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.
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
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
It can be shown that is not Riemann integrable on
Let be given.
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,
R.G. Bartle, D.R. Sherbert, Introduction to Real Analysis, John Wiley & sons
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