Conditional Probabilities obtained by Probabilities with values on the Field of Bilateral Series

Abstract

Abstract A generalization of the notion of probability is considered. This generalization has been introduced in [7] and takes values on an intensification B of the field R of the real numbers. The new notions of “probabilities which single out the elementary events” and of “pseudodensity which is completely compatible with a given comparative probability” are introduced and we prove that, if S is the set of elementary events and p 0 is a finitely additive probability on P(S) with real values, compatible with a given comparative probability, under suitable conditions there is an algebra A containing S and a finitely additive probability p on A with values in B which singles out the elementary events and which, on A, has the real part equal to p 0. The results achieved are exploited in order to define the conditional probability p(A/E), with values in B, for each non-impossible event E such that p 0(E)=0 and a ∩E∈A. Moreover, the notion of “complete algebra” with respect to a finitely additive probability ...