Probabilistic Powerdomains and Quasi-Continuous Domains

Abstract

The probabilistic powerdomain VX on a space X is the space of all continuous valuations on X. We show that, for every quasi-continuous domain X, VX is again a quasi-continuous domain, and that the Scott and weak topologies then agree on VX. This also applies to the subspaces of probability and subprobability valuations on X. We also show that the Scott and weak topologies on the VX may differ when X is not quasicontinuous, and we give a simple, compact Hausdorff counterexample.