## Constructive definitions of fuzzy random variables.(English)Zbl 0929.60005

Summary: When we deal with a random experiment, we are often interested in functions of the experimental outcomes rather than the outcomes themselves. Fuzzy random variables formalize fuzzy-valued functions of the outcomes in a random experiment, that is, existing imprecise quantification processes. The concepts of fuzzy random variable and its fuzzy expected value have been introduced by M. L. Puri and D. A. Ralescu [J. Math. Anal. Appl. 114, 409-422 (1986; Zbl 0592.60004)] by means of descriptive definitions. Nevertheless, constructive definitions of fuzzy random variables would play an essential role in the constructive definition of their integrals, which will be especially valuable to perform practical computations and to develop further results concerning the integration of these variables. We present constructive definitions of fuzzy random variables and integrably bounded fuzzy random variables based on the Hausdorff convergence. The use of the last definition to obtain a constructive definition of the fuzzy expected value of an integrably bounded fuzzy random variable is finally discussed.

### MSC:

 60A99 Foundations of probability theory 03E72 Theory of fuzzy sets, etc. 60D05 Geometric probability and stochastic geometry 28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections

Zbl 0592.60004
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