Abstract
Let F*(K) be the set of all fuzzy complex numbers. In this paper some classical and measure-theoretical notions are extended to the case of complex fuzzy sets. They are fuzzy complex number-valued distance on F*(K), fuzzy complex number-valued measure on F*(K), and some related notions, such as null-additivity, pseudo-null-additivity, null-subtraction, pseudo-null-subtraction, autocontionuous from above, autocontionuous from below, and autocontinuity of the defined fuzzy complex number-valued measures. Properties of fuzzy complex number-valued measures are studied in detail.
Highlights
It is well known that additivity of a classical measure primly depicted measure problems under error-free condition
Research on fuzzy measures was very deep in those aspects: research based on a certain number of subsets of a classic set and a real value nonaddable measure, research based on fuzzy sets and a real value measure (e.g., Zadeh’s addable measure [3]), and especially the research on fuzzy value measures which generalizes the set value measure theory
In 1986 Zhang [7] defined a kind of fuzzy set measure on Rn, in 1998 Wu et al generalized the codomain of fuzzy measure to fuzzy real number field and defined the Sugeno integral of fuzzy number fuzzy measure [8], and Guo et al defined the G fuzzy value measure integral of fuzzy value function [9] which generalized the Sugeno integral about fuzzy value fuzzy measure to fuzzy set [10]
Summary
It is well known that additivity of a classical measure primly depicted measure problems under error-free condition. Wang and Li [28] gave the fuzzy complex valued measure based on the fuzzy complex number concept of Buckley, studied Lebesgue integral of fuzzy complex valued function, and obtained some important results. We defined, based on Ha’s work [38], the concepts of fuzzy complex distance and complex fuzzy set value complex fuzzy measure (an extension of fuzzy measure) on fuzzy complex number field. We present, based on Zhang’s work [21], the concepts of nulladditivity, pseudo-null-additivity, null-subtraction, pseudonull-subtraction, autocontinuous from above, autocontinuous from below, and autocontinuity of fuzzy complex value fuzzy complex measure on complex fuzzy number set (this measure has the properties PGP and SA/SB).
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