generalized Riemann integral

A gauge $\delta$ is a function which assigns to every real number $x$ an interval $\delta(x)$ such that $x\in\delta(x)$ .

Given a gauge $\delta$ , a partition ${U_{i}}_{i=1}^{n}$ of an interval $[a,b]$ is said to be $\delta$ -fine if, for every point $x\in[a,b]$ , the set $U_{i}$ containing $x$ is a subset of $\delta(x)$

A function $f:[a,b]\rightarrow\mathbb{R}$ is said to be generalized Riemann integrable on $[a,b]$ if there exists a number $L\in\mathbb{R}$ such that for every $\epsilon>0$ there exists a gauge $\delta_{\epsilon}$ on $[a,b]$ such that if $\dot{\mathcal{P}}$ is any $\delta_{\epsilon}$ -fine partition of $[a,b]$ , then

|S(f;\dot{\mathcal{P}})-L|<\epsilon,

where $S(f;\dot{\mathcal{P}})$ is any Riemann sum for $f$ using the partition $\dot{\mathcal{P}}$ . The collection of all generalized Riemann integrable functions is usually denoted by $\mathcal{R}^{*}[a,b]$ .

If $f\in\mathcal{R}^{*}[a,b]$ 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 $\delta(x)=[x-y,x+y]$ for some number $y$ , we recover the Riemann integral as a special case.

Figure 1: Riemann sum over a $\delta$ -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