# Henstock-Kurzweil integral

The **Henstock-Kurzweil integral** (also known as the **Generalized Riemann integral**) is one of the most widely applicable generalizations of the Riemann integral, but it also uses a strikingly simple and elegant idea. It was developed independently by Ralph Henstock and Jaroslav Kurzweil.

## Contents

## Definition

Let

Let

We say that is *Generalized Riemann Integrable* on if and only if, , there exists a gauge such that,

if is a -fine tagged partition on , then

Here, is the Riemann sum of on with respect to

The elegance of this integral lies in in the ability of a gauge to 'measure' a partition more accurately than its norm

## Illustration

The utility of the Henstock-Kurzweil integral is demonstrated by this function, which is not Riemann integrable but is Generalized Riemann Integrable.

Consider the function $f:[0,1]\rightarrow\mathh{R}$ (Error compiling LaTeX. ! Undefined control sequence.)

everywhere else.

It can be shown that is not Riemann integrable on

Let be given.

Consider gauge

everywhere else.

Let be a -fine partition on

The Riemann sum will have maximum value only when the tags are of the form , . Also, each tag can be shared by at most two divisions.

But as is arbitrary, we have that is Generalized Riemann integrable or,

## References

R.G. Bartle, D.R. Sherbert, *Introduction to Real Analysis*, John Wiley & sons

## See Also

*This article is a stub. Help us out by expanding it.*