Riemann–Stieltjes integral
E259762
The Riemann–Stieltjes integral is a generalization of the Riemann integral in which integration is taken with respect to a function of bounded variation rather than just the identity function, allowing more flexible treatment of sums and distributions.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Riemann–Stieltjes integral canonical | 1 |
| Riemann–Stieltjes integration | 1 |
| Stieltjes integrals | 1 |
| Young integral | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2364435 — 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: Riemann–Stieltjes integral Context triple: [Riemann integral, hasVariant, Riemann–Stieltjes integral]
-
A.
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.
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B.
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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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.
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D.
Riemann sums
Riemann sums are a fundamental method in calculus for approximating the area under a curve by summing the areas of a sequence of rectangles, forming the basis of the definition of the definite integral.
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E.
Selberg integral
The Selberg integral is a fundamental multidimensional generalization of Euler’s beta integral that plays a central role in random matrix theory, combinatorics, and special functions.
- 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: Riemann–Stieltjes integral Target entity description: The Riemann–Stieltjes integral is a generalization of the Riemann integral in which integration is taken with respect to a function of bounded variation rather than just the identity function, allowing more flexible treatment of sums and distributions.
-
A.
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.
-
B.
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.
-
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.
Riemann sums
Riemann sums are a fundamental method in calculus for approximating the area under a curve by summing the areas of a sequence of rectangles, forming the basis of the definition of the definite integral.
-
E.
Selberg integral
The Selberg integral is a fundamental multidimensional generalization of Euler’s beta integral that plays a central role in random matrix theory, combinatorics, and special functions.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
generalization of Riemann integral
ⓘ
integral ⓘ mathematical concept ⓘ |
| allowsIntegrationWithRespectTo |
cumulative distribution functions
ⓘ
monotone functions ⓘ step functions ⓘ |
| alsoKnownAs |
Riemann–Stieltjes integral
ⓘ
surface form:
Riemann–Stieltjes integration
|
| application |
moment calculations via distribution functions
ⓘ
probability theory ⓘ spectral theory ⓘ stochastic processes (in simple settings) ⓘ |
| canBeDefinedFor | complex measures via Stieltjes measures ⓘ |
| captures | sums weighted by jumps of the integrator ⓘ |
| comparison |
less general than Lebesgue–Stieltjes integral
ⓘ
more flexible than Riemann integral ⓘ |
| definitionMethod | limit of Riemann–Stieltjes sums ⓘ |
| domain | closed interval [a,b] ⓘ |
| extends | Riemann integration with respect to measures induced by distribution functions ⓘ |
| field |
measure theory
ⓘ
real analysis ⓘ |
| generalizes | Riemann integral ⓘ |
| historicalDevelopment | introduced in late 19th century ⓘ |
| integrandType |
complex-valued function
ⓘ
real-valued function ⓘ |
| integratorType | function of bounded variation ⓘ |
| linearityIn |
integrand
ⓘ
integrator when combined appropriately ⓘ |
| motivation |
to generalize sums of the form Σ f(x_i)(α(x_i)−α(x_{i−1}))
ⓘ
to integrate with respect to distribution functions ⓘ |
| namedAfter |
Bernhard Riemann
ⓘ
Thomas Joannes Stieltjes ⓘ |
| property |
depends on values of integrand at points where integrator has variation
ⓘ
sensitive to discontinuities of integrator ⓘ |
| reducesTo |
Riemann integral when integrator is identity function
ⓘ
Riemann integral when integrator is x ↦ x ⓘ |
| relatedConcept |
Stieltjes measure
ⓘ
surface form:
Lebesgue–Stieltjes integral
Stieltjes measure ⓘ Riemann–Stieltjes integral self-linksurface differs ⓘ
surface form:
Young integral
|
| requires | integrator of bounded variation on [a,b] ⓘ |
| satisfies | integration by parts formula ⓘ |
| sufficientConditionForExistence |
continuous integrand and integrator of bounded variation
ⓘ
integrand with only jump discontinuities and continuous integrator ⓘ |
| textbookTreatment |
commonly appears in advanced undergraduate analysis courses
ⓘ
commonly appears in introductory graduate real analysis courses ⓘ |
| uses |
Riemann–Stieltjes sums
ⓘ
tagged partitions of an interval ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Riemann–Stieltjes integral Description of subject: The Riemann–Stieltjes integral is a generalization of the Riemann integral in which integration is taken with respect to a function of bounded variation rather than just the identity function, allowing more flexible treatment of sums and distributions.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Stieltjes integrals
this entity surface form:
Riemann–Stieltjes integration
this entity surface form:
Young integral