Henstock-Kurzweil integral

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

Definition

Let $f:[a,b]\rightarrow\mathbb{R}$

Let $L\in\mathbb{R}$

We say that $f$ is Generalised Reimann Integrable on $[a,b]$ if and only if, $\forall\epsilon>0$ , there exists a gauge $\delta:[a,b]\rightarrow\mathbb{R}^+$ such that,

if $\mathcal{\dot{P}}$ is a $\delta$ -fine tagged partition on $[a,b]$ , then $|L-S(f,\mathcal{\dot{P}})|<\epsilon$

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

Page

Toolbox

Search

Henstock-Kurzweil integral

Definition

See Also