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Possibility theory. I: The measure- and integral-theoretic groundwork. (English) Zbl 0955.28012
Summary: In Part I, I provide the basis for a measure- and 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 semi-normed 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 for these variables. The almost everywhere equality and dominance of fuzzy variables is defined and studied. The proof is given for a Radon-NikodĂ˝m-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 measure- and integral-theoretic treatment of a conditional possibility and possibilistic independence, discussed in more detail in Parts II and III (summarized below) of this series of three papers.

MSC:
28E10 Fuzzy measure theory
03B48 Probability and inductive logic
03E72 Theory of fuzzy sets, etc.
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