The Evolution of the Concept of Fuzzy Measure

Abstract

Most information discovery processes need to understand the reasons of the success of the inference methods or the usability of the new information, which can sometimes be somehow explained with a few useful measures on the premises or on the chosen relations, logics and implications. This chapter presents a state of art on the latest concepts of measure, from the additive measures, to monotone fuzzy measures and the latest monotone measures in relation to a preorder that gives an ordering for a measurable characteristic. A few measures on fuzzy sets and fuzzy relations are proposed. Namely, the entropy measures on fuzzy sets, which are a measure of fuzziness in itself, and the specificity measures on fuzzy sets, which cam be understood as measures of utility of the information contained in a fuzzy set. Some measures on fuzzy relations to be considered are the conditionality measures, which are degrees of generalization of modus ponens when making fuzzy inference. Those measures on fuzzy sets and fuzzy relations provide interesting information on fuzzy inference and approximate reasoning processes.