Integral with Respect to Fuzzy Measure in Finite Dimensional Banach Spaces
نویسندگان
چکیده
Fuzzy valued measure ([7], [10], [3], [5]) is a natural generalization of a set valued measure ([6]). Infinite addition is defined ([12]) as a Zadeh’s extension principle of continuity. In [9], using an additive fuzzy valued measure, an integral of single valued function is defined and some basic properties are given. In this paper we proceed to investigate specific properties of integrals in fuzzy analysis. Basic space is finite dimensional Banach space. We shall prove some results concerning the fundamental property of the integral the property that the integral is a new fuzzy valued measure on the measurable space (Ω,A) on which the basic fuzzy valued measure is defined. The range of the fuzzy valued measure is a set of fuzzy sets with closed levels. From the definition of the integral it is easy to notice that the ”fuzzy” integral is closely related to the ”set” integral defined with respect to the set valued measure derived from fuzzy valued measure taking its level set. That connection will be used often in the sequel. The organization of the paper is as follows: Section 2 contains some basic definitions, notions and notations. In Section 3 the main result of the paper concerning the integral with respect to fuzzy valued measure is given.
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