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	4
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 measure extension theorem ⓘ linked to: Carathéodory’s extension theorem Carathéodory’s theorem on extension of measures ⓘ linked to: Carathéodory’s extension theorem
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	Hahn–Kolmogorov theorem ⓘ linked to: Kolmogorov extension 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