Vitali convergence theorem

E898498

convergence theorem theorem in measure theory

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.

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All labels observed (1)

Label	Occurrences
Vitali convergence theorem canonical	2

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 ⓘ

Referenced by (2)

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

Montel theorem → relatedTo → Vitali convergence theorem ⓘ

subject surface form: Montel's theorem

dominated convergence theorem → relatedTo → Vitali convergence theorem ⓘ