Stieltjes measure

E898477

Borel measure Lebesgue–Stieltjes measure measure

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.

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All labels observed (2)

Label	Occurrences
Lebesgue–Stieltjes integral	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 NERFINISHED ⓘ
providesFoundationFor	Lebesgue–Stieltjes integral NERFINISHED ⓘ Riemann–Stieltjes integral NERFINISHED ⓘ
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

Referenced by (2)

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

Riemann–Stieltjes integral → relatedConcept → Stieltjes measure ⓘ

this entity surface form: Lebesgue–Stieltjes integral