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$$\mathbb{B}$$-fuzzy probabilities. (English) Zbl 0868.03023
Summary: We treat $$\mathbb{B}$$-fuzzy probabilities in nonstandard frameworks by methods based on Boolean-valued models and especially Boolean powers. These formal methods, apart from the difficulties associated with the formal model itself, give a simple and powerful theory of fuzzy probability spaces and fuzzy random variables. Essentially everything is reduced to a transfer principle. However, one can also work using classical methods of proof. We first give the necessary background on nonstandard frameworks. Within these frameworks we develop Boolean fuzzy probabilities and fuzzy random variables.

##### MSC:
 03E72 Theory of fuzzy sets, etc. 60B99 Probability theory on algebraic and topological structures 03H05 Nonstandard models in mathematics 03E40 Other aspects of forcing and Boolean-valued models
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##### References:
 [1] Bell, J.L., () [2] Chang, C.C.; Keisler, H.J., () [3] Drossos, C.A., Foundations of fuzzy sets. A nonstandard approach, Fuzzy sets and systems, 37, 287-307, (1990) · Zbl 0712.03045 [4] C.A. Drossos, Random and fuzzy sets: A nonstandard approach, in: S. Weber and E.P. Klement, Eds., Uncertainty Measures (Kluwer Academic Publishers, Dordrecht, to appear). · Zbl 0712.03045 [5] Drossos, C.A.; Markakis, G., Boolean fuzzy sets, Fuzzy sets and systems, 46, 81-95, (1992) · Zbl 0760.03016 [6] Drossos, C.A.; Markakis, G., Boolean representations of fuzzy sets, Kybernetes, 22, 3, 35-40, (1993) · Zbl 0925.03202 [7] Drossos, C.A.; Markakis, G., Boolean powers and stochastic spaces, Math. slovaca, 44, 1, 1-19, (1994) · Zbl 0789.03038 [8] Drossos, C.A.; Markakis, G.; Tzavelas, G., A nonstandard approach to a general theory of random and $$B$$-fuzzy sets, (1989), unpublished manuscript [9] Kwakernaak, H., Fuzzy random variables: definitions and theorems, Inform. sci., 15, 1-29, (1978) · Zbl 0438.60004 [10] Mansfield, R., The theory of Boolean ultrapowers, J. math. logic, 2, 3, 297-323, (1971) · Zbl 0216.29401 [11] Markakis, G., A Boolean generalization of nonstandard analysis with applications in fuzzy sets, (), (in Greek) [12] Puri, M.L.; Ralescu, D.A., Fuzzy random variables, J. math. anal. appl., 111, 406-422, (1986) · Zbl 0605.60038 [13] Stroyan, K.; Luxembourg, W.A.J., Introduction to the theory of infinitesimals, (1976), Academic Press New York · Zbl 0336.26002 [14] Theodoropoulos, P., Nonstandard foundations of $$B$$-fuzzy probability and statistics of vague data, (), (in Greek) [15] Truesdell, C., A first course in rational continuum mechanics: vol. 1 general concepts, () · Zbl 0357.73011
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