On Probability Domains II

Abstract

We continue our studies of the foundation of probability theory using elementary category theory. We propose a classification scheme of probability domains in terms of cogenerators and their algebraic and topological properties and use the scheme to describe the transition from classical to fuzzy probability. We show that Łukasiewicz tribes form a category of natural probability domains in which σ-fields of sets are “minimal” and measurable [0,1]-valued functions are “maximal” objects. The maximal objects form an epireflective subcategory in which both the classical and fuzzy probability can be modelled. This leads to a better understanding of the transition.