Hindawi Publishing Corporation e Scientific World Journal Volume 2014, Article ID 493703, 7 pages http://dx.doi.org/10.1155/2014/493703

Research Article Complex -Valued Complex Fuzzy Measures and Their Properties

Shengquan Ma1,2 and Shenggang Li1

1 College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, China 2 School of Information and Technology, Hainan Normal University, Haikou 571158, China

Correspondence should be addressed to Shengquan Ma; [email protected]

Received 21 April 2014; Revised 27 May 2014; Accepted 29 May 2014; Published 24 June 2014

Academic Editor: Jianming Zhan

Copyright © 2014 S. Ma and S. Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

∗ Let 𝐹 (𝐾) be the set of all fuzzy complex . In this paper some classical and measure-theoretical notions are extended to the ∗ case of complex fuzzy sets. They are fuzzy complex -valued distance on 𝐹 (𝐾), fuzzy -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.

1. Introduction fuzzy measure to fuzzy set [10]. In 1989, Buckley presented the concept of fuzzy complex number [11] which inspired that It is well known that additivity of a classical measure primly people needed to consider the measure and integral problem depicted measure problems under error-free condition. But about fuzzy complex number. when measure error was unavoidable, additivity could not At the beginning of the 90s, Guang-Quan [12–21]intro- fully depict the measure problems under certain condition. duced fuzzy real distance and discussed the fuzzy real To overcome such difficulties, fuzzy measure has been devel- measure based on fuzzy sets and then gave the fuzzy real oped. Research on fuzzy measures was very deep in those value fuzzy integral and established fuzzy real valued measure aspects: research based on a certain number of subsets of theory on fuzzy set space. During 1991-1992, Buckley and Qu a classic set and a real value nonaddable measure (such as [22, 23] studied the problems of fuzzy complex analysis: fuzzy Choquet’s content theory [1], Sugeno’s measure theory [2]), complex function differential and fuzzy complex function research based on fuzzy sets and a real value measure (e.g., integral. During 1996–2001, Qiu et al. studied serially basic Zadeh’s addable measure [3]), and especially the research on problems of fuzzy complex analysis theory, including the fuzzy value measures which generalizes the set value measure continuity of fuzzy complex numbers and fuzzy complex theory. valued series [24], fuzzy complex valued functions and their Being a newly developing theory developed in the later differentiability25 [ ], and fuzzy complex valued measure and 1960s, set value measure had been applied in many fields [4– fuzzy complex valued integral function [26, 27]. Wang and 6]. After the appearance of fuzzy numbers, people naturally Li [28]gavethefuzzycomplexvaluedmeasurebasedonthe thought of related measure and integral. In 1986 Zhang [7] fuzzy complex number concept of Buckley, studied Lebesgue 𝑛 defined a kind of fuzzy set measure on 𝑅 ,in1998Wu integraloffuzzycomplexvaluedfunction,andobtainedsome et al. generalized the codomain of fuzzy measure to fuzzy important results. field and defined the Sugeno integral of fuzzy Asforapplicationsoffuzzycomplexnumbertheory, number fuzzy measure [8], and Guo et al. also defined the Ramotetal.[29, 30]studiedcomplexfuzzysetsandcomplex 𝐺 fuzzy value measure integral of fuzzy value function [9] fuzzy logic, Dick [31] studied fuzzy complex logic more pro- which generalized the Sugeno integral about fuzzy value foundly, Ha et al. [32] applied fuzzy complex set in statistical 2 The Scientific World Journal

