monotone convergence theorem

E284671

result in measure theory 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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All labels observed (1)

Label	Occurrences
monotone convergence theorem canonical	6

How this entity was disambiguated

This entity first appeared as the object of triple T2631196 — 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: monotone convergence theorem
Context triple: [Lebesgue integration, characterizedBy, monotone convergence theorem]

A. 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.
B. Cauchy sequence
A Cauchy sequence is a sequence whose terms become arbitrarily close to each other as the sequence progresses, providing a fundamental criterion for convergence in metric and normed spaces.
C. 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.
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. Banach fixed-point theorem
The Banach fixed-point theorem is a fundamental result in metric space theory that guarantees the existence and uniqueness of a fixed point for any contraction mapping and provides a method for finding it via iterative approximation.
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: monotone convergence theorem
Target entity description: 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.

A. 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.
B. Cauchy sequence
A Cauchy sequence is a sequence whose terms become arbitrarily close to each other as the sequence progresses, providing a fundamental criterion for convergence in metric and normed spaces.
C. 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.
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. Banach fixed-point theorem
The Banach fixed-point theorem is a fundamental result in metric space theory that guarantees the existence and uniqueness of a fixed point for any contraction mapping and provides a method for finding it via iterative approximation.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	result in measure theory ⓘ theorem ⓘ
allows	passing limit inside integral for monotone sequences ⓘ
alsoKnownAs	Beppo Levi theorem ⓘ linked to: Fatou's lemma
appliesTo	extended real-valued functions ⓘ nonnegative measurable functions ⓘ pointwise increasing sequence of functions ⓘ
assumes	measurable space ⓘ measure ⓘ monotone increasing sequence ⓘ nonnegative functions ⓘ sequence of measurable functions ⓘ σ-finite measure (optional but common) ⓘ
category	convergence theorem in integration theory ⓘ
conclusion	integral of pointwise limit equals limit of integrals ⓘ limit of integrals is finite or +∞ consistently with limit function ⓘ ∫ lim f_n dμ = lim ∫ f_n dμ for nonnegative increasing f_n ⓘ
contrastsWith	bounded convergence theorem ⓘ linked to: dominated convergence theorem dominated convergence theorem ⓘ
field	measure theory ⓘ probability theory ⓘ real analysis ⓘ
hasVersion	version for expectations of random variables ⓘ version for sums with counting measure ⓘ
holdsFor	complete measures ⓘ σ-algebras ⓘ
implies	Fatou lemma (in some formulations) ⓘ
importance	fundamental theorem of Lebesgue integration ⓘ key tool in modern analysis ⓘ
isSpecialCaseOf	Lebesgue dominated convergence theorem (with monotone domination) ⓘ linked to: dominated convergence theorem
isStrongerThan	Fatou lemma in the monotone case ⓘ linked to: Fatou's lemma
namedAfter	Beppo Levi ⓘ
relatesConcept	Lebesgue integral ⓘ linked to: Lebesgue integration increasing sequence of functions ⓘ measurable function ⓘ nonnegative function ⓘ pointwise convergence ⓘ
requires	countable monotonicity of measure ⓘ
typicalStatement	If 0 ≤ f_1 ≤ f_2 ≤ … and f_n → f pointwise, then ∫ f_n dμ → ∫ f dμ ⓘ
usedIn	construction of Lebesgue integral ⓘ expectation of random variables ⓘ interchanging limit and integral for nonnegative increasing sequences ⓘ probability theory ⓘ linked to: Probability Theory stochastic processes ⓘ
usedToProve	Fubini theorem (components of proofs) ⓘ linked to: Fubini's theorem Lebesgue dominated convergence theorem ⓘ linked to: dominated convergence theorem Tonelli theorem ⓘ linked to: Tonelli's theorem

How these facts were elicited

Referenced by (6)

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

Lebesgue integration → characterizedBy → monotone convergence theorem ⓘ

dominated convergence theorem → comparedTo → monotone convergence theorem ⓘ

Fatou's lemma → relatedTo → monotone convergence theorem ⓘ

Tonelli's theorem → relatedConcept → monotone convergence theorem ⓘ

Dini's theorem → relatedTo → monotone convergence theorem ⓘ

Vitali convergence theorem → comparedTo → monotone convergence theorem ⓘ

All labels observed (1)

How this entity was disambiguated Show

Statements (47)

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Referenced by (6)

How this entity was disambiguated

How these facts were elicited