Structural Characteristics for Set Functions

Abstract

Up to now, we have used some structural characteristics such as nonnegativity, monotonicity, additivity, subadditivity, λ-rule, f-additivity, continuity, etc. to describe the features of a set function. Since the fuzzy measures in general lose additivity, they appear much looser than the classical measures. Thus it is quite difficult to develop a general theory of fuzzy measures without any additional condition. Before 1981, it was thought that fuzzy measures additionally possessed subadditivity (even f-additivity), or satisfied the λ-rule. But these conditions are so strong that the essence of the problem is concealed in most propositions. Since 1981, many new concepts on structural characteristics, which fuzzy measures (or the monotone set functions) may possess (e.g., null-additivity, autocontinuity and uniform autocontinuity) have been introduced. As we shall see later, they are substantially weaker than subadditivity or the λ-rule, but can effectively depict most important fuzzy measures and are powerful enough to guarantee that many important theorems presented in the following two chapters will be justified. In several theorems, they go so far as to be a necessary and sufficient condition. In fuzzy measure theory, these new concepts replace additivity and thus play important roles.