Constructing the space of valuations of a quasi-Polish space as a space of ideals

TL;DR

This paper constructs the space of valuations on a quasi-Polish space using ideals of a countable relation, linking domain theory and quasi-Polish spaces with implications for probabilistic semantics.

Contribution

It introduces a novel construction of valuations on quasi-Polish spaces via ideals, bridging domain theory and topology with formalizable methods.

Findings

Constructs valuations using ideals of countable relations.

Connects domain theory with quasi-Polish space topology.

Aligns with computable measures and formal systems.

Abstract

We construct the space of valuations on a quasi-Polish space in terms of the characterization of quasi-Polish spaces as spaces of ideals of a countable transitive relation. Our construction is closely related to domain theoretical work on the probabilistic powerdomain, and helps illustrate the connections between domain theory and quasi-Polish spaces. Our approach is consistent with previous work on computable measures, and can be formalized within weak formal systems, such as subsystems of second order arithmetic.