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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.