Paolo Mancosu (Berkeley)

A probability space is given by a triple (Ω, K, P) where Ω is a set called the sample space, K is a σ-algebra of subsets of Ω, and P is a probability function from K to the interval [0, 1]. The standard Kolmogorovian approach to probability theory on infinite sample spaces is neither regular nor total. Totality, expressed set-theoretically, is the request that every subset of the sample space Ω is measurable, i.e., has a probability value. Regularity, expressed set-theoretically, is the request that only the empty set gets probability 0. Mathematical and philosophical interest in non-Kolmogorovian approaches to probability theory in the last decade has been motivated by the possibility to satisfy totality and regularity in non-Archimedean contexts (Wenmackers and Horsten 2013, Benci, Horsten, Wenmackers 2018). Much of the mathematical discussion has been focused on the cardinalities of the sample space, the algebra of events, and the range (Hájek 2011, Pruss 2013, Hofweber 2014). In this talk I will present some new results characterizing the relation between completeness and regularity in a variety of probabilistic settings and I will give necessary and sufficient conditions relating regularity and the cardinalities of the sample space, the algebra of events, and the range of the probability function, thereby improving on the results hitherto available in the literature. This is joint work with Guillaume Massas (Group in Logic, UC Berkeley).