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 (5)

Label	Occurrences
dominated convergence theorem canonical	8
Lebesgue dominated convergence theorem	3
Lebesgue dominated convergence theorem (with monotone domination)	1
bounded convergence theorem	1
dominated convergence theorem (Bochner version)	1

How this entity was disambiguated

This entity first appeared as the object of triple T2631197 — 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: dominated convergence theorem
Context triple: [Lebesgue integration, characterizedBy, dominated convergence theorem]

A. Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
B. 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.
C. 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.
D. Cauchy condensation test
The Cauchy condensation test is a convergence criterion in mathematical analysis that determines whether an infinite series with positive, nonincreasing terms converges by comparing it to a related series formed by powers of two.
E. Cauchy convergence criterion
The Cauchy convergence criterion is a fundamental concept in mathematical analysis that characterizes convergence of sequences (and series) by requiring that their terms become arbitrarily close to each other beyond some index.
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: dominated convergence theorem
Target entity description: 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.

A. Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
B. 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.
C. 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.
D. Cauchy condensation test
The Cauchy condensation test is a convergence criterion in mathematical analysis that determines whether an infinite series with positive, nonincreasing terms converges by comparing it to a related series formed by powers of two.
E. Cauchy convergence criterion
The Cauchy convergence criterion is a fundamental concept in mathematical analysis that characterizes convergence of sequences (and series) by requiring that their terms become arbitrarily close to each other beyond some index.
F. None of above. chosen

Statements (48)

Predicate	Object
instanceOf	convergence theorem ⓘ theorem in measure theory ⓘ
alsoKnownAs	Lebesgue dominated convergence theorem ⓘ linked to: dominated convergence theorem
appliesTo	Lebesgue integral ⓘ linked to: Lebesgue integration 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 ⓘ

How these facts were elicited