monotone convergence theorem
E284671
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| monotone convergence theorem canonical | 4 |
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.
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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.
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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.
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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.
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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 |
Fatou's lemma
ⓘ
surface form:
Beppo Levi theorem
|
| 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 |
dominated convergence theorem
ⓘ
surface form:
bounded 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 |
dominated convergence theorem
ⓘ
surface form:
Lebesgue dominated convergence theorem (with monotone domination)
|
| isStrongerThan |
Fatou's lemma
ⓘ
surface form:
Fatou lemma in the monotone case
|
| namedAfter | Beppo Levi ⓘ |
| relatesConcept |
Lebesgue integration
ⓘ
surface form:
Lebesgue integral
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 ⓘ
surface form:
probability theory
stochastic processes ⓘ |
| usedToProve |
Fubini's theorem
ⓘ
surface form:
Fubini theorem (components of proofs)
dominated convergence theorem ⓘ
surface form:
Lebesgue dominated convergence theorem
Tonelli's theorem ⓘ
surface form:
Tonelli theorem
|
How these facts were elicited
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Instruction
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Input
Subject: monotone convergence theorem Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.