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.

Definition

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

Let $L\in\mathbb{R}$

We say that $f$ is Generalized Riemann 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 $|L-S(f,\mathcal{\dot{P}})|<\epsilon$

Here, $S(f,\mathcal{\dot{P}})$ is the Riemann sum of $f$ on $[a,b]$ with respect to $\mathcal{\dot{P}}$

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 $f:[0,1]\rightarrow\mathbb{R}$

$f\left( \frac{1}{n}\right) =n\forall n\in\mathbb{N}$

$f(x)=0$ everywhere else.

It can be shown that $f$ is not Riemann integrable on $[0,1]$

Let $\varepsilon>0$ be given.

Consider gauge $\delta:[0,1]\rightarrow\mathbb{R}^+$

$\delta\left( \frac{1}{n}\right) =\frac{\varepsilon}{k2^{k+1}}$

$\delta(x)=1$ everywhere else.

Let $\mathcal{\dot{P}}$ be a $\delta$ -fine partition on $[0,1]$

The Riemann sum will have maximum value only when the tags are of the form $t_i=\frac{1}{n}$ , $n\in \mathbb{N}$ . Also, each tag can be shared by at most two divisions.