Since the concepts of fuzzy measures and fuzzy integrals were proposed, there have been reported several examples of applications which are concerned with subjective evaluation problems of fuzzy objects. This paper defines conditional fuzzy measures and gives the foundation for developing the theory of fuzzy measures.
First, the inverse operation of fuzzy integrals is defined corresponding to the differentiation of set functions. A theorem which is similar to Radon-Nikodym's theorem in the theory of Lebesgue integrals is proved by using this operation. This theorem clarifies that a monotone set function can be expressed by a fuzzy integral under suitable conditions. Here, fuzzy integrals find a new significance of their existence.
Next, by using this theorem, the existence of conditional fuzzy measures is proved. The concept of conditional fuzzy measures corresponding to conditional probabilities is expected to give the theory of fuzzy measures wider regions of applications.
Finally, fuzzy decision-making problems are briefly discussed as an example of applications of conditional fuzzy measures. A theorem is also proved which corresponds to Bayes' theorem in the probability theory. Fuzzy decision-making problems are similar to stochastic decision-making ones and are solved by finding the a posteriori fuzzy measure.