Fubini's theorem

E284675

mathematical theorem theorem in measure theory

Fubini's theorem is a fundamental result in measure theory that allows the evaluation of double integrals as iterated integrals under suitable integrability conditions.

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All labels observed (12)

Label	Occurrences
Fubini's theorem canonical	2
Fubini theorem	1
Fubini theorem (components of proofs)	1
Fubini theorem for Banach-valued functions	1
Fubini theorem for integrals	1
Fubini's theorem for Bochner integrals	1
Fubini's theorem for Lebesgue integrals	1
Fubini's theorem for improper Riemann integrals under additional conditions	1
Fubini's theorem for integrable (not necessarily non‑negative) functions	1
Fubini–Tonelli theorem	1
Fubini–Tonelli type results	1
stochastic Fubini theorems	1

How this entity was disambiguated

This entity first appeared as the object of triple T2631219 — 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: Fubini's theorem
Context triple: [Lebesgue integration, relatedTo, Fubini's theorem]

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. 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).
C. 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.
D. Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
E. Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus links differentiation and integration by showing that the definite integral of a function can be computed using any of its antiderivatives.
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: Fubini's theorem
Target entity description: Fubini's theorem is a fundamental result in measure theory that allows the evaluation of double integrals as iterated integrals under suitable integrability conditions.

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. 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).
C. 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.
D. Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
E. Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus links differentiation and integration by showing that the definite integral of a function can be computed using any of its antiderivatives.
F. None of above. chosen

Statements (43)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem in measure theory ⓘ
allows	interchanging the order of integration under suitable conditions ⓘ
appliesTo	integrable functions on product measure spaces ⓘ σ-finite measure spaces ⓘ
assumes	measurability of the function on the product space ⓘ
category	theorems about integrals ⓘ theorems in analysis ⓘ
comparedWith	Tonelli's theorem for nonnegative functions ⓘ linked to: Tonelli's theorem
concerns	integration over product of measurable spaces ⓘ iterated integration with respect to different variables ⓘ
ensures	almost-everywhere equality of sections of integrable functions ⓘ measurability of sections of measurable functions on product spaces ⓘ
field	integration theory ⓘ measure theory ⓘ real analysis ⓘ
generalizes	interchange of summation and integration in some contexts ⓘ
hasConsequence	iterated integrals exist and are finite for almost all sections when the function is integrable ⓘ order of integration does not affect the value of the integral under its hypotheses ⓘ
hasVersion	Fubini's theorem for Bochner integrals ⓘ linked to: Fubini's theorem Fubini's theorem for Lebesgue integrals ⓘ linked to: Fubini's theorem Fubini's theorem for improper Riemann integrals under additional conditions ⓘ linked to: Fubini's theorem
historicalPeriod	early 20th century mathematics ⓘ
implies	equality of double integral and iterated integrals for integrable functions ⓘ
isToolFor	changing variables in multiple integrals together with the change of variables theorem ⓘ computing expectations of functions of several random variables ⓘ separating variables in integrals ⓘ
namedAfter	Guido Fubini ⓘ
relatesTo	Lebesgue integral ⓘ linked to: Lebesgue integration Tonelli's theorem ⓘ double integrals ⓘ iterated integrals ⓘ multiple integrals ⓘ product measure ⓘ
requiresCondition	integrability of the function on the product space ⓘ σ-finiteness of the underlying measure spaces ⓘ
states	the integral of an integrable function over a product space equals the iterated integrals almost everywhere ⓘ
usedIn	functional analysis ⓘ harmonic analysis ⓘ mathematical physics ⓘ partial differential equations ⓘ probability theory ⓘ stochastic processes ⓘ

How these facts were elicited

Referenced by (13)

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

Lebesgue integration → relatedTo → Fubini's theorem ⓘ

monotone convergence theorem → usedToProve → Fubini theorem (components of proofs) ⓘ

linked to: Fubini's theorem

Fubini's theorem → hasVersion → Fubini's theorem for Lebesgue integrals ⓘ

linked to: Fubini's theorem

Fubini's theorem → hasVersion → Fubini's theorem for Bochner integrals ⓘ

linked to: Fubini's theorem

Fubini's theorem → hasVersion → Fubini's theorem for improper Riemann integrals under additional conditions ⓘ

linked to: Fubini's theorem

Tonelli's theorem → concerns → Fubini–Tonelli type results ⓘ

linked to: Fubini's theorem

Tonelli's theorem → relatesTo → Fubini's theorem ⓘ

Tonelli's theorem → isSpecialCaseOf → Fubini–Tonelli theorem ⓘ

linked to: Fubini's theorem

Tonelli's theorem → comparedWith → Fubini's theorem for integrable (not necessarily non‑negative) functions ⓘ

linked to: Fubini's theorem

Itô isometry → foundationFor → stochastic Fubini theorems ⓘ

linked to: Fubini's theorem

measure theory → usesConcept → Fubini theorem ⓘ

linked to: Fubini's theorem

Young inequality for convolutions → requires → Fubini theorem for integrals ⓘ

linked to: Fubini's theorem

Bochner integral → satisfies → Fubini theorem for Banach-valued functions ⓘ

linked to: Fubini's theorem

All labels observed (12)

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Statements (43)

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Referenced by (13)

How this entity was disambiguated

How these facts were elicited