On the product of M-measures in l-groups

In this paper we continue the investigation dealt with in A. BOCCUTO - B. RIECAN, On extension theorems for M-measures in l-groups, Math. Slovaca (2008). Starting from an extension-type existence theorem for M-measures with values in l-groups, we obtain existence results in the countably compact case for M-measures and product of M-measures. The motivation of this study in due to the fact that in probability theory, in many applications it is advisable to deal with set functions, which are not necessarily additive, but satisfy other properties: for example, continuity from below and from above for sequences of sets and "compatibility" with respect to the operations of finite suprema and infima. These functions are called M-measures. For example, in decision making, this is the case of the theory of intuitionistic fuzzy events (shortly IF-events), which are pairs A = (μA, nu A) of measurable functions μA, nu A : Omega -->[0, 1] such that μA+nu A <= 1. Another application is the theory of joint random variables: in this context theM-measure extension theorem plays a crucial role in the construction of joint observables. Moreover, to consider latticegroup or Riesz space-valued set functions allows to get applications in stochastic processes and in probabilities depending on the time and/or on the informations of the individual.