An analysis of the Henstock-Kurzweil integral
Let I = [a; b] R be a compact interval. Let f : I ! R be a real-valued function. We investigate the so-called Henstock-Kurzweil or HK integral and compare it to the integrals of Riemann and Lebesgue. In the classic version of the Fundamental Theorem of Calculus, given a compact interval I = [a; b] R and functions f : I ! R and F : I ! R with F0(x) = f(x) for all x 2 I; neither the Riemann, or R; integral nor the Lebesgue, or L; integral guarantees that Z b a f = F(b) F(a): However, the Henstock-Kurzweil integral integrates every derivative, making integration and di erentiation truly inverse processes. We look at some of the consequences of this result. In addition, we investigate speci c properties of the Henstock-Kurzweil integral which serve to illustrate how a relatively small change in the de nition of the Riemann integral can have far reaching consequences, and how the Lebesgue integral can be seen as a special case of the Henstock-Kurzweil integral. In particular, we show that R(I) $ L(I) $ HK(I) where R(I); L(I) and HK(I) denote the classes of Riemann integrable, Lebesgue integrable and Henstock- Kurzweil integrable functions over I; respectively. Finally, we discuss the consequences of using the Henstock- Kurzweil integral in various areas of applied mathematics.