Measure and Integration in Topological Order
by
Abbas Edalat
Imperial College

Suppose we have a probability distribution on a compact metric space, given by its value on a countable base, and suppose we have a real-valued Lipschitz function on the space. How can we compute, up to any given threshold of accuracy, the expected value, i.e. the integral, of the this function with respect to the probability distribution? We present a domain-theoretic foundation for measure and integration theory to solve this problem. We work with the upper space of the metric space, which consists of the set of non-empty compact subsets of the space ordered by reverse inclusion. This is an \omega-continuous dcpo (directed complete partial order) equipped with its Scott topology. The set of probability distributions on this dcpo can be ordered to give another \omega-continuous dcpo called the probabilistic power domain of the upper space. The crucial point is that the set of probability distributions on the metric space, equipped with the weak topology, can be embedded into the set of maximal elements of the probabilistic power domain. We obtain an algorithm to express any probability distribution on the metric space as the supremum of an increasing sequence of simple distributions on the upper space. This algorithm and the domain-theoretic generalised Riemann integral are then used to compute integrals up to any desired accuracy.

Date received: April 12, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caae-18.