General Riemann Integrals and Their Computation via Domain.

Abstract

In this work we extend the domain-theoretic approach of the generalized Riemann integral introduced by A. Edalat in 1995. We begin by laying down a related theory of general Riemann integration for bounded real-valued functions on an arbitrary set X with a finitely additive measure on an algebra of subsets of X. Based on the theory developed we obtain a formula to calculate integral of a bounded function in terms of the regular Riemann integral. By classical extension theorems on set functions we can further extend this generalized Riemann integral to more general set functions such as valuations on lattices of subsets. For the setting of bounded functions defined on a continuous domain D with a Borel measure for the Scott topology, we can compute the Riemann integral of a function effectively and so the value of the integral can be obtained up to a given accuracy. By invoking the results of J. Lawson we can extend this type of Riemann integral to maximal point spaces, as a special case of the Polish spaces. Furthermore we show that when X is a compact metric space, our approach of Riemann integration is equivalent to the generalized Riemann integration introduced by A. Edalat in the sense that the two integrals yield the same value. Finally we prove that the approach of integration taken by R. Heckmann is equivalent to our approach, and the values of the integrals are the same.