Probability of fuzzy events defined as denumerable additivity measure.

*(English)*Zbl 0604.60005In this paper the probability of fuzzy events is defined as a denumerable additivity measure. This definition is based on a non-conventional approach of separativity between fuzzy subsets. The presented measure fulfils all properties analogous to the properties of classical probability in the crisp case. Further, the notion of conditional probability of fuzzy events, complete fuzzy repartition and independent fuzzy events are defined by means of the probability measure considered here. Connections between all the above notions are presented in this paper, too. Finally, the Bayes formula is proved for the fuzzy case.

##### Keywords:

probability of fuzzy events; separativity between fuzzy subsets; conditional probability of fuzzy events; independent fuzzy events
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##### References:

[1] | Kabala, Z.; Wrociński, I., A short notice on Bayes rule, Busefal, 18, 92-98, (1984) · Zbl 0598.60010 |

[2] | Khalili, S., Fuzzy measures and mappings, J. math. anal. appl., 68, 92-99, (1979) · Zbl 0448.28002 |

[3] | Klement, E.P.; Lowen, R.; Schwychla, W., Fuzzy probability measures, Fuzzy sets and systems, 5, 21-30, (1981) |

[4] | K. Piasecki, New concept of separated fuzzy subsets, Proc. Polish Symposium on Interval and Fuzzy Mathematics 1983 (to appear). · Zbl 0593.04011 |

[5] | K. Piasecki, Probability space defined by means of fuzzy relation ‘less than’, Fuzzy Sets and Systems, to appear. · Zbl 0604.60006 |

[6] | Sugeno, M., Theory of fuzzy integrals and its applications, () · Zbl 0316.60005 |

[7] | Zadeh, L.A., Probability measures of fuzzy events, J. math. anal. appl., 23, 421-427, (1968) · Zbl 0174.49002 |

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