Fuzzy probabilities

We introduce the notion of fuzzy probabilities bound to the fuzzy sets. 1 The fuzzy sets A fuzzy set is a function with values in [0, 1]. The fuzzy set is open if the function is lower semi continuous. 2 The measure of a fuzzy set If we take a fuzzy set f , then its measure is: m(f) = ∫ fdμ 3 The fuzzy probabilities We consider the fuzzy sets of a space X with a measure of probability. The fuzzy probability of X is the measure over the fuzzy sets. We have the theorem: Theorem The fuzzy Lebesgue measurable sets are the Lebesgue measurable functions with values in [0, 1]. Proof: We show first that the step functions are fuzzy Lebesgue measurable sets. ∑ i ai1Ei Then the superior limit of a countable set of step functions is a fuzzy Lebesgue measurable set so that we can approximate a Lebesgue measurable function by the limit and the difference is positiv with zero integral and so is a fuzzy Lebesgue measurable set. 4 Bibliography A.Kaufmann, ”Introduction à la théorie des sous-ensembles flous”, Masson et cie, 1973.