Tonelli's theorem
E284676
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.
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]
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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.
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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 for integrable (not necessarily non‑negative) functions
ⓘ
linked to:
Fubini's theorem
|
| concerns |
Fubini–Tonelli type results
ⓘ
linked to:
Fubini's theorem
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 on Euclidean spaces
ⓘ
linked to:
Lebesgue measure
general σ‑finite measure spaces ⓘ |
| implies | monotone convergence of partial integrals in some applications ⓘ |
| influenced |
functional analysis
ⓘ
modern probability theory ⓘ |
| isSpecialCaseOf |
Fubini–Tonelli theorem
ⓘ
linked to:
Fubini's 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
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
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Input
Subject: Tonelli's theorem Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Tonelli's theorem
linked to: Tonelli's theorem
linked to: Tonelli's theorem