Possibility Theory I the Measure- and Integral-theoretic Groundwork

In this paper, I provide the basis for a measureand integral-theoretic formulation of possibility theory. It is shown that, using a general definition of possibility measures, and a generalization of Sugeno’s fuzzy integral – the seminormed fuzzy integral, or possibility integral –, a unified and consistent account can be given of many of the possibilistic results extant in the literature. The striking formal analogy between this treatment of possibility theory, using possibility integrals, and Kolmogorov’s measure-theoretic formulation of probability theory, using Lebesgue integrals, is explored and exploited. I introduce and study possibilistic and fuzzy variables as possibilistic counterparts of stochastic and real stochastic variables respectively, and develop the notion of a possibility distribution for these variables. The almost everywhere equality and dominance of fuzzy variables is defined and studied. The proof is given for a Radon-Nikodym-like theorem in possibility theory. Following the example set by the classical theory of integration, product possibility measures and multiple possibility integrals are introduced, and a Fubini-like theorem is proven. In this way, the groundwork is laid for a unifying measureand integral-theoretic treatment of conditional possibility and possibilistic independence, discussed in more detail in Parts II and III of this series of three papers.