Riemann-Stieltjes integral

Let f and α be boundedPlanetmathPlanetmathPlanetmath, real-valued functions defined upon a closed finite interval I=[a,b] of (ab), P={x0,,xn} a partitionMathworldPlanetmathPlanetmath of I, and ti a point of the subinterval [xi-1,xi]. A sum of the form


is called a Riemann-Stieltjes sum of f with respect to α. f is said to be Riemann Stieltjes integrable with respect to α on I if there exists A such that given any ϵ>0 there exists a partition Pϵ of I for which, for all P finer than Pϵ and for every choice of points ti, we have


If such an A exists, then it is unique and is known as the Riemann-Stieltjes integral of f with respect to α. f is known as the integrand and α the integrator. The integral is denoted by

Title Riemann-Stieltjes integral
Canonical name RiemannStieltjesIntegral
Date of creation 2013-03-22 12:51:13
Last modified on 2013-03-22 12:51:13
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 11
Author Mathprof (13753)
Entry type Definition
Classification msc 26A42
Related topic RiemannSum
Related topic IntegralSign
Defines Riemann-Stieltjes sum
Defines integrator