Henstock–Kurzweil integral
E259763
The Henstock–Kurzweil integral is a highly general integration theory that extends and refines the Riemann integral, capable of integrating a broader class of functions while retaining many of the intuitive properties of Riemann integration.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Denjoy integral | 1 |
| Henstock–Kurzweil integral canonical | 1 |
| generalized Riemann–Henstock integral | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2364437 — 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: Henstock–Kurzweil integral Context triple: [Riemann integral, contrastedWith, Henstock–Kurzweil integral]
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A.
Riemann–Liouville integral
The Riemann–Liouville integral is a fundamental operator in fractional calculus that generalizes the concept of an n-fold repeated integral to non-integer (fractional) orders.
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B.
Riemann integral
The Riemann integral is a fundamental concept in calculus that defines the integral of a function as the limit of sums of function values over increasingly fine partitions of an interval.
-
C.
Lebesgue integration
Lebesgue integration is a foundational measure-theoretic framework for defining and analyzing integrals, particularly suited to handling limits, convergence, and more general functions than those allowed by Riemann integration.
-
D.
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.
-
E.
Hardy–Littlewood maximal function
The Hardy–Littlewood maximal function is a fundamental operator in real analysis and harmonic analysis that controls the local averages of a function and plays a key role in differentiation theorems and singular integral theory.
- 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: Henstock–Kurzweil integral Target entity description: The Henstock–Kurzweil integral is a highly general integration theory that extends and refines the Riemann integral, capable of integrating a broader class of functions while retaining many of the intuitive properties of Riemann integration.
-
A.
Riemann–Liouville integral
The Riemann–Liouville integral is a fundamental operator in fractional calculus that generalizes the concept of an n-fold repeated integral to non-integer (fractional) orders.
-
B.
Riemann integral
The Riemann integral is a fundamental concept in calculus that defines the integral of a function as the limit of sums of function values over increasingly fine partitions of an interval.
-
C.
Lebesgue integration
Lebesgue integration is a foundational measure-theoretic framework for defining and analyzing integrals, particularly suited to handling limits, convergence, and more general functions than those allowed by Riemann integration.
-
D.
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.
-
E.
Hardy–Littlewood maximal function
The Hardy–Littlewood maximal function is a fundamental operator in real analysis and harmonic analysis that controls the local averages of a function and plays a key role in differentiation theorems and singular integral theory.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
generalized Riemann integral
ⓘ
integration theory ⓘ mathematical integral ⓘ |
| admitsExtensionTo | improper integrals over unbounded intervals ⓘ |
| allows | fine partitions adapted to local behavior of the integrand ⓘ |
| alsoKnownAs |
gauge integral
ⓘ
Henstock–Kurzweil integral ⓘ
surface form:
generalized Riemann–Henstock integral
|
| canIntegrate |
all Lebesgue integrable functions
ⓘ
all improper Riemann integrable functions ⓘ some functions not Lebesgue integrable ⓘ |
| characterizedBy | gauge condition on partitions instead of uniform mesh size ⓘ |
| compatibleWith | classical techniques of Riemann integration ⓘ |
| contrastsWith | measure-theoretic approach of Lebesgue integration ⓘ |
| definitionBasedOn |
gauges
ⓘ
tagged partitions ⓘ |
| domain | real-valued functions on intervals of the real line ⓘ |
| extends |
Lebesgue integration
ⓘ
surface form:
Lebesgue integral
Riemann integral ⓘ |
| field | real analysis ⓘ |
| generalizes |
Lebesgue integral on measurable functions
ⓘ
Riemann integral on bounded intervals ⓘ |
| hasProperty |
every Lebesgue integrable function has the same Henstock–Kurzweil and Lebesgue integrals
ⓘ
every derivative is Henstock–Kurzweil integrable ⓘ integral is unique when it exists ⓘ integral of a derivative recovers the original function up to a constant under mild conditions ⓘ |
| hasTextbookTreatmentIn |
Jaroslav Kurzweil’s works on generalized integration
ⓘ
Ralph Henstock’s books on non-absolute integration ⓘ |
| implies | Denjoy integral on many functions ⓘ |
| includesAsSpecialCase | Riemann–Stieltjes integral for suitable integrators ⓘ |
| is | nonabsolute integral ⓘ |
| isDefinedOn | bounded intervals of the real line ⓘ |
| isEquivalentTo | Denjoy integral for functions with certain regularity conditions ⓘ |
| isNot | countably additive measure integral ⓘ |
| isStrongerThan | Lebesgue integral in terms of integrable functions ⓘ |
| isWeakerThan | Denjoy integral in generality ⓘ |
| namedAfter |
Jaroslav Kurzweil
ⓘ
Ralph Henstock ⓘ |
| preservesProperty | fundamental theorem of calculus for all differentiable functions with regulated derivative ⓘ |
| relatedConcept |
fine partition
ⓘ
gauge ⓘ nonabsolute integration ⓘ |
| retains | intuitive partition-sum interpretation of the integral ⓘ |
| satisfies |
additivity over intervals
ⓘ
linearity ⓘ translation invariance on the real line ⓘ |
| subsumes |
Lebesgue integration
ⓘ
surface form:
Lebesgue integral on the real line
|
| usedIn |
probability theory and stochastic integration generalizations
ⓘ
theory of differential equations ⓘ |
| yearIntroduced | 1957 ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Henstock–Kurzweil integral Description of subject: The Henstock–Kurzweil integral is a highly general integration theory that extends and refines the Riemann integral, capable of integrating a broader class of functions while retaining many of the intuitive properties of Riemann integration.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Denjoy integral
this entity surface form:
generalized Riemann–Henstock integral