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A Daniell-Kolmogorov theorem for supremum preserving upper probabilities. (English) Zbl 0941.60005

The paper investigates uncertain processes for which the given imprecise probability model is a possibility measure, interpreted as upper probabilities that are in particular supremum preserving. The aim of the paper is to prove, for such possibilistic processes, a counterpart to a fundamental result in the theory of stochastic processes, namely the Daniell-Kolmogorov theorem. To attain this purpose, the authors define a possibilistic process as a special family of possibilistic variables, and show how its possibility distribution functions can be constructed. The notions of inner and outer regularity for possibility measures are introduced and investigated. Within this framework, an analog for possibilistic processes (and possibility measures) of the well-known probabilistic Daniell-Kolmogorov theorem is proved, in the important special case that the variables assume values in compact spaces and that the possibility measures involved are regular.

MSC:

60A10 Probabilistic measure theory
60B05 Probability measures on topological spaces
60G05 Foundations of stochastic processes
03B52 Fuzzy logic; logic of vagueness
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[1] Birkhoff, G., Lattice Theory (1973), American Mathematical Society Colloquium Publications: American Mathematical Society Colloquium Publications Providence · Zbl 0126.03801
[2] Burrill, C. W., Measure, Integration and Probability (1972), McGrawHill: McGrawHill New York · Zbl 0248.28001
[3] de Cooman, G., Evaluatieverzamelingen en-afbeeldingen - Een ordetheoretische benadering van vaagheid en onzekerheid, (Ph.D. Thesis (1993), Universiteit Gent: Universiteit Gent Ghent)
[4] de Cooman, G., Possibility theory I: The measure- and integral-theoretic groundwork, Internat. J. General Systems, 25, 291-323 (1997) · Zbl 0955.28012
[5] de Cooman, G., Possibility theory II: Conditional possibility, Internat. J. General Systems, 25, 325-351 (1997) · Zbl 0955.28013
[6] de Cooman, G., Possibility theory III: Possibilistic independence, Internat. J. General Systems, 25, 353-371 (1997) · Zbl 0955.28014
[7] de Cooman, G.; Kerre, E. E., Ample fields, Simon Stevin, 67, 235-244 (1993) · Zbl 0799.28012
[8] Davey, B. A.; Priestley, H. A., Introduction to Lattices and Order (1990), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0701.06001
[9] Doob, J. L., Stochastic Processes (1967), Wiley: Wiley New York
[10] Gierz, G.; Hofmann, K. H.; Keimel, K.; Lawson, J. D.; Mislove, M.; Scott, D. S., A Compendium of Continuous Lattices (1980), Springer: Springer New York · Zbl 0452.06001
[11] Halmos, P. R., Measure Theory (1974), Springer: Springer New York · Zbl 0117.10502
[12] H.J. Janssen, G. de Cooman, E.E. Kerre, Ample fields as a basis for possibilistic processes, Fuzzy Sets and Systems, submitted for publication.; H.J. Janssen, G. de Cooman, E.E. Kerre, Ample fields as a basis for possibilistic processes, Fuzzy Sets and Systems, submitted for publication. · Zbl 1023.94027
[13] Kelley, J. L., General Topology (1959), D. Van Nostrand: D. Van Nostrand Princeton, NJ
[14] Boyen, L.; de Cooman, G.; Kerre, E. E., On the extension of P-consistent mappings, (De Cooman, G.; Ruan, D.; Kerre, E. E., Foundations and Applications of Possibility Theory (1995), World Scientific: World Scientific Singapore), 88-98 · Zbl 0981.28501
[15] Pap, E., Null-Additive Set Functions (1995), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0856.28001
[16] Suzuki, H., Atoms of fuzzy measures and fuzzy integrals, Fuzzy Sets and Systems, 41, 329-342 (1991) · Zbl 0759.28015
[17] Walley, P., Statistical Reasoning with Imprecise Probabilities (1991), Chapman & Hall: Chapman & Hall London · Zbl 0732.62004
[18] Wang, P.-Z., Fuzzy contactability and fuzzy variables, Fuzzy Sets and Systems, 8, 81-92 (1982) · Zbl 0493.28001
[19] Wang, Z., Extension of possibility measures defined on an arbitrary nonempty class of sets, (Proc. 1st IFSA Congress. Proc. 1st IFSA Congress, Palma de Mallorca, Spain (1985))
[20] Wang, Z.; Klir, G. J., Fuzzy Measure Theory (1992), Plenum Press: Plenum Press New York · Zbl 0812.28010
[21] Zadeh, L. A., Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems, 1, 3-28 (1978) · Zbl 0377.04002
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