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

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 ⓘ

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