Vitali convergence theorem
E898498
The Vitali convergence theorem is a result in measure theory that gives conditions under which pointwise convergence of a sequence of integrable functions implies convergence of their integrals, strengthening the dominated convergence theorem via uniform integrability.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Vitali convergence theorem canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T10991924 — 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: Vitali convergence theorem Context triple: [Montel's theorem, relatedTo, Vitali convergence theorem]
-
A.
dominated convergence theorem
The dominated convergence theorem is a fundamental result in measure theory that provides conditions under which one can interchange limits and integrals for sequences of functions bounded by an integrable dominating function.
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B.
Lebesgue differentiation theorem
The Lebesgue differentiation theorem is a fundamental result in real analysis stating that, for an integrable function, the averages over shrinking neighborhoods converge almost everywhere to the function’s pointwise value.
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C.
monotone convergence theorem
The monotone convergence theorem is a fundamental result in measure theory stating that the integral of a pointwise increasing sequence of nonnegative measurable functions equals the limit of their integrals.
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D.
Fatou's lemma
Fatou's lemma is a fundamental result in measure theory that provides an inequality relating the integral of the pointwise limit inferior of a sequence of nonnegative measurable functions to the limit inferior of their integrals.
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E.
Vitali covering lemma
The Vitali covering lemma is a fundamental result in measure theory that provides conditions under which a collection of sets can be reduced to a disjoint subcollection that still covers almost all of the original set, and it underpins many key theorems in real analysis and differentiation.
- 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: Vitali convergence theorem Target entity description: The Vitali convergence theorem is a result in measure theory that gives conditions under which pointwise convergence of a sequence of integrable functions implies convergence of their integrals, strengthening the dominated convergence theorem via uniform integrability.
-
A.
dominated convergence theorem
The dominated convergence theorem is a fundamental result in measure theory that provides conditions under which one can interchange limits and integrals for sequences of functions bounded by an integrable dominating function.
-
B.
Lebesgue differentiation theorem
The Lebesgue differentiation theorem is a fundamental result in real analysis stating that, for an integrable function, the averages over shrinking neighborhoods converge almost everywhere to the function’s pointwise value.
-
C.
monotone convergence theorem
The monotone convergence theorem is a fundamental result in measure theory stating that the integral of a pointwise increasing sequence of nonnegative measurable functions equals the limit of their integrals.
-
D.
Fatou's lemma
Fatou's lemma is a fundamental result in measure theory that provides an inequality relating the integral of the pointwise limit inferior of a sequence of nonnegative measurable functions to the limit inferior of their integrals.
-
E.
Vitali covering lemma
The Vitali covering lemma is a fundamental result in measure theory that provides conditions under which a collection of sets can be reduced to a disjoint subcollection that still covers almost all of the original set, and it underpins many key theorems in real analysis and differentiation.
- F. None of above. chosen
Statements (42)
| Predicate | Object |
|---|---|
| instanceOf |
convergence theorem
ⓘ
theorem in measure theory ⓘ |
| appliesTo |
Lebesgue integrable functions
ⓘ
sequences of integrable functions ⓘ |
| assumption |
integrability of each function in the sequence
ⓘ
pointwise almost everywhere convergence ⓘ uniform integrability of the sequence ⓘ |
| characterizes | uniform integrability via convergence of integrals ⓘ |
| comparedTo |
dominated convergence theorem
NERFINISHED
ⓘ
monotone convergence theorem NERFINISHED ⓘ |
| conclusion |
L1 convergence under suitable hypotheses
ⓘ
convergence of integrals ⓘ |
| field |
integration theory
ⓘ
measure theory ⓘ real analysis ⓘ |
| generalizationOf | dominated convergence theorem NERFINISHED ⓘ |
| historicalPeriod | early 20th century mathematics ⓘ |
| implies |
integrals of the sequence converge to the integral of the limit
ⓘ
limit function is integrable ⓘ |
| languageOfOriginalPublication | Italian ⓘ |
| namedAfter | Giuseppe Vitali NERFINISHED ⓘ |
| relatedTo |
Dunford–Pettis theorem
NERFINISHED
ⓘ
Vitali covering theorem NERFINISHED ⓘ Vitali–Hahn–Saks theorem NERFINISHED ⓘ |
| strengthens | dominated convergence theorem NERFINISHED ⓘ |
| topic |
L1 convergence
ⓘ
Lebesgue integration ⓘ almost everywhere convergence ⓘ convergence of integrals ⓘ pointwise convergence ⓘ uniform integrability ⓘ |
| typicalFormulation |
for sequences bounded in L1 and uniformly integrable
ⓘ
on finite measure spaces ⓘ |
| usedIn |
ergodic theory
ⓘ
functional analysis ⓘ martingale theory NERFINISHED ⓘ probability theory ⓘ theory of Banach function spaces ⓘ |
| usesConcept |
L1-boundedness
ⓘ
absolute continuity of integrals ⓘ tightness of measures ⓘ uniform integrability ⓘ |
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
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Vitali convergence theorem Description of subject: The Vitali convergence theorem is a result in measure theory that gives conditions under which pointwise convergence of a sequence of integrable functions implies convergence of their integrals, strengthening the dominated convergence theorem via uniform integrability.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Montel's theorem