󸀠 ∗ learning theory and obtained a key theorem of statistical 𝑐 ∈𝐹(𝐾) is said to be a fuzzy infinity [21](writtenas∞̃) 󸀠 ̃ 󸀠 learning theory, Fu and Shen [33] studied modeling problems if one of the supports of 𝑎=̃ Re 𝑐 and 𝑏=Im 𝑐 is an 󸀠 󸀠 ∗ of fuzzy complex number, and Fu and Shen [34]applied unbound set. For any 𝑐1,𝑐2 ∈𝐹 (𝐾),onemakesthefollowing fuzzy complex in pattern recognition and classification and appointments: obtained important results. Please see [35–37]forother 󸀠 󸀠 󸀠 󸀠 󸀠 󸀠 applications. 𝑐1 ≤𝑐2 if and only if Re 𝑐1 ≤ Re 𝑐2 and Im 𝑐1 ≤ Im 𝑐2; 󸀠 󸀠 󸀠 󸀠 󸀠 󸀠 This paper will extend the classical measure to fuzzy 𝑐 =𝑐 if and only if 𝑐 ≤𝑐 and 𝑐 ≤𝑐; complex number-valued measure, which can better express 1 2 1 2 2 1 󸀠 󸀠 󸀠 󸀠 󸀠 󸀠 the interactions among the attributes (cf. [32, 34–37]) and, 𝑐1 <𝑐2 if and only if 𝑐1 ≤𝑐2 and Re 𝑐1 ≤ Re 𝑐2 or 󸀠 󸀠 thus, is expected to have extensive applications in informa- Im 𝑐1 < Im 𝑐2; tion fusion technology, classification technology, machine 𝑐󸀠 ≥0 𝑐󸀠 ≥0 𝑐󸀠 ≥0 learning, pattern recognition, and other fields. Section 2 is if and only if Re ,Im . ∗ some preliminary notions (including fuzzy complex number, One uses A to denote a family (which is obviously non- ∗ real fuzzy distance between two fuzzy real numbers, and empty) of subsets of 𝐹 (𝐾) that satisfies the following two fuzzy complex numbers) and some basic operations conditions: and order relation of fuzzy complex. Section 3 is prepared ̃ ∗ ̃ ̃ ̃ ̃ for the next section. We defined, based on Ha’s work (1) for each 𝐴∈A ,if𝐵={inf 𝐴0 | 𝐴0 ⊂ 𝐴} has upper ∗ [38], the concepts of fuzzy complex distance and complex bound, then sup 𝐵∈𝐹̃ (𝐾); fuzzy set value complex fuzzy measure (an extension of ̃ ∗ ̃ ̃ ̃ ̃ (2) for each 𝐴∈A ,if𝐶={sup 𝐴0 | 𝐴0 ⊂ 𝐴} has lower fuzzy measure) on fuzzy complex number field. We also ̃ ∗ present, based on Zhang’s work [21], the concepts of null- bound, then sup 𝐵∈𝐹 (𝐾). additivity, pseudo-null-additivity, null-subtraction, pseudo- null-subtraction, autocontinuous from above, autocontinu- 2.2. Fuzzy Distance of Fuzzy Numbers ous from below, and autocontinuity of fuzzy complex value ∗ ∗ fuzzy complex measure on complex fuzzy number set (this Definition 2 (see cf. [21]). A mapping 𝜌:𝐹̃ (𝑅) × 𝐹 (𝑅) → ∗ measurehasthepropertiesPGPandSA/SB).InSection 4, 𝐹 (𝑅) satisfying the following conditions is called a fuzzy ∗ we deduced some important properties on complex fuzzy set metric or a fuzzy distance on 𝐹 (𝑅): value complex fuzzy measure which are generalizations of the ̃ ∗ ̃ ̃ corresponding results in measure theory; we also obtain some (1) for any 𝑎,̃ 𝑏∈𝐹(𝑅), 𝜌(̃ 𝑎,̃ 𝑏) ≥,and 0 𝜌(̃ 𝑎,̃ 𝑏) = 0 if ̃ results on related integral theory. and only if 𝑎=̃ 𝑏; ̃ ∗ ̃ ̃ (2) for any 𝑎,̃ 𝑏∈𝐹 (𝑅), 𝜌(̃ 𝑎,̃ 𝑏) = 𝜌(̃ 𝑏, 𝑎)̃ ; 2. Preliminaries ̃ ∗ ̃ ̃ (3) for any 𝑎,̃ 𝑏, 𝑐∈𝐹̃ (𝑅), 𝜌(̃ 𝑎,̃ 𝑏) ≤ 𝜌(̃ 𝑎,̃ 𝑐)̃ + 𝜌(̃ 𝑐,̃ 𝑏). 2.1. Fuzzy Complex Numbers. In this paper, 𝑅 is the set of all ̃ ̃ ̃ ̃ real numbers set, 𝐾 is the set of all complex numbers, 𝑋 is an 𝜌(𝑎, 𝑏) is called the fuzzy distance of fuzzy real numbers 𝑎 and ∗ ̃ ordinary set, 𝐹 (𝑅) isthesetofallrealfuzzynumberson𝑅, 𝑏. Δ(𝑅) is the set of all interval numbers, (𝑋, A) is a measurable ∗ ∗ ∗ ∗ space (thus A is a 𝜎-algbra), and 𝐹 (𝐾) is the set of all fuzzy Example 3. The mapping 𝜌:𝐹̃ (𝑅) × 𝐹 (𝑅) → 𝐹 (𝑅) ∗ ∗ complex numbers on 𝐾.Let𝐹+ (𝑅) =𝑎: {̃ 𝑎≥0,̃ 𝑎∈𝐹̃ (𝑅)} defined by ∗ ̃ ̃ ̃ ̃ ∗ √ and let 𝐹+ (𝐾) =𝐴+𝑖 { 𝐵|𝐴, 𝐵∈𝐹+ (𝑅), 𝑖 = −1}. 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝜌̃(𝑎,̃ ̃𝑏) = ⋃ 𝜆 [󵄨𝑎− −𝑏−󵄨 , 󵄨𝑎− −𝑏−󵄨 ∨ 󵄨𝑎+ −𝑏+󵄨] ̃ ∗ 󵄨 1 1 󵄨 sup 󵄨 𝜂 𝜂 󵄨 󵄨 𝜂 𝜂 󵄨 Definition 1 (see [12]). Let 𝑎,̃ 𝑏∈𝐹(𝑅).Thenthemapping 𝜆∈[0,1] 𝜆≤𝜂≤1 ̃ ̃ ̃ (𝑎,̃ 𝑏) : 𝐾 → [0, 1] defined by (𝑎,̃ 𝑏)(𝑥+𝑖𝑦)=𝑎(𝑥)̃ ∧ 𝑏(𝑦) is (1) called a fuzzy complex number, where 𝑎̃ is called the real part ̃ ̃ ̃ ∗ of (𝑎,̃ 𝑏) (written as Re(𝑎,̃ 𝑏))and𝑏 is called the imaginary is a fuzzy distance on 𝐹 (𝑅). ̃ √ part (written as Im(𝑎,̃ 𝑏)), and 𝑖= −1. One will identify ∗ ∗ ̃ Remark 4. Analogously, a mapping 𝜌:𝐹̃ (𝐾) × 𝐹 (𝐾) → (𝑎,̃ 0) with 𝑎̃ and, thus, think fuzzy complex numbers are an ∗ 𝐹 (𝑅) satisfying the following conditions is called a fuzzy extension of fuzzy real numbers. The set of all fuzzy complex ∗ ∗ 𝐹 (𝐾) numbers on 𝐾 is denoted by 𝐹 (𝐾). metric or a fuzzy distance on : 𝐴, 𝐵 𝑅 (𝐴, 𝐵) ≜ 𝐴 + 𝑖𝐵 = {𝑥+ 𝑖𝑦| For any subsets of ,write 𝑎,̃ ̃𝑏∈𝐹∗(𝐾) 𝜌(̃ 𝑎,̃ ̃𝑏) ≥ 0 𝜌(̃ 𝑎,̃ ̃𝑏) = 0 𝑥 ∈ 𝐴, 𝑦 ∈𝐵} ∘ ∈ {+,−,⋅} (1) for any , ,and if .Theoperation is described as 𝑎=̃ ̃𝑏 follows: and only if ; 𝑎,̃ ̃𝑏∈𝐹∗(𝐾) 𝜌(̃ 𝑎,̃ ̃𝑏) = 𝜌(̃ ̃𝑏, 𝑎)̃ 󸀠 󸀠 󸀠 󸀠 󸀠 󸀠 󸀠 󸀠 (2) for any , ; (1) 𝑐1 ∘𝑐2 =(Re 𝑐1 ∘ Re 𝑐2, Im 𝑐1 ∘ Im 𝑐2) for any 𝑐1,𝑐2 ∈ ∗ 𝑎,̃ ̃𝑏, 𝑐∈𝐹̃ ∗(𝐾) 𝜌(̃ 𝑎,̃ ̃𝑏) ≤ 𝜌(̃ 𝑎,̃ 𝑐)̃ + 𝜌(̃ 𝑐,̃ ̃𝑏) 𝐹 (𝐾); (3) for any , . 󸀠 󸀠 󸀠 󸀠 ∗ ̃ ̃ ̃ (2) 𝑐⋅𝑐1 =(𝑎Re 𝑐1,𝑏Im 𝑐1) for any 𝑐1 ∈𝐹(𝐾) and 𝑐= 𝜌(𝑎, 𝑏) is called the fuzzy distance of fuzzy complex numbers ̃ (𝑎, 𝑏) ∈𝐾 (𝑐=𝑎+𝑖𝑏,𝑎,𝑏∈𝑅,𝑖=√−1). 𝑎̃ and 𝑏. The Scientific World Journal 3

