Resumen de Radon-Nikodym derivatives and conditioning in fuzzy measure theory.
Domenico Candeloro, Sabrina Pucci
In the last twenty years many papers have appeared dealing with fuzzy theory. In particular, fuzzy integration theory had its origin in the well-known Thesis of Sugeno [7]. More recently, some authors faced this topic by means of some binary operations (see for instance [3], [8] and references): a fuzzy measure must be additive with respect to one of them, an the integral is to define in a way, which is very similar to the construction of the Lebesgue integral. On the contrary, we are interested in the original theory of fuzzy integral as formulated by Sugeno. Even if this theory doesn't generalize the Lebesgue integral, it is an interesting alternative. However we must point out that Sugeno's Thesis, even if full of impressive and innovative ideas and of interesting applications, sometimes is harder than necessary, and not quite correct. Batle and Trillas, in [2], discover and correct some imperfections of it, but we feel that the greatest problem arise in connection with Radon-Nikodym derivatives, and with the definition of conditional fuzzy measure. In this note we try to supply easier theorems, and more correct, than those carried out in [7]: this is done in section 2. In section 3 we study a particular kind of fuzzy measure, which is well-suited to our conditions for the existence of Radon-Nikodym derivative: in this case our results are contained in those of [3], [8], but are much more direct. In section 4 we develop, for the same kind of measures, a theory of conditioning very similar to the classical one.
