Power Spaces of Valuations
by
Reinhold Heckmann
Universität des Saarlandes

Valuations are measure-like functions which assign non-negative real numbers to the open sets of a topological space. The probabilistic power domain of Jones and Plotkin, which is used to model the semantics of probabilistic programs, consists of Scott continuous valuations which are bounded by 1. The lower bag domain, which is used to specify a bag semantics for non-deterministic programs, consists of Scott continuous integer-valued valuations. Abbas Edalat uses Scott continuous valuations to obtain a domain-theoretic treatment of measures and Riemann-like integrals.

Here, we introduce point continuous valuations as a subclass of the class of Scott continuous valuations. On continuous domains, every Scott continuous valuation is point continuous. We show that the space of point continuous valuations is the sobrification of the space of finite valuations (finite linear combinations of point valuations), and present universal properties of these two spaces of valuations. Using these properties, integration of a continuous real-valued function w.r.t. a Scott continuous (not necessarily point continuous) valuation can be elegantly defined. Finally, we specialize the general results to the case of integer-valued valuations.

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-32.