Two Probabilistic Powerdomains in Topological Domain Theory

We present two probabilistic powerdomain constructions in topological domain theory. The first is given by a free ”convex space” construction, fitting into the theory of modelling computational effects via free algebras for equational theories, as proposed by Plotkin and Power. The second is given by an observationally induced approach, following Schröder and Simpson. We show the two constructions coincide when restricted to ω-continuous dcppos, in which case they yield the space of (continuous) probability valuations equipped with the Scott topology. Thus either construction generalises the classical domain-theoretic probabilistic powerdomain. On more general spaces, the constructions differ, and the second seems preferable. Indeed, for countably-based spaces, we characterise the observationally induced powerdomain as the space of probability valuations with weak topology. However, we show that such a characterisation does not extend to non countably-based spaces.