Domain theory and integration

We present a domain-theoretic framework for measure theory and integration of bounded read-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 ω-chain of linear combinations of point valuations (simple valuations) on the zipper space, thus providing a constructive setup for these measures. We use this setting to develop a theory of integration based on a new notion of integral which generalises and shares all the basic properties of the Riemann integral. The theory provides a new technique for computing the Lebesgue integral. It also leads to a new algorithm for integration over fractals of iterated function systems