∗ ̃ Example 5. Let 𝜌̃be a fuzzy distance on 𝐹 (𝑅).Thenthemap- to be convergent to 𝑍 according to distance 𝜌̃, denoted by 𝜌󸀠 :𝐹∗(𝐾) × 𝐹∗(𝐾) →∗ 𝐹 (𝑅) ̃ ̃ ping defined by (𝜌)̃ lim𝑛→∞𝑍𝑛 = 𝑍. 󸀠 󸀠 󸀠 󸀠 󸀠 󸀠 󸀠 𝜌 (𝑐1,𝑐2)=𝜌(̃ Re 𝑐1, Re 𝑐2)∨𝜌(̃ Im 𝑐1, Im 𝑐2) (2) Definition 11. Let 𝑍 be a nonempty complex number set, let 𝐹(𝑍) be the set of all complex fuzzy sets on 𝑍, and let 𝜌̃ be a ∗ is a fuzzy distance on 𝐹 (𝐾). fuzzy complex value metric on 𝐹(𝑍). A complex fuzzy set- value complex fuzzy measure is a mapping 𝜇̃ : 𝐹(𝑍) → {𝑐󸀠}⊂𝐹∗(𝐾) 𝑐󸀠 ∈ ∗ ∗ ̃ ̃ ̃ ̃ ∗ √ Definition 6 (see cf. [12]). Let 𝑛 and let 𝐹+ (𝐾) (where 𝐹+ (𝐾) =𝐴+𝑖 { 𝐵|𝐴, 𝐵∈𝐹+ (𝑅)}, 𝑖= −1) 𝐹∗(𝐾) {𝑐󸀠} 𝑐󸀠 . 𝑛 is said to converge to according to a fuzzy metric which satisfies the following conditions: 󸀠 ∗ 󸀠 󸀠 𝜌 on 𝐹 (𝐾) (written as lim𝑛→∞𝑐𝑛 =𝑐) if, for each 𝜀>0, 󸀠 󸀠 󸀠 (1) 𝜇(⌀)̃ =; 0 there exists a positive 𝑁 such that 𝜌 (𝑐𝑛,𝑐 )<𝜀for all 𝑛≥𝑁 . (2) for any 𝐴,̃ 𝐵̃ ∈ 𝐹(𝑍) satisfying 𝐴⊂̃ 𝐵̃, 𝜇(̃ 𝐴)̃ ≤ 𝜇(̃ 𝐵)̃ (i.e., Re 𝜇(̃ 𝐴)̃ ≤ Re 𝜇(̃ 𝐵)̃ and Im 𝜇(̃ 𝐴)̃ ≤ Im 𝜇(̃ 𝐵)̃ ); 3. Complex Fuzzy Set-Valued Complex ̃ (3) (lower semicontinuous) if {𝐴𝑛} ⊂ 𝐹(𝑍) with 𝐴𝑛 ⊂ Fuzzy Measures ̃ ̃ 𝐴𝑛+1, (𝑛 = 1,2,...),then(𝜌)̃ lim𝑛→∞𝜇(̃ 𝐴𝑛)= 𝜇(⋃̃ ∞ 𝐴̃ ) The notion of complex fuzzy measure on family of classical 𝑛=1 𝑛 ; ̃ ̃ sets was given in [26]. (4) (upper semicontinuous) if {𝐴𝑛} ⊂ 𝐹(𝑍) with 𝐴𝑛 ⊃ 𝐴̃ (𝑛=1,2,...) 𝜇(̃ 𝐴̃ ) ≠ ∞̃ 𝑛 ∧ ∧ 𝑛+1, and 𝑛0 for some 0,then 𝑅+=[0,+∞) 𝐶+={𝑥+𝑖𝑦| ̃ ∞ ̃ Definition 7 (see [26]). Let and let (𝜌)̃ lim𝑛→∞𝜇(̃ 𝐴𝑛)=𝜇(⋂̃ 𝑛=1 𝐴𝑛). ∧ + 𝑥, 𝑦 ∈ 𝑅 }. A fuzzy measure on a 𝜎-algebra 𝐴 composed of Apparently, a complex fuzzy set-value complex fuzzy ∧ + measure is also a kind of special generalized fuzzy measures. subsets of 𝑋 is a mapping 𝜇:𝐴𝐶 → which satisfies the following conditions: ∗ Definition 12. Amapping𝜇̃ : 𝐹(𝑍)+ →𝐹 (𝐾) is said to be (1) 𝜇(𝜙) =0; (1) 0-add if 𝜇(̃ 𝐴∪̃ 𝐵)̃ = 𝜇(̃ 𝐴)̃ for any 𝐴,̃ 𝐵̃ ∈ 𝐹(𝑍) 𝐴⊂𝐵 |𝜇(𝐴)| ≤ |𝜇(𝐵)| (2) if ,then ; satisfying 𝐴∪̃ 𝐵̃ ∈ 𝐹(𝑍) and 𝜇(̃ 𝐵)̃ = 0; {𝐴 }∞ ↑ 𝜇(⋃∞ 𝐴 )= 𝜇(𝐴 ) (3) if 𝑛 1 ,then 𝑛=1 𝑛 lim𝑛→∞ 𝑛 ; 𝜇(̃ 𝐴∩̃ 𝐵̃𝑐)=𝜇(̃ 𝐴)̃ ∞ (2) null-additive (briefly, 0-sub) if for (4) if {𝐴𝑛} ↓ and |𝜇(𝐴𝑛 )| < +∞ for some 𝑛0,then ̃ ̃ ̃ ̃𝑐 ̃ 1 0 any 𝐴, 𝐵 ∈ 𝐹(𝑍) satisfying 𝐴∩𝐵 ∈ 𝐹(𝑍) and 𝜇(̃ 𝐵) = 𝜇(⋂∞ 𝐴 )= 𝜇(𝐴 ) 𝑛=1 𝑛 lim𝑛→∞ 𝑛 . 