Stably Compact Spaces and the Probabilistic Powerspace construction

We put forward a revised definition of stably compact spaces which allows us to show their equivalence with Nachbin’s compact ordered spaces in an entirely elementary fashion. We then exhibit some constructions for stably compact spaces which apparently have not appeared in the literature before. These constructions allow us to show that the set of (sub-)probability valuations can be equipped with a topology which turns this set into another stably compact space. The topology chosen is not random; it is the weakest topology which makes integration of lower semicontinuous functions a continuous operation.