An axiomatic approach to extensional probability measures

Abstract

Replacing the demand of countable additivity (σ-additivity), imposed on probability measures by the classical Kolmogorov axiomatic, by a stronger axiom, and considering only probability measures taking their values in the Cantor subset of the unit interval of real numbers, we obtain such an axiomatic system that each probability measure satisfying these axioms is extensional in the sense that probability values ascribed to measurable unions and intersections of measurable sets are functions of probability values ascribed to particular sets in question. Moreover, each such probability measure can be set into a one-to-one correspondence with a boolean-valued probability measure taking its values in the set of all subsets of an infinite countable space, e.g., the space of all natural numbers.