0; In this paper we need an expansion of this notion. First (3) autocontinuous from above (briefly, autoc. ↓)if ̃ ̃ ̃ ̃ we defined the concept of fuzzy complex value distance. (𝜌)̃ lim𝑛𝜇(̃ 𝐴𝑛 ∪ 𝐵) = 𝜇(̃ 𝐵) for any for all {𝐴𝑛}⊂ ̃ ̃ ̃ ∗ ∗ ∗ 𝐹(𝑍) and 𝐵 ∈ 𝐹(𝑍) satisfying 𝐴𝑛 ∪ 𝐵 ∈ 𝐹(𝑍),and Definition 8. Amapping𝜌:𝐹̃ (𝐾) × 𝐹 (𝐾) → 𝐹 (𝐾) ̃ ̃ (𝜌)̃ lim𝑛𝜇(̃ 𝐴𝑛)=0; satisfying the following conditions is called a fuzzy complex ∗ value metric or a fuzzy complex value distance on 𝐹 (𝐾): (4) autocontinuous from below (briefly, autoc. ↑)if ̃ ̃ ̃ ̃ (𝜌)̃ lim𝑛𝜇(̃ 𝐴𝑛 ∩ 𝐵) = 𝜇(̃ 𝐵) for any for all {𝐴𝑛}⊂ 𝑎,̃ ̃𝑏∈𝐹∗(𝐾) 𝜌(̃ 𝑎,̃ ̃𝑏) ≥ 0 𝜌(̃ 𝑎,̃ ̃𝑏) = 0 ̃ ̃𝑐 ̃ (1) for any , ,and if 𝐹(𝑍) and 𝐵 ∈ 𝐹(𝑍) satisfying 𝐴𝑛 ∩ 𝐵 ∈ 𝐹(𝑍),and 𝑎=̃ ̃𝑏 ̃ ̃ and only if ; (𝜌)̃ lim𝑛𝜇(̃ 𝐴𝑛)=0; ̃ ∗ ̃ ̃ (2) for any 𝑎,̃ 𝑏∈𝐹 (𝐾), 𝜌(̃ 𝑎,̃ 𝑏) = 𝜌(̃ 𝑏, 𝑎)̃ ; (5)autocontinuousifitisbothautoc.↓ and autoc. ↑. ̃ ∗ ̃ ̃ (3) for any 𝑎,̃ 𝑏, 𝑐∈𝐹̃ (𝐾), 𝜌(̃ 𝑎,̃ 𝑏) ≤ 𝜌(̃ 𝑎,̃ 𝑐)̃ + 𝜌(̃ 𝑐,̃ 𝑏). Definition 13. A complex fuzzy set-value complex fuzzy ̃ 𝜇̃ 𝐹(𝑍) 𝜌(̃ 𝑎,̃ 𝑏) is called the fuzzy complex value distance of fuzzy measure on is said to be pseudo-null-additive (briefly, ̃ /𝐴̃ 𝐴̃ ∈ 𝐹(𝑍) 𝜇(̃ 𝐴)̃ ≠ ∞̃ complex numbers 𝑎̃ and 𝑏. P.0-add ,where with ) if it satisfies 𝜇(̃ 𝐵∩̃ 𝐶)̃ = 𝜇(̃ 𝐶)̃ for all 𝐵̃ ∈ 𝐹(𝑍) and all 𝐶∈̃ 𝐴̃ ∩ 𝐹(𝑍) = ∗ Remark 9. Itcanbeeasilyseenthatamapping𝜌:𝐹̃ (𝐾) × {𝐴∩̃ 𝐷|̃ 𝐷̃ ∈ 𝐹(𝑍)} with 𝜇(̃ 𝐴∩̃ 𝐵)̃ = 𝜇(̃ 𝐴)̃ .Itissaidtobe ∗ ∗ ∗ 𝐹 (𝐾)→𝐹(𝐾) is a fuzzy complex value metric on 𝐹 (𝐾) pseudo-null-subtraction (briefly, P.0-sud/𝐴̃,where𝐴̃ ∈ 𝐹(𝑍) 𝜌=̃ 𝜌̃ +𝑖𝜌̃ 𝜌̃ ̃ ̃ ̃ ̃ ̃ if and only if 1 2 for some two fuzzy metrics 1 and with 𝜇(̃ 𝐴) ≠ ∞̃) if it satisfies 𝜇(̃ 𝐵∩𝐶) = 𝜇(̃ 𝐶) for all 𝐵 ∈ 𝐹(𝑍) 𝜌̃ 𝐹∗(𝐾) 2 on . and all 𝐶∈̃ 𝐴̃ ∩ 𝐹(𝑍) with 𝜇(̃ 𝐴∩̃ 𝐵)̃ = 𝜇(̃ 𝐴)̃ . ̃ ∗ ̃ ∗ ∗ Definition 10. Let {𝑍𝑛}⊂𝐹 (𝐾), 𝑍∈𝐹 (𝐾),and𝜌:𝐹̃ (𝐾) × ∗ ∗ Definition 14. A complex fuzzy set-value complex fuzzy 𝐹 (𝐾)→𝐹(𝐾) be a fuzzy complex value distance, and let measure 𝜇̃ on fuzzy 𝜎-algebra F is said to have property ̃ ̃ ̃ ̃ ̃ ̃ √ 𝜌(̃ 𝑍𝑛, 𝑍) = 𝜌̃1(𝑍𝑛, 𝑍) +𝜌 𝑖̃2(𝑍𝑛, 𝑍) (𝑖= −1). If for each (PGP) if, for each 𝜀=𝜀1+𝑖𝜀2 >0, there exists a 𝛿=𝛿1+𝑖𝛿2 >0 ̃ ̃ ̃ ̃ ̃ ̃ 𝜀>0, there exists a positive integer 𝑁 such that 𝜌̃1(𝑍𝑛, 𝑍) < 𝜀 such that 𝜇(̃ 𝐸∪𝐹) < 𝜀 whenever 𝜇(̃ 𝐸) ∨ 𝜇(̃ 𝐹) <.It 𝛿 ̃ ̃ ̃ ̃ and 𝜌̃2(𝑍𝑛, 𝑍) < 𝜀 for all 𝑛≥𝑁hold, then {𝑍𝑛} is said is said to have property (S/A) if, for any {𝐵𝑛}⊂F with 4 The Scientific World Journal

