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Domain Theory and Integration?

<_author_search_(abbas edalat)>Abbas Edalat

Department of Computing
Imperial College of Science, Technology and Medicine
180 Queen's Gate
London SW7 2BZ UK

Abstract
We present a domain-theoretic framework for measure theory and integration of bounded real-valued functions with respect to bounded Borel measures on compact metric spaces. The set of normalised Borel measures of the metric space can be embedded into the maximal elements of the normalised probabilistic power domain of its upper space. Any bounded Borel measure on the compact metric space can then be obtained as the least upper bound of an !-chain of linear combinations of point valuations (simple valuations) on the upper space, thus providing a constructive setup for these measures. We use this setting to define a new notion of integral of a bounded real-valued function with respect to a bounded Borel measure on a compact metric space. By using an !-chain of simple valuations, whose lub is the given Borel measure, we can then obtain increasingly better approximations to the value of the integral, similar to the way the Riemann integral is obtained in calculus by using step functions. We show that all the basic results in the theory of Riemann integration can be extended in this more general setting. Furthermore, with this new notion of integration, the value of the integral, when it exists, coincides with the Lebesgue integral of the function. An immediate area for application is in the theory of iterated function systems with probabilities on compact metric spaces, where we obtain a simple approximating sequence for the integral of a real-valued continuous function with respect to the invariant measure.

1 Introduction

The theory of Riemann integration of real-valued functions was developed by Cauchy, Riemann, Stieltjes, and Darboux, amongst other mathematicians of the 19th century. With its simple, elegant and constructive nature, it soon became, as it is today, a solid basis of calculus; it is now used in all branches of science. The theory, however, has its limitations in the following main areas, listed here not in any particular order of significance: ?To appear in Theoretical Computer Science, 1995.