Riemann–Stieltjes integral

E259762

generalization of Riemann integral integral mathematical concept

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.

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All labels observed (4)

Label	Occurrences
Riemann–Stieltjes integral canonical	1
Riemann–Stieltjes integration	1
Stieltjes integrals	1
Young integral	1

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 ⓘ

Referenced by (4)

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

Riemann integral → hasVariant → Riemann–Stieltjes integral ⓘ

Divergent Series → topic → Riemann–Stieltjes integral ⓘ

this entity surface form: Stieltjes integrals

Riemann–Stieltjes integral → alsoKnownAs → Riemann–Stieltjes integral ⓘ

this entity surface form: Riemann–Stieltjes integration

Riemann–Stieltjes integral → relatedConcept → Riemann–Stieltjes integral self-linksurface differs ⓘ

this entity surface form: Young integral