dominated convergence theorem

E284672

convergence theorem theorem in measure theory

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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All labels observed (4)

Label	Occurrences
dominated convergence theorem canonical	5
Lebesgue dominated convergence theorem	2
Lebesgue dominated convergence theorem (with monotone domination)	1
bounded convergence theorem	1

Statements (48)

Predicate	Object
instanceOf	convergence theorem ⓘ theorem in measure theory ⓘ
alsoKnownAs	dominated convergence theorem ⓘ surface form: Lebesgue dominated convergence theorem
appliesTo	Lebesgue integration ⓘ surface form: Lebesgue integral sequence of measurable functions ⓘ
assumes	each function in the sequence is dominated in absolute value by the dominating function almost everywhere ⓘ existence of an integrable dominating function ⓘ pointwise almost everywhere convergence ⓘ
category	result about interchanging limits and integrals ⓘ
comparedTo	Fatou's lemma ⓘ monotone convergence theorem ⓘ
concludes	integral of the limit equals limit of the integrals ⓘ interchange of limit and integral is valid under its hypotheses ⓘ limit function is integrable ⓘ
field	integration theory ⓘ measure theory ⓘ real analysis ⓘ
generalizationOf	bounded convergence theorem on finite measure spaces ⓘ
hasCondition	dominating function bounds the absolute value of each function in the sequence almost everywhere ⓘ dominating function is integrable ⓘ functions in the sequence are measurable ⓘ pointwise almost everywhere convergence of the sequence to the limit function ⓘ
holdsFor	complex-valued integrable functions ⓘ real-valued integrable functions ⓘ
implies	convergence of integrals ⓘ uniform integrability of the sequence under its hypotheses ⓘ
importance	fundamental tool in modern analysis ⓘ
logicalForm	(f_n→f a.e. and \|f_n\|≤g integrable) ⇒ lim_n ∫ f_n dμ = ∫ f dμ ⓘ
namedAfter	Henri Lebesgue ⓘ
relatedTo	Vitali convergence theorem ⓘ uniform convergence and integration ⓘ
requires	integrable dominating function ⓘ measurable functions ⓘ measure space ⓘ
requiresTypeOfConvergence	pointwise almost everywhere convergence rather than uniform convergence ⓘ
strongerThan	Fatou's lemma under additional hypotheses ⓘ
taughtIn	advanced undergraduate analysis courses ⓘ graduate real analysis courses ⓘ
typicalFormulation	If f_n are measurable, f_n→f almost everywhere, and \|f_n\|≤g with g integrable, then f is integrable and ∫f_n→∫f ⓘ
usedFor	establishing continuity of parameter-dependent integrals ⓘ interchanging expectation and limit in probability ⓘ justifying passage of limits under the integral sign ⓘ proving convergence of series of integrable functions ⓘ
usedIn	Fourier analysis ⓘ functional analysis ⓘ partial differential equations ⓘ probability theory ⓘ stochastic processes ⓘ

Referenced by (9)

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

Lebesgue integration → characterizedBy → dominated convergence theorem ⓘ

monotone convergence theorem → usedToProve → dominated convergence theorem ⓘ

this entity surface form: Lebesgue dominated convergence theorem

monotone convergence theorem → isSpecialCaseOf → dominated convergence theorem ⓘ

this entity surface form: Lebesgue dominated convergence theorem (with monotone domination)

monotone convergence theorem → contrastsWith → dominated convergence theorem ⓘ

this entity surface form: bounded convergence theorem

dominated convergence theorem → alsoKnownAs → dominated convergence theorem ⓘ

this entity surface form: Lebesgue dominated convergence theorem

Fatou's lemma → relatedTo → dominated convergence theorem ⓘ

Tonelli's theorem → relatedConcept → dominated convergence theorem ⓘ

measure theory → usesConcept → dominated convergence theorem ⓘ