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Possibility theory, 1: the measure- and integral-theoretic groundwork

Gert De Cooman (UGent)
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Abstract
In this paper, 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 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 measure- and integral-theoretic treatment of conditional possibility and possibilistic independence, discussed in more detail in Parts II and III of this series of three papers.
Keywords
Radon-Nikodym-like theorem, possibility distribution, Fubini-like theorem, UNCERTAINTY, possibilistic variable, possibility measure, seminormed fuzzy integral, fuzzy variable

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Chicago
De Cooman, Gert. 1997. “Possibility Theory, 1: The Measure- and Integral-theoretic Groundwork.” International Journal of General Systems 25 (4): 291–323.
APA
De Cooman, Gert. (1997). Possibility theory, 1: the measure- and integral-theoretic groundwork. INTERNATIONAL JOURNAL OF GENERAL SYSTEMS, 25(4), 291–323.
Vancouver
1.
De Cooman G. Possibility theory, 1: the measure- and integral-theoretic groundwork. INTERNATIONAL JOURNAL OF GENERAL SYSTEMS. 1997;25(4):291–323.
MLA
De Cooman, Gert. “Possibility Theory, 1: The Measure- and Integral-theoretic Groundwork.” INTERNATIONAL JOURNAL OF GENERAL SYSTEMS 25.4 (1997): 291–323. Print.
@article{182365,
  abstract     = {In this paper, 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 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 measure- and integral-theoretic treatment of conditional possibility and possibilistic independence, discussed in more detail in Parts II and III of this series of three papers.},
  author       = {De Cooman, Gert},
  issn         = {0308-1079},
  journal      = {INTERNATIONAL JOURNAL OF GENERAL SYSTEMS},
  keyword      = {Radon-Nikodym-like theorem,possibility distribution,Fubini-like theorem,UNCERTAINTY,possibilistic variable,possibility measure,seminormed fuzzy integral,fuzzy variable},
  language     = {eng},
  number       = {4},
  pages        = {291--323},
  title        = {Possibility theory, 1: the measure- and integral-theoretic groundwork},
  url          = {http://dx.doi.org/10.1080/03081079708945160},
  volume       = {25},
  year         = {1997},
}

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