathéodory’s theorem on extension of measures	1

How this entity was disambiguated

This entity first appeared as the object of triple T998591 — 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’s extension theorem
Context triple: [Constantin Carathéodory, notableWork, Carathéodory’s extension theorem]

A. Cantor’s theorem
Cantor’s theorem is a fundamental result in set theory stating that the power set of any set has a strictly greater cardinality than the set itself, implying there is no largest infinity.
B. Radon–Nikodym derivative
The Radon–Nikodym derivative is a function that represents how one measure changes with respect to another absolutely continuous measure, playing a central role in modern probability theory and measure theory.
C. Cameron–Martin theorem
The Cameron–Martin theorem is a fundamental result in probability theory and functional analysis that characterizes how Gaussian measures on infinite-dimensional spaces change under shifts by elements of a special Hilbert subspace (the Cameron–Martin space).
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. Kakutani fixed-point theorem
The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.
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’s extension theorem
Target entity description: 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.

A. Cantor’s theorem
Cantor’s theorem is a fundamental result in set theory stating that the power set of any set has a strictly greater cardinality than the set itself, implying there is no largest infinity.
B. Radon–Nikodym derivative
The Radon–Nikodym derivative is a function that represents how one measure changes with respect to another absolutely continuous measure, playing a central role in modern probability theory and measure theory.
C. Cameron–Martin theorem
The Cameron–Martin theorem is a fundamental result in probability theory and functional analysis that characterizes how Gaussian measures on infinite-dimensional spaces change under shifts by elements of a special Hilbert subspace (the Cameron–Martin space).
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. Kakutani fixed-point theorem
The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.
F. None of above. chosen

Statements (44)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in measure theory ⓘ
alternativeName	Carathéodory’s extension theorem ⓘ surface form: Carathéodory’s measure extension theorem Carathéodory’s extension theorem ⓘ surface form: Carathéodory’s theorem on extension of measures
appearsIn	graduate textbooks on measure theory ⓘ graduate textbooks on probability theory ⓘ
appliesTo	algebra of sets ⓘ pre-measure ⓘ
assumes	algebra contains the empty set ⓘ underlying set is arbitrary ⓘ
characterizedBy	construction of an outer measure from a pre-measure ⓘ restriction of outer measure to Carathéodory-measurable sets ⓘ
concludesAbout	complete measure ⓘ measure ⓘ σ-algebra generated by an algebra ⓘ
ensures	extension agrees with the pre-measure on the original algebra ⓘ extension is a measure on the σ-algebra generated by the algebra ⓘ extension is complete with respect to null sets of the outer measure ⓘ measure of empty set is zero in the extension ⓘ
field	measure theory ⓘ probability theory ⓘ real analysis ⓘ
guarantees	existence of a measure extending a pre-measure ⓘ uniqueness of a measure extending a pre-measure ⓘ
holdsUnderCondition	pre-measure is σ-finite (for uniqueness on generated σ-algebra in some formulations) ⓘ
implies	existence of Lebesgue measure on ℝ ⓘ existence of probability measures from consistent finite-dimensional distributions ⓘ existence of product measures ⓘ
isToolFor	building measure spaces from simpler set functions ⓘ formalizing probability spaces from set functions on algebras ⓘ
namedAfter	Constantin Carathéodory ⓘ
partOf	foundations of modern measure theory ⓘ
relatedTo	Kolmogorov extension theorem ⓘ surface form: Hahn–Kolmogorov theorem Kolmogorov extension theorem ⓘ
requires	pre-measure is defined on an algebra of subsets of a set ⓘ pre-measure is non-negative ⓘ pre-measure is σ-additive on the algebra ⓘ
usedFor	construction of Borel measures ⓘ construction of Lebesgue measure ⓘ construction of product measures on product spaces ⓘ extension of probability pre-measures to probability measures ⓘ
usesConcept	Carathéodory measurability criterion ⓘ outer measure ⓘ σ-algebra generated by a collection of sets ⓘ

How these facts were elicited

Referenced by (7)

Full triples — surface form annotated when it differs from this entity's canonical label.

Constantin Carathéodory → notableWork → Carathéodory’s extension theorem ⓘ

this entity surface form: Carathéodory’s criterion for measurability

Carathéodory’s extension theorem → alternativeName → Carathéodory’s extension theorem ⓘ

this entity surface form: Carathéodory’s measure extension theorem

Carathéodory’s extension theorem → alternativeName → Carathéodory’s extension theorem ⓘ

this entity surface form: Carathéodory’s theorem on extension of measures

Lebesgue measure → constructedBy → Carathéodory’s extension theorem ⓘ

this entity surface form: Carathéodory extension theorem

Kolmogorov extension theorem → relatedTo → Carathéodory’s extension theorem ⓘ

this entity surface form: Carathéodory extension theorem

measure theory → usesConcept → Carathéodory’s extension theorem ⓘ

this entity surface form: Carathéodory extension theorem

All labels observed (5)

How this entity was disambiguated Show

Statements (44)

How these facts were elicited Show

Referenced by (7)

How this entity was disambiguated

How these facts were elicited