Multivalent probability spaces

Abstract

We give an abstract version of some systems of sets similar to λ-systems discussed by Dynkin and others as a useful auxiliary tool. Our abstract version has a “Boolean” nature. This means that its elements have no internal set-theoretic structure. A natural system of axioms is formulated. 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. Particularly addition and subtraction are in some exactly formulated sense mutually inverse. We state some properties of these abstract Dynkin algebras and investigate enlargements of such algebras via some limit transitions (we referred to them as to free enlargements). The free enlargement of an abstract Dynkin algebra is closed under limits of monotonic sequences of its elements. We prove that every (additive) probability on an abstract Dynkin algebra has a unique continuous (= contably additive) extension to the corresponding free enlargement. This result contradicting to the usual difference between additivity and countable additivity can be explained by freeness of the enlargement under review. Refs 7. Keywords: Dynkin algebra, free enlargement, continuous extension of probability.

Keywords:

Dynkin algebra, free enlargement, continuous extension of probability