(𝜌)̃ 𝜇(̃ 𝐵̃ )=0 {𝐵̃ } {𝐵̃ } {𝐵̃ }⊂F ⊂ 𝐹(𝑍) 𝐵̃ ↗ ⋃∞ 𝐵̃ ∈ F lim𝑛 𝑛 , there exists a subsequence 𝑛𝑘 of 𝑛 such Assume 𝑛 and 𝑛 𝑛=1 𝑛 . ∞ ∞ ̃ (⋃∞ 𝐵̃ )⊝𝐵̃ that 𝜇(̃ ⋂ ⋃ 𝐵𝑛 )=0.Itissaidtohaveproperty(S/B)if Then 𝑛=1 𝑛 𝑛 is a monotonic decrease sequence and 𝑗=1 𝑘=𝑗 𝑘 ∞ ̃ ̃ ̃ ̃ (⋃𝑛=1 𝐵𝑛)⊝𝐵𝑛 ↘⌀,so (𝜌)̃ lim𝑛𝜇(̃ 𝐵𝑛)=0for any {𝐵𝑛}⊂F. ∞ ̃ Re 𝜇(̃ ⋃𝐵𝑛) 4. Main Results 𝑛=1 Let 𝑋 be a set and 𝐹(𝑋) the set of all fuzzy sets on 𝑋.Thena ∞ ̃ ̃ ̃ subfamily F ⊂ 𝐹(𝑋) is addable if and only if it satisfies the = Re 𝜇̃ (((⋃𝐵𝑛)⊝𝐵𝑛)⊕𝐵𝑛) following conditions (see [21]): 𝑛=1 ∞ 𝑋∈F =(𝜌)̃ 𝜇((̃ ⋃𝐵̃ )⊝𝐵̃ )+(𝜌)̃ 𝜇(̃ 𝐵̃ ) (1) ; lim𝑛 Re 𝑛 𝑛 lim𝑛 Re 𝑛 𝑛=1 (2) if 𝐴,̃ 𝐵∈̃ F,then𝐴⊕̃ 𝐵̃, 𝐴⊝̃ 𝐵∈̃ F, =0+(𝜌)̃ 𝜇(̃ 𝐵̃ ) lim𝑛 Re 𝑛 where (𝐴⊕̃ 𝐵)(𝑥)̃ = min(1, 𝐴(𝑥)̃ + 𝐵(𝑥))̃ and (𝐴⊝̃ 𝐵)(𝑥)̃ = (0, 𝐴(𝑥)̃ − 𝐵(𝑥))̃ ( 𝑥∈𝑋) =(𝜌)̃ 𝜇(̃ 𝐵̃ ). max for all . lim𝑛 Re 𝑛 We first have the following result. (5) Theorem 15. 𝜇̃ Every fuzzy complex measure on a addable 𝜇(̃ ⋃∞ 𝐵̃ )=(𝜌)̃ 𝜇(̃ 𝐵̃ ) F ⊂ 𝐹(𝑍) Similarly we can prove Im 𝑛=1 𝑛 lim𝑛 Im 𝑛 ; class is a complex fuzzy value fuzzy complex 𝜇̃ 𝜇̃ measure on F. thus is also lower continuous. In summary, is a complex fuzzy set-value complex fuzzy measure. Proof. We only prove the upper continuity and lower conti- Theorem 16. 𝜇̃ {𝐴̃ }⊂F ⊂ 𝐹(𝑍) 𝐴̃ ↘⋂∞ 𝐴̃ ∈ F Every complex fuzzy set-value complex fuzzy nuity of .Suppose 𝑛 , 𝑛 𝑛=1 𝑛 , 𝜇̃ 𝜎 F ⊂ 𝐹(𝑍) 𝜇(̃ 𝐴̃ ) ≠ ∞̃ 𝑛 𝜇̃ measure on a fuzzy -algebra is exhaustive. and 𝑛0 for some 0. By monotonicity of ,wehave 0≤ 𝜇(̃ 𝐴̃ )≤ 𝜇(̃ 𝐴̃ ) 0≤ 𝜇(̃ 𝐴̃ )≤ 𝜇(̃ 𝐴̃ ) {𝐴̃ }⊂F Re 𝑛 Re 𝑛0 and Im 𝑛 Im 𝑛0 Proof. Suppose 𝑛 is a disjoin sequence; then ̃ ̃ ̃ ∞ ̃ for any 𝑛≥𝑛0.Since𝐴𝑛 ⊝ 𝐴𝑛 ↗ 𝐴𝑛 ⊝(⋂ 𝐴𝑛),wehave 0 0 𝑛=1 ∞ ∞ ∞ ̃ ̃ ⋃𝐴𝑘 ↘ ⋂⋃𝐴𝑘 =⌀. (6) 𝜇(̃ 𝐴̃ )=(𝜌)̃ 𝜇((̃ 𝐴̃ ⊝ 𝐴̃ )⊕𝐴̃ ) 𝑘=𝑛 𝑛=1𝑘=𝑛 Re 𝑛0 lim𝑛 Re 𝑛0 𝑛 𝑛 ⋂∞ ⋃∞ 𝐴̃ =⌀̸ (∧∞ ∨∞ 𝐴̃ (𝑧 )) > 0 ̃ ̃ ̃ Assume 𝑛=1 𝑘=𝑛 𝑘 ;thenRe 𝑛=1 𝑘=𝑛 𝑘 0 for =(𝜌)̃ lim Re 𝜇(̃ 𝐴𝑛 ⊝ 𝐴𝑛)+(𝜌)̃ lim Re 𝜇(̃ 𝐴𝑛) ∞ ∞ ̃ 𝑛 0 𝑛 some 𝑧0 ∈𝑍or Im(∧𝑛=1∨𝑘=𝑛𝐴𝑘(𝑧0)) > 0 for some 𝑧0 ∈𝑍; ∞ ̃ ∞ ̃ that is, Re(∨𝑘=𝑛𝐴𝑘(𝑧0)) > 0 for all 𝑛≥1or Im(∨𝑘=𝑛𝐴𝑘(𝑧0)) > ∞ 0 𝑛≥1 = 𝜇(̃ 𝐴̃ ⊝(⋂𝐴̃ )) + (𝜌)̃ 𝜇(̃ 𝐴̃ ). for all . Without loss of generality, we assume the Re 𝑛0 𝑛 lim𝑛 Re 𝑛 𝑘 𝑘 𝑛=1 first. Then there are two distinct indexes 𝑛1 and 𝑛2 such that ̃ ̃ ̃ Re 𝐴𝑘 (𝑧0)>0and Re 𝐴𝑘 (𝑧0)>0(and, thus, Re(𝐴𝑘 (𝑧0)∧ (3) 𝑛1 𝑛2 𝑛1 ̃ ̃ 𝐴𝑘 (𝑧0)) > 0), which conflicts with the fact that {𝐴𝑛} is a 𝑛2 Thereby ∞ ̃ ∞ ∞ ̃ disjoin sequence. Therefore ⋃𝑘=𝑛 𝐴𝑘 ↘ ⋂𝑛=1 ⋃𝑘=𝑛 𝐴𝑘 =⌀ holds. Thus ∞ 𝜇(̃ 𝐴̃ )+ 𝜇(̃ ⋂𝐴̃ ) ∞ Re 𝑛0 Re 𝑛 𝑛=1 (𝜌)̃ 𝜇(̃ 𝐴̃ )≤(𝜌)̃ 𝜇(̃ ⋃𝐴̃ ) lim𝑛 Re 𝑛 lim𝑛 Re 𝑘 𝑘=𝑛 ∞ = 𝜇(̃ 𝐴̃ ⊝(⋂𝐴̃ )) + (𝜌)̃ 𝜇(̃ 𝐴̃ ) Re 𝑛0 𝑛 Re 𝑛 = Re (⌀) =0, 𝑛=1 (4) (7) ∞ ∞ ̃ ̃ ̃ (𝜌)̃ lim Im 𝜇(̃ 𝐴𝑛)≤(𝜌)̃ lim Im 𝜇(̃ ⋃𝐴𝑘) + 𝜇(̃ ⋂𝐴𝑛) 𝑛 𝑛 𝑛=1 𝑘=𝑛

