Fuzzy Measures and Coherent Join Measures

In assigning weights and scores in a decision problem usually we assume that they are finitely additive normalized measures, i.e., from the formal point of view, finitely additive probabilities. In some cases, however, this may be too restrictive. For example, the weights and scores could be assigned with not precise numbers, or some logic and numerical considerations on the problem at hand can lead to conditions different from additivity. We must then consider extensions of the concept of probability. One, introduced by Zadeh, is to express the probabilities with fuzzy numbers; another extension, considered by Sugeno, Weber and others, is to first replace the additivity with the condition of monotonicity, much weaker, and then identify conditions "intermediate" between monotonicity and additivity. In any case, by assigning weights and scores, they must be consistent with the point of view considered. The conditions of consistency of finitely additive probabilities and their generalizations were discussed in several papers. This paper proposes an extension of the concept of finitely additive probability from a purely geometric point of view. Specifically, the environment of Euclidean Geometry, by de Finetti used to define the consistency of an assignment of probabilities, is replaced by the more general environment of Join Geometry by Prenowitz and Jantosciak. In this context we introduce the concept of coherent join measure, i.e., normalized measure that is consistent with a join system, in particular a join space or a join geometry. We show that decomposable measures with respect to a t-conorm are special cases of join coherent measures. Finally, we present some applications, significant special cases and possible lines of research.