Stieltjes measure
E898477
A Stieltjes measure is a measure on the real line constructed from a nondecreasing, right-continuous function, providing the measure-theoretic foundation for the Riemann–Stieltjes and Lebesgue–Stieltjes integrals.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Lebesgue–Stieltjes integral | 2 |
| Lebesgue–Stieltjes measures | 1 |
| Stieltjes measure canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10991499 — 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: Stieltjes measure Context triple: [Riemann–Stieltjes integral, relatedConcept, Stieltjes measure]
-
A.
Lebesgue measure
Lebesgue measure is the standard way of assigning a consistent notion of "length," "area," or "volume" to subsets of Euclidean space, forming the foundation of modern measure theory and integration.
-
B.
Wiener measure
Wiener measure is the canonical probability measure on the space of continuous paths that models standard Brownian motion in probability theory and stochastic analysis.
-
C.
Carathéodory measurability criterion
The Carathéodory measurability criterion is a fundamental condition in measure theory that characterizes measurable sets via an outer measure by requiring additivity over intersections and complements.
-
D.
Riemann–Stieltjes integral
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.
-
E.
Haar measure
Haar measure is a fundamental concept in harmonic analysis and topological group theory, providing a translation-invariant way to assign measures to subsets of locally compact groups.
- 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: Stieltjes measure Target entity description: A Stieltjes measure is a measure on the real line constructed from a nondecreasing, right-continuous function, providing the measure-theoretic foundation for the Riemann–Stieltjes and Lebesgue–Stieltjes integrals.
-
A.
Lebesgue measure
Lebesgue measure is the standard way of assigning a consistent notion of "length," "area," or "volume" to subsets of Euclidean space, forming the foundation of modern measure theory and integration.
-
B.
Wiener measure
Wiener measure is the canonical probability measure on the space of continuous paths that models standard Brownian motion in probability theory and stochastic analysis.
-
C.
Carathéodory measurability criterion
The Carathéodory measurability criterion is a fundamental condition in measure theory that characterizes measurable sets via an outer measure by requiring additivity over intersections and complements.
-
D.
Riemann–Stieltjes integral
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.
-
E.
Haar measure
Haar measure is a fundamental concept in harmonic analysis and topological group theory, providing a translation-invariant way to assign measures to subsets of locally compact groups.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
Borel measure
ⓘ
Lebesgue–Stieltjes measure ⓘ measure ⓘ |
| absolutelyContinuousPartHasDensity | Radon–Nikodym derivative of F with respect to Lebesgue measure ⓘ |
| associatedWith |
cumulative distribution function
ⓘ
distribution function ⓘ |
| canBe |
finite measure
ⓘ
sigma-finite measure ⓘ |
| canBeDecomposedInto |
absolutely continuous part
ⓘ
discrete part ⓘ singular continuous part ⓘ |
| canHaveAtomsCorrespondingTo | jump discontinuities of F ⓘ |
| characterizedBy |
mu((a,b]) = F(b) - F(a)
ⓘ
mu([a,b)) = F(b-) - F(a-) ⓘ |
| compatibleWith | change-of-variables formulas in integration ⓘ |
| constructedFrom |
nondecreasing function
ⓘ
right-continuous function ⓘ |
| correspondsBijectivelyTo | nondecreasing right-continuous functions modulo constants ⓘ |
| definedOn | real line ⓘ |
| determines | Lebesgue–Stieltjes integral with respect to F ⓘ |
| domain | Borel sigma-algebra on the real line ⓘ |
| extends | Riemann–Stieltjes integration to Lebesgue integration framework ⓘ |
| generalizes |
Lebesgue measure on the real line
ⓘ
counting measure on discrete subsets of the real line ⓘ |
| is |
complete if completed with respect to null sets
ⓘ
inner regular on open sets ⓘ outer regular on Borel sets ⓘ sigma-additive ⓘ |
| isContinuousOnIntervalsWhere | F is continuous ⓘ |
| isExampleOf | Radon measure on the real line ⓘ |
| mayHavePropertyOfGeneratingFunction | bounded variation ⓘ |
| namedAfter | Thomas Joannes Stieltjes ⓘ |
| providesFoundationFor |
Lebesgue–Stieltjes integral
ⓘ
linked to:
Stieltjes measure
Riemann–Stieltjes integral ⓘ |
| requiresPropertyOfGeneratingFunction |
nondecreasing
ⓘ
right-continuous ⓘ |
| specialCaseOf | Borel measure induced by a monotone function ⓘ |
| usedIn |
functional analysis
ⓘ
measure theory ⓘ probability theory ⓘ real analysis ⓘ spectral theory ⓘ |
| usedToDefine | distribution of a real-valued random variable ⓘ |
| usedToModel |
cumulative mass distributions on the real line
ⓘ
cumulative probability distributions on the real line ⓘ |
| valuesIn | extended nonnegative reals ⓘ |
| zeroOn | intervals where F is constant ⓘ |
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: Stieltjes measure Description of subject: A Stieltjes measure is a measure on the real line constructed from a nondecreasing, right-continuous function, providing the measure-theoretic foundation for the Riemann–Stieltjes and Lebesgue–Stieltjes integrals.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Stieltjes measure
linked to: Stieltjes measure
linked to: Stieltjes measure