Some remarks on generalized Riemann integral

Abstract

Because of the desire to calculate the areas of elementary figures, a variety of integrals has been established, popular among which are the Lebesgue integral and the Riemann integral. It is clear that Riemann integral is fundamental in elementary calculus and it can be used to define and calculate many geometric and physical quantities, such as area, volume, and work. However, the Riemann integral has its limitations. The theory of the Lebesgue integral reveals that the Riemann integral is basically used for continuous functions. In fact, f: [a, b] -+ R is Riemann integrable iff S is continuous a.e. on [a, b]. Also, as is known, the convergence theorems for this integral are severely restricted. With the motivation of generalizing the Riemann integral so as to enlarge the class of integrable functions for which the convergence theorems hold, the Lebesgue integral was successfully established. Generalizations of the Lebesgue integral, such as the Perron integral and special Denjoy integral, appeared later. The most interesting generalization is the generalized Riemann integral (GR integral for short), discovered by Kurzweil and Henstock independently, although it is equivalent to Perron and special Denjoy integrals. Contrary to the classical exposition of the Lebesgue integral which needs the concepts of measurable sets and measurable functions, before defining the integral, one can define the GR integral directly based on Riemann sums; therefore, the definition is constructive. Then, if one wishes, one can find all the Lebesgue measurable sets and measurable functions via the definition of GR integral. Furthermore, all the convergence theorems can be proved using the definition of the integral [l-3]. It is known that f is Lebesgue integrable iff both f and If] are GR integrable. Hence, the Lebesgue integral can be introduced through the GR integral, avoiding measure theory. Since the definition of the GR integral is very similar to that of the Riemann integral, one can easily grasp the 515 0022-247X/89 $3.00