On the measure of a fuzzy set based on continuous t-conorms

The original aim of this research was the introduction of a compositive measure of information (see e.g. [5]) on a class (an algebra) of fuzzy subsets of an universe f2. However, there is a bijection between the class of compositive informations and the class of decomposable measures (see e.g. [2]). Therefore, we restated our problem as the one of the construction of a decomposable measure over the space (f2,~7), starting from a crisp measure defined on (~,~¢). Here ~ and ,4 are suitable algebras of, respectively, crisp and fuzzy subsets of f2 (the measurable ones). This problem has been analyzed by many authors in some special cases; in particular if the crisp measure is a possibility (Sugeno; see e.g. [6]) or an archimedean decomposable measure [7]. Here we present an approach which permits us to construct such a fuzzy measure in the case where the crisp one has an arbitrary continuous composition law. The main results and the detailed proofs are contained in [4]. Finally, the original information problem can be solved by a symmetrization of the obtained results. ~) 1997 Elsevier Science B.V. 1. Preliminaries In this section we recall some definitions and some notations regarding fuzzy sets (Section 1.1) and decomposable measures (Section 1.2). In the following definition we identify a fuzzy set with its membership function. We remark that this identification will not cause any problem. Definition 1.1. A fuzzy subset of a space f2 is a map A:(2-+ [0, 1], where, V~oef2, /~(o0) represents the membership degree of ~o with respect to the subset J.. Definition 1.2. Union, intersection and complemen-tation are the classical ones: (J.voB) (~o) = max [A(~), B(~o)], (~m~) (~o) = min [/~((o), ~(co)], (c~)(~o) = t-Y,(~).