Multivalent probability spaces

Abstract

We give an abstract version of systems of sets similar to the λ-systems considered by Dynkin and other authors as a useful auxiliary tool. Our abstract version is of “Boolean” nature. This means, in particular, that the elements of the corresponding abstract algebra have no intrinsic set-theoretic structure. A natural system of axioms is specified. This system describes properties of two binary relations (inclusion and disjointness) and properties of two partial binary operations (addition and subtraction) closely connected with these binary relations. In particular, addition and subtraction are mutually inverse in a certain precisely specified sense. We state simple properties of these abstract Dynkin algebras and investigate extensions of such algebras by means of passages to certain limits (we refer to them as free extensions). The free extension of an abstract Dynkin algebra is closed under taking limits of monotone sequences of its elements. We prove that every (additive) probability on an abstract Dynkin algebra has a unique continuous (= countably additive) extension to the corresponding free extension of the algebra. This result, which disagrees with the usual difference between additivity and countable additivity, can be explained by the freeness of the extension under consideration.