generalized Riemann integral

A gauge δ is a function which assigns to every real number x an interval δ(x) such that xδ(x).

Given a gauge δ, a partitionPlanetmathPlanetmath Uii=1n of an interval [a,b] is said to be δ-fine if, for every point x[a,b], the set Ui containing x is a subset of δ(x)

A function f:[a,b] is said to be generalized Riemann integrable on [a,b] if there exists a number L such that for every ϵ>0 there exists a gauge δϵ on [a,b] such that if 𝒫˙ is any δϵ-fine partition of [a,b], then


where S(f;𝒫˙) is any Riemann sumMathworldPlanetmath for f using the partition 𝒫˙. The collectionMathworldPlanetmath of all generalized Riemann integrable functions is usually denoted by *[a,b].

If f*[a,b] then the number L is uniquely determined, and is called the generalized Riemann integral of f over [a,b].

The reason that this is called a generalized Riemann integral is that, in the special case where δ(x)=[x-y,x+y] for some number y, we recover the Riemann integral as a special case.

Figure 1: Riemann sum over a δ-fine partition
Title generalized Riemann integral
Canonical name GeneralizedRiemannIntegral
Date of creation 2013-03-22 13:40:03
Last modified on 2013-03-22 13:40:03
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 12
Author rspuzio (6075)
Entry type Definition
Classification msc 26A42
Synonym Kurzweil-Henstock integral
Synonym gauge integral
Defines generalized Riemann integrable
Defines gauge