Abstract
Lately, the way to express systems consisting of objects with no sharp boundaries and more generally the method to measure such objects have attracted attention of the people in the field of systems engineering. The concept of fuzzy sets defined by Zadeh gives us an important clue for an approach to a study of the above mentioned systems and objects. This paper is concerned with the idea of fuzzy meassure and fuzzy integral and attempts to represent fuzzy systems.
At first, the definition of fuzzy measure is given which has only monotonicity left after removing additivity from the characteristics of measure. Next, the fuzzy integral is proposed by using fuzzy measure. The fuzzy integral is an extention of the Lebesgue integral in a certain sense.
The characteristics, the measurability of functions in a fuzzy measure space and especially the difference between the Lebesque integral and the fuzzy integral are clarified.
Furthermore, a theorem on fuzzy integral in a product space is proved corresponding to Fubini's theorem in the theory of Lebesgue integral.
Finally, this paper discusses, as an example, an application of the fuzzy integral to the problems of board games and suggests possibilities for further applications.
