Fuzzy measures and measures of fuzziness

In this paper, the fuzzy integral defined by Z.-X. Wang (Fuzzy Math. Wuhan, China, in press), which is different from that defined by M. Sugeno (“Theory of Fuzzy Integrals and Its Applications,” Ph. D., Tokyo Inst. of Technology, 1974), is further considered, and it is shown that the fuzzy measures of ordinary sets and fuzzy sets can be determined by each other. Summing up the results on the measure of fuzziness by A. DeLuca and S. A. Termini (Inform. and Control20 (1972), 301–312), Z.-X. Wang (op. cit.) and R. R. Yager (Internat. J. Gen. Systems5 (1979), 221–229; Inform. and Control44 (1980), 236–260), the axioms for measures of fuzziness are given. Furthermore, as an application of the furry integrals, a measure of fuzziness is defined. Inversely, it is proven that a measure of fuzziness satisfying some conditions can surely be expressed as a fuzzy integral with respect to some fuzzy measure.