= 𝜇(̃ 𝐴̃ )+(𝜌)̃ 𝜇(̃ 𝐴̃ ). = Im (⌀) =0. Re 𝑛0 lim𝑛 Re 𝑛 ̃ Thenweget(𝜌)̃ lim𝑛𝜇(̃ 𝐴𝑛)≤0. ̃ ∞ ̃ It follows that (𝜌)̃ lim𝑛 Re 𝜇(̃ 𝐴𝑛)=Re 𝜇(⋂̃ 𝑛=1 𝐴𝑛).Similarly, On the other hand, since 𝜇̃ is complex fuzzy set- ̃ ∞ ̃ 𝜎 F we can prove (𝜌)̃ lim𝑛 Im 𝜇(̃ 𝐴𝑛)=Im 𝜇(̃ ⋂ 𝐴𝑛),which value complex fuzzy measure on the fuzzy -algebra , 𝑛=1 ̃ ̃ means that 𝜇̃ is upper continuous. (𝜌)̃ lim𝑛𝜇(̃ 𝐴𝑛)≥0,so(𝜌)̃ lim𝑛𝜇(̃ 𝐴𝑛)=0. The Scientific World Journal 5

̃ ̃ ∗ ̃ ̃ ̃ ̃ Theorem 17. If 𝜇(̃ F)={𝜇(̃ 𝐴) | 𝐴∈F}∈𝐴 ,then As 𝐸∪𝐴𝑛 ↘ 𝐸∪𝐴 and 𝜇̃ is upper continuous and 0-addable, we have 𝜇(̃ 𝐴̃ )≤(𝜌)̃ 𝜇(̃ 𝐴̃ ) Re lim 𝑛 lim Re 𝑛 (𝜌)̃ 𝜇(̃ 𝐸∪̃ 𝐴̃ )= 𝜇(̃ 𝐸∪̃ 𝐴)̃ = 𝜇(̃ 𝐸)̃ , 𝑛→∞ 𝑛→∞ lim𝑛 Re 𝑛 Re Re ̃ (11) ≤(𝜌)̃ lim Re 𝜇(̃ 𝐴𝑛) (𝜌)̃ 𝜇(̃ 𝐸∪̃ 𝐴̃ )= 𝜇(̃ 𝐸∪̃ 𝐴)̃ = 𝜇(̃ 𝐸)̃ . 𝑛→∞ lim𝑛 Im 𝑛 Im Im ≤ 𝜇(̃ 𝐴̃ ), (𝜌)̃ 𝜇(̃ 𝐸∪̃ 𝐴̃ )=𝜇(̃ 𝐸)̃ Re 𝑛→∞lim 𝑛 So lim𝑛 𝑛 holds. Similar to Theorem 18,wehavethefollowing. ̃ ̃ Im 𝜇(̃ lim 𝐴𝑛)≤(𝜌)̃ lim Im 𝜇(̃ 𝐴𝑛) 𝑛→∞ 𝑛→∞ ∗ ̃ ̃ ̃ ̃ Theorem 19. Let 𝜇:𝐹(𝑍)→̃ + 𝐹(𝐾) =𝐴+𝑖 { 𝐵:𝐴, 𝐵∈ ̃ ̃ 𝐹∗(𝑅), 𝑖 = √−1} 𝐸̃ ∈ 𝐹(𝑍) 𝜇̃ ≤(𝜌)̃ lim Im 𝜇(̃ 𝐴𝑛)≤Im 𝜇(̃ lim 𝐴𝑛). + and .If is 0-subtractable and 𝑛→∞ 𝑛→∞ ̃ ̃ continuous on 𝐹(𝑍),then,forany{𝐸𝑛} ⊂ 𝐹(𝑍) satisfying 𝐸𝑛 ⊃ (8) ̃ ̃ ̃ ̃ ̃𝑐 𝐸𝑛+1 (𝑛=1,2,...)and (𝜌)̃ lim𝑛𝜇(̃ 𝐸𝑛)=0, (𝜌)̃ lim𝑛𝜇(̃ 𝐸∩𝐸𝑛)= ∞ ̃ ∞ ∞ ̃ ̃ Proof. Since ⋂𝑛=𝑘 𝐴𝑛 ↗ about 𝑘,Re𝜇(̃ ⋃𝑘=1 ⋂𝑛=𝑘 𝐴𝑛)= 𝜇(̃ 𝐸). ∞ ̃ ∞ ∞ ̃ (𝜌)̃ lim𝑛 Re 𝜇(⋂̃ 𝑛=𝑘 𝐴𝑛) and Im 𝜇(⋃̃ 𝑘=1 ⋂𝑛=𝑘 𝐴𝑛)= ∞ ̃ Theorem 20. 𝐴∈̃ F ⊂ 𝐹(𝑍) 𝜇̃ (𝜌)̃ lim𝑛 Im 𝜇(⋂̃ 𝑛=𝑘 𝐴𝑛). Assume that , is a complex 𝜇(̃ ⋂∞ 𝐴̃ )≤ 𝜇(̃ 𝐴̃ ) 𝜇(̃ ⋂∞ 𝐴̃ )≤ fuzzy set-value complex fuzzy measure on 𝐹(𝑍) which is As Re 𝑛=𝑘 𝑛 Re 𝑛 and Im 𝑛=𝑘 𝑛 ̃ ̃ ̃ ̃ ∞ ̃ ̃ pseudo-zero addable about 𝐴,and𝜇(̃ 𝐴) ≠ ∞̃.If(𝜌)̃ lim𝑛𝜇(̃ 𝐴∩ Im 𝜇(̃ 𝐴𝑛) for all 𝑛≥𝑘,Re𝜇(⋂̃ 𝐴𝑛)≤inf𝑛≥𝑘 Re 𝜇(̃ 𝐴𝑛) and 𝑛=𝑘 𝐵̃ )=𝜇(̃ 𝐴)̃ {𝐵̃ }⊂F 𝐵̃ ↑ 𝜇(̃ ⋂∞ 𝐴̃ )≤ 𝜇(̃ 𝐴̃ ) 𝑛 for any 