Measure-free conditioning and extensions of additive measures on finite MV-algebras

Abstract

Using the well known representation of any finite MV-algebra as a product of finite MV-chains as factors, we obtain a representation of its canonical extension as a Girard algebra product of the canonical extensions of the MV-chain factors. Based on this representation and using the results from our last paper, we characterize the additive measures on any finite MV-algebra resp. the weakly and the strongly additive measures on its canonical Girard algebra extension, and that as convex combinations of the corresponding measures on the respective factors. After that we apply the results to measure-free defined conditional events which for this reason are considered as elements of the canonical extension of a finite MV-algebra of events. This definition for conditional events is, naturally, equivalent to the well known definition as lattice intervals of events. Here, the weak additivity is considered as a reasonable notion for a “global additivity” property of a measure extension. These extensions are essentially different from the measure extension which has a form analogous to the classical conditional probability but which is only “levelwise additive”. Moreover, the measure-free approach of this paper is completely different from the classical approach in which “conditional measures of events (given a fixed condition event)” are considered based on additive measures (states) of events without having defined conditional events.