Abstract not found

In a recent paper, probabilistic processes are used to generate Borel probability measures on topological spaces X that are equipped with a representation in the sense of Type-2 Theory of Effectivity. This gives rise to a natural representation of the set M(X) of Borel probability measures on X. We compare this representation to a canonically constructed representation which encodes a Borel probability measure as a lower semicontinuous function from the open sets to the unit interval. This canonical representation turns out to be admissible with respect to the weak topology on M(X). Moreover, we prove that for countably based topological spaces X the representation via probabilistic processes is equivalent to the canonical representation and thus admissible with respect to the weak topology on M(X).

In the Type-2 Theory of Effectivity, one considers representations of topological spaces in which infinite words are used as “names ” for the elements they represent. Given such a representation, we show that probabilistic processes on infinite words generate Borel probability measures on the represented space. Conversely, for several well-behaved types of space, every Borel probability measure is represented by a corresponding probabilistic process. Accordingly, we consider probabilistic processes as providing “probabilistic names ” for Borel probability measures. We show that integration is computable with respect to the induced representation of measures. 1

Abstract not found

domain theoretic approach to effective

We present a method for constructing from a given domain representation of a space X with underlying domain D, a domain representation of a subspace of compact subsets of X where the underlying domain is the Plotkin powerdomain of D. We show that this operation is functorial over a category of domain representations with a natural choice of morphisms. We study the topological properties of the space of representable compact sets and isolate conditions under which all compact subsets of X are representable. Special attention is paid to admissible representations and representations of metric spaces.