𝑛 satisfying 𝑛 ,then Im 𝑛=𝑘 𝑛 inf𝑛≥𝑘 Im 𝑛 . Therefore ̃ ̃ ̃𝑐 ̃ ̃ ̃ (𝜌)̃ lim𝑛𝜇(̃ 𝐶∪(𝐴∩𝐵𝑛)) = 𝜇(̃ 𝐶) for any 𝐶∈𝐴∩F. ∞ ∞ ̃ ̃ Re 𝜇(̃ lim 𝐴𝑛)=Re 𝜇(̃ ⋃⋂𝐴𝑛) ̃ ∞ ̃ Proof. Let 𝐵=⋃𝑛=1 𝐵𝑛.As𝜇̃ is lower continuous, we have 𝑛→∞ 𝑘=1𝑛=𝑘 ∞ ̃ ̃ ̃ ̃ ̃ ≤(𝜌)̃ lim inf Re 𝜇(̃ 𝐴𝑛) Re 𝜇(̃ 𝐴∩𝐵) = Re 𝜇(̃ 𝐴∩⋃𝐵𝑛) 𝑘→∞𝑛≥𝑘 𝑛=1 =(𝜌)̃ 𝜇(̃ 𝐴̃ ), ∞ lim Re 𝑛 = 𝜇(̃ ⋃ (𝐴∩̃ 𝐵̃ )) = (𝜌)̃ (𝐴∩̃ 𝐵̃ ), 𝑛→∞ Re 𝑛 lim𝑛 Re 𝑛 (9) 𝑛=1 ∞ ∞ ̃ ̃ ∞ Im 𝜇(̃ lim 𝐴𝑛)=Im 𝜇(̃ ⋃⋂𝐴𝑛) 𝜇(̃ 𝐴∩̃ 𝐵)̃ = 𝜇(̃ 𝐴∩̃ ⋃𝐵̃ ) 𝑛→∞ Im Im 𝑛 𝑘=1𝑛=𝑘 𝑛=1 ̃ ∞ ≤(𝜌)̃ lim inf Im 𝜇(̃ 𝐴𝑛) ̃ ̃ 𝑘→∞𝑛≥𝑘 = Im 𝜇(̃ ⋃ (𝐴∩𝐵𝑛)) 𝑛=1 ̃ =(𝜌)̃ lim Im 𝜇(̃ 𝐴𝑛), =(𝜌)̃ (𝐴∩̃ 𝐵̃ ). 𝑛→∞ lim𝑛 Im 𝑛 ̃ ̃ (12) which means Re 𝜇(̃ lim𝑛→∞𝐴𝑛)≤(𝜌)̃ lim𝑛→∞ Re 𝜇(̃ 𝐴𝑛). 𝜇(̃ 𝐴̃ )≤(𝜌)̃ 𝜇(̃ 𝐴̃ ) ̃ ̃ ̃ ̃ ̃𝑐 ̃ ̃𝑐 ̃ ̃ ̃𝑐 Similarly, Im lim𝑛→∞ 𝑛 lim𝑛→∞ Im 𝑛 . Therefore 𝜇(̃ 𝐴∩𝐵) = 𝜇(̃ 𝐴).As𝐴∩𝐵𝑛 ↘ 𝐴∩𝐵 , 𝐶∪(𝐴∩𝐵𝑛)↘ 𝑐 𝐶∪(̃ 𝐴∩̃ 𝐵̃ ) for any 𝐶∈̃ 𝐴∩̃ F.ByuppercontinuityandP.0- /𝐴̃ 𝜇̃ From properties of upper limits and lower limits we can add of ,wehave see the following theorem holds. (𝜌)̃ 𝜇(̃ 𝐶∪(̃ 𝐴∩̃ 𝐵̃𝑐 )) lim𝑛 Re 𝑛 ∗ ̃ ̃ ̃ ̃ Theorem 18. Let 𝜇̃ : 𝐹(𝑍)+ →𝐹 (𝐾) =𝐴+𝑖 { 𝐵:𝐴, 𝐵∈ ∞ 𝐹∗(𝑅), 𝑖 = √−1} 𝐸̃ ∈ 𝐹(𝑍) 𝜇̃ ̃ ̃ ̃𝑐 + and .If is 0-addable and upper = Re 𝜇(̃ ⋂ (𝐶∪(𝐴∩𝐵𝑛))) ̃ ̃ 𝑛=1 continuous on 𝐹(𝑍),then,forany{𝐴𝑛} ⊂ 𝐹(𝑍) satisfying 𝐴𝑛 ⊃ ̃ ̃ ̃ ̃ ̃ 𝐴𝑛+1 (𝑛= 1,2,...), (𝜌)̃ lim𝑛𝜇(̃ 𝐴𝑛)=0 and 𝜇(̃ 𝐸∪𝐴𝑛 ) ≠ ∞̃, ̃ ̃ ̃𝑐 ̃ 0 = Re 𝜇(̃ 𝐶∪(𝐴∩𝐵 )) = Re 𝜇(̃ 𝐶) , (𝜌)̃ 𝜇(̃ 𝐸∪̃ 𝐴̃ )=𝜇(̃ 𝐸)̃ lim𝑛 𝑛 . (13) (𝜌)̃ 𝜇(̃ 𝐶∪(̃ 𝐴∩̃ 𝐵̃𝑐 )) ̃ ∞ ̃ lim𝑛 Im 𝑛 Proof. Let 𝐴=⋂𝑛=1 𝐴𝑛.Since𝜇̃ is upper continuous, ∞ ∞ 𝑐 ̃ ̃ ̃ ̃ = Im 𝜇(̃ ⋂ (𝐶∪(̃ 𝐴∩̃ 𝐵̃ ))) Re 𝜇(̃ 𝐴) = Re 𝜇(̃ ⋂𝐴𝑛)=(𝜌)̃ lim Re 𝜇(̃ 𝐴𝑛)=0, 𝑛 𝑛 𝑛=1 𝑛=1 (10) 𝑐 ∞ = Im 𝜇(̃ 𝐶∪(̃ 𝐴∩̃ 𝐵̃ )) = Im 𝜇(̃ 𝐶)̃ . 𝜇(̃ 𝐴)̃ = 𝜇(̃ ⋂𝐴̃ )=(𝜌)̃ 𝜇(̃ 𝐴̃ )=0.̃ Im Im 𝑛 lim𝑛 Im 𝑛 𝑛=1 ̃ ̃ ̃𝑐 ̃ Hence (𝜌)̃ lim𝑛𝜇(̃ 𝐶∪(𝐴∩𝐵𝑛)) = 𝜇(̃ 𝐶). 6 The Scientific World Journal

󸀠 󸀠󸀠 󸀠 󸀠󸀠 Similarly, we have the following. For 𝛿2 =𝛿2 +𝑖𝛿2 >0and there exists 𝛿3 =𝛿3 +𝑖𝛿3 >0, 󸀠 󸀠 2 󸀠󸀠 󸀠󸀠 2 ̃ 𝛿3 ∈(0,𝛿2 ∧𝜀/2 ),and𝛿3 ∈(0,𝛿2 ∧𝜀/2 ) such that Re(𝜇(̃ 𝐸)∨ Theorem 21. Suppose that 𝐴∈̃ F ⊂ 𝐹(𝑍), 𝜇̃ is a ̃ 󸀠 ̃ ̃ 󸀠󸀠 𝜇(̃ 𝐹)) <3 𝛿 ,Im(𝜇(̃ 𝐸) ∨ 𝜇(̃ 𝐹)) <3 𝛿 ⇒ complex fuzzy set-value complex fuzzy measure on 𝐹(𝑍) 𝐴̃ 𝜇(̃ 𝐴)̃ ≠ ∞̃ ̃ ̃ 󸀠 ̃ ̃ 󸀠󸀠 which is pseudo-zero subtractable about ,and .If Re 𝜇(̃ 𝐸∪𝐹) <2 𝛿 , Im 𝜇(̃ 𝐸∪𝐹) <2 𝛿 . (16) ̃ ̃ ̃ ̃ ̃ (𝜌)̃ lim𝑛𝜇(̃ 𝐴∩𝐵𝑛)=𝜇(̃ 𝐴) for any {𝐵𝑛}⊂F satisfying 𝐵𝑛 ↑, ̃ ̃ ̃ ̃ ̃ 𝛿 =𝛿󸀠 +𝑖𝛿󸀠󸀠 >0𝛿󸀠 ∈(0,𝛿󸀠 ∧𝜀/22) 𝛿󸀠󸀠 ∈(0,𝛿󸀠󸀠 ∧ then (𝜌)̃ lim𝑛𝜇(̃ 𝐶∩𝐵𝑛)=𝜇(̃ 𝐶) for any 𝐶∈𝐴∩F. For 3 3 3 , 3 2 , 3 2 2 ̃ 𝜀/2 ) since (𝜌)̃ lim𝑛𝜇(̃ 𝐸𝑛)=0, there exists 𝑛3 >𝑛2,suchthat Theorem 22. 𝜇̃ ̃ 󸀠 ̃ 󸀠󸀠 ̃ ̃ Let be a complex fuzzy set-value complex fuzzy Re 𝜇(̃ 𝐸𝑛 )<𝛿 and Im 𝜇(̃ 𝐸𝑛 )<𝛿 . Therefore 𝜇(̃ 𝐸𝑛 ∪𝐹) <2 𝛿 , ̃ 3 3 3 3 3 measure on a fuzzy 𝜎-algebra F ⊂ 𝐹(𝑍) and 𝜇(̃ F)={𝜇(̃ 𝐴) : 𝜇(̃ 𝐸̃ ∪ 𝐸̃ ∪ 𝐹)̃ < 𝛿 𝜇(̃ 𝐸̃ ∪ 𝐸̃ ∪ 𝐸̃ ∪ 𝐹)̃ < 𝜀 ∗ 𝑛3 𝑛2 1,and 𝑛3 𝑛2 𝑛1 . 𝐴∈̃ F}∈𝐴 .If𝜇̃ is 𝑎𝑢𝑡𝑜𝑐. ↓,then𝜇̃ possesses (PGP) property. Generally we can get 𝑛𝑘+1 >𝑛𝑘 >𝑛𝑘−1 >⋅⋅⋅>𝑛1, 𝑘−1 𝑟+1 ̃ and 𝛿𝑘 <𝛿𝑘−1 ∧𝜀/2 ,suchthat𝜇(̃ ⋃ 𝐸𝑛 )<𝛿𝑘−1, (𝑘 = Proof. Suppose that 𝜇̃ does not possess (P.G.P)property; then 𝑖=𝑘 𝑖 𝜀 =𝜀󸀠 +𝑖𝜀󸀠󸀠 𝜀󸀠 ,𝜀󸀠󸀠 1,2,3,...,𝑟+1,𝑟≥1). there exists an 0 0 0 (where 0 0 are positive real ̃ ∞ ̃ ̃ ∞ ∞ ̃ ∞ ̃