Complex Fuzzy Set-Valued Complex Fuzzy Measures and Their Properties

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.


Introduction
It is well known that additivity of a classical measure primly depicted measure problems under error-free condition. But when measure error was unavoidable, additivity could not fully depict the measure problems under certain condition. To overcome such difficulties, fuzzy measure has been developed. 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 (such as Choquet's content theory [1], Sugeno's measure theory [2]), 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.
Being a newly developing theory developed in the later 1960s, set value measure had been applied in many fields [4][5][6]. After the appearance of fuzzy numbers, people naturally thought of related measure and integral. In 1986 Zhang [7] defined a kind of fuzzy set measure on , 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. also defined the fuzzy value measure integral of fuzzy value function [9] which generalized the Sugeno integral about fuzzy value fuzzy measure to fuzzy set [10]. In 1989, Buckley presented the concept of fuzzy complex number [11] which inspired that people needed to consider the measure and integral problem about fuzzy complex number.
At the beginning of the 90s, Guang-Quan [12][13][14][15][16][17][18][19][20][21] introduced fuzzy real distance and discussed the fuzzy real measure based on fuzzy sets and then gave the fuzzy real value fuzzy integral and established fuzzy real valued measure theory on fuzzy set space. During 1991-1992, Buckley and Qu [22,23] studied the problems of fuzzy complex analysis: fuzzy complex function differential and fuzzy complex function integral. During 1996-2001, Qiu et al. studied serially basic problems of fuzzy complex analysis theory, including the continuity of fuzzy complex numbers and fuzzy complex valued series [24], fuzzy complex valued functions and their differentiability [25], and fuzzy complex valued measure and fuzzy complex valued integral function [26,27]. 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.
As for applications of fuzzy complex number theory, Ramot et al. [29,30] studied complex fuzzy sets and complex fuzzy logic, Dick [31] studied fuzzy complex logic more profoundly, Ha et al. [32] applied fuzzy complex set in statistical 2 The Scientific World Journal learning theory and obtained a key theorem of statistical learning theory, Fu and Shen [33] studied modeling problems of fuzzy complex number, and Fu and Shen [34] applied fuzzy complex in pattern recognition and classification and obtained important results. Please see [35][36][37] for other applications.
This paper will extend the classical measure to fuzzy complex number-valued measure, which can better express the interactions among the attributes (cf. [32,[34][35][36][37]) and, thus, is expected to have extensive applications in information fusion technology, classification technology, machine learning, pattern recognition, and other fields. Section 2 is some preliminary notions (including fuzzy complex number, real fuzzy distance between two fuzzy real numbers, and two fuzzy complex numbers) and some basic operations and order relation of fuzzy complex. Section 3 is prepared for the next section. 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 also 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). In Section 4, we deduced some important properties on complex fuzzy set value complex fuzzy measure which are generalizations of the corresponding results in measure theory; we also obtain some results on related integral theory.
(̃,̃) is called the fuzzy distance of fuzzy real numbers̃and .
is a fuzzy distance on * ( ).
is a fuzzy distance on * ( ).

Complex Fuzzy Set-Valued Complex Fuzzy Measures
The notion of complex fuzzy measure on family of classical sets was given in [26].
Definition 7 (see [26]). Let (2) if ⊂ , then | ( )| ≤ | ( )|; In this paper we need an expansion of this notion. First we defined the concept of fuzzy complex value distance.  Apparently, a complex fuzzy set-value complex fuzzy measure is also a kind of special generalized fuzzy measures.
We first have the following result.