facts about Riemann–Stieltjes integral

  • If the integrator g of the http://planetmath.org/node/3187Riemann–Stieltjes integralabf(x)𝑑g(x) is the identity function, then the integral reduces to the Riemann integral abf(x)𝑑x.

  • If the integrand of the Riemann–Stieltjes integral is a constant function, one has

  • If the integrand f is continuousMathworldPlanetmathPlanetmath and the integrator g monotonically nondecreasing on the interval[a,b],  then there exists a number ξ on the interval such that


    Cf. the integral mean value theorem.

  • If f is continuous, g monotonically nondecreasing and differentiableMathworldPlanetmathPlanetmath on the interval  [a,b],  then

    ddxaxf(t)𝑑g(t)=f(x)g(x)for a<x<b.
Title facts about Riemann–Stieltjes integral
Canonical name FactsAboutRiemannStieltjesIntegral
Date of creation 2013-03-22 18:55:03
Last modified on 2013-03-22 18:55:03
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 5
Author pahio (2872)
Entry type Topic
Classification msc 26A42
Synonym properties of Riemann–Stieltjes integral
Related topic PropertiesOfRiemannStieltjesIntegral