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.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Lebesgue–Stieltjes integral | 1 |
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 ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Lebesgue–Stieltjes integral