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 (11)

Label	Occurrences
Fubini's theorem canonical	2
Fubini theorem	1
Fubini theorem (components of proofs)	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

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 ⓘ surface form: Tonelli's theorem for nonnegative functions
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 self-linksurface differs ⓘ surface form: Fubini's theorem for Bochner integrals Fubini's theorem self-linksurface differs ⓘ surface form: Fubini's theorem for Lebesgue integrals Fubini's theorem self-linksurface differs ⓘ surface form: Fubini's theorem for improper Riemann integrals under additional conditions
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 integration ⓘ surface form: Lebesgue integral 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 ⓘ

Referenced by (12)

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's theorem ⓘ

this entity surface form: Fubini theorem (components of proofs)

Fubini's theorem → hasVersion → Fubini's theorem self-linksurface differs ⓘ

this entity surface form: Fubini's theorem for Lebesgue integrals

Fubini's theorem → hasVersion → Fubini's theorem self-linksurface differs ⓘ

this entity surface form: Fubini's theorem for Bochner integrals

Fubini's theorem → hasVersion → Fubini's theorem self-linksurface differs ⓘ

this entity surface form: Fubini's theorem for improper Riemann integrals under additional conditions

Tonelli's theorem → concerns → Fubini's theorem ⓘ

this entity surface form: Fubini–Tonelli type results

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

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

this entity surface form: Fubini–Tonelli theorem

Tonelli's theorem → comparedWith → Fubini's theorem ⓘ

this entity surface form: Fubini's theorem for integrable (not necessarily non‑negative) functions

Itô isometry → foundationFor → Fubini's theorem ⓘ

this entity surface form: stochastic Fubini theorems

measure theory → usesConcept → Fubini's theorem ⓘ

this entity surface form: Fubini theorem

Young inequality for convolutions → requires → Fubini's theorem ⓘ

this entity surface form: Fubini theorem for integrals