Tonelli's theorem

E284676

theorem in measure theory

Tonelli's theorem is a fundamental result in measure theory that justifies interchanging the order of integration for non-negative measurable functions in iterated Lebesgue integrals.

Try in SPARQL Jump to: Surface forms Disambiguation Statements Elicitation Referenced by

All labels observed (3)

Label	Occurrences
Tonelli theorem	2
Tonelli's theorem canonical	2
Tonelli's theorem for nonnegative functions	1

How this entity was disambiguated

This entity first appeared as the object of triple T2631220 — 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: Tonelli's theorem
Context triple: [Lebesgue integration, relatedTo, Tonelli'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. Girsanov theorem
Girsanov theorem is a fundamental result in stochastic calculus that describes how the dynamics of stochastic processes, particularly Brownian motion, change under an equivalent change of probability measure.
C. de Bruijn–van Aardenne–Ehrenfest theorem
The de Bruijn–van Aardenne–Ehrenfest theorem is a fundamental result in combinatorics that characterizes the number of Eulerian circuits in directed graphs, particularly de Bruijn graphs, and underpins constructions in coding theory and discrete mathematics.
D. Kronecker’s lemma
Kronecker’s lemma is a result in real analysis and summability theory that links the convergence of series with weighted averages of their partial sums, often used in the study of Fourier series and ergodic theorems.
E. 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.
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: Tonelli's theorem
Target entity description: Tonelli's theorem is a fundamental result in measure theory that justifies interchanging the order of integration for non-negative measurable functions in iterated Lebesgue integrals.

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. Girsanov theorem
Girsanov theorem is a fundamental result in stochastic calculus that describes how the dynamics of stochastic processes, particularly Brownian motion, change under an equivalent change of probability measure.
C. de Bruijn–van Aardenne–Ehrenfest theorem
The de Bruijn–van Aardenne–Ehrenfest theorem is a fundamental result in combinatorics that characterizes the number of Eulerian circuits in directed graphs, particularly de Bruijn graphs, and underpins constructions in coding theory and discrete mathematics.
D. Kronecker’s lemma
Kronecker’s lemma is a result in real analysis and summability theory that links the convergence of series with weighted averages of their partial sums, often used in the study of Fourier series and ergodic theorems.
E. 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.
F. None of above. chosen

Statements (45)

Predicate	Object
instanceOf	theorem in measure theory ⓘ
appliesTo	Lebesgue integrable functions ⓘ non‑negative measurable functions ⓘ
assumes	measurability of the function on the product space ⓘ σ‑finite measure spaces ⓘ
category	theorem about Lebesgue integration ⓘ theorem about product measures ⓘ
clarifies	conditions under which iterated integrals are well defined ⓘ
comparedWith	Fubini's theorem ⓘ surface form: Fubini's theorem for integrable (not necessarily non‑negative) functions
concerns	Fubini's theorem ⓘ surface form: Fubini–Tonelli type results iterated Lebesgue integrals ⓘ product measures ⓘ
conclusion	x↦∫_Y f(x,y) dν(y) is μ‑measurable ⓘ y↦∫_X f(x,y) dμ(x) is ν‑measurable ⓘ
field	integration theory ⓘ measure theory ⓘ real analysis ⓘ
guarantees	equality of iterated integrals for non‑negative measurable functions ⓘ possibility of interchanging the order of integration for non‑negative measurable functions ⓘ that iterated integrals equal the integral over the product space for non‑negative measurable functions ⓘ
historicalPeriod	20th‑century mathematics ⓘ
holdsFor	Lebesgue measure ⓘ surface form: Lebesgue measure on Euclidean spaces general σ‑finite measure spaces ⓘ
implies	monotone convergence of partial integrals in some applications ⓘ
influenced	functional analysis ⓘ modern probability theory ⓘ
isSpecialCaseOf	Fubini's theorem ⓘ surface form: Fubini–Tonelli theorem
namedAfter	Leonida Tonelli ⓘ
oftenPresentedWith	Fubini's theorem in analysis textbooks ⓘ
relatedConcept	absolute convergence of integrals ⓘ dominated convergence theorem ⓘ monotone convergence theorem ⓘ product σ‑algebra ⓘ
relatesTo	Fubini's theorem ⓘ
requires	measurability with respect to the product σ‑algebra ⓘ non‑negativity of the integrand ⓘ
statedIn	terms of integrals over X×Y and iterated integrals over X and Y ⓘ
typicalAssumption	(X,Σ,μ) and (Y,T,ν) are σ‑finite measure spaces ⓘ f:X×Y→[0,∞] is Σ⊗T‑measurable ⓘ
typicalFormulation	∫_{X×Y} f d(μ×ν) = ∫_X (∫_Y f dν) dμ = ∫_Y (∫_X f dμ) dν for non‑negative measurable f ⓘ
usedFor	analysis of multiple integrals ⓘ establishing convergence of integrals ⓘ justifying change of order of integration ⓘ probability theory on product spaces ⓘ stochastic processes ⓘ

How these facts were elicited

Referenced by (5)

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

Lebesgue integration → relatedTo → Tonelli's theorem ⓘ

monotone convergence theorem → usedToProve → Tonelli's theorem ⓘ

this entity surface form: Tonelli theorem

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

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

this entity surface form: Tonelli's theorem for nonnegative functions

measure theory → usesConcept → Tonelli's theorem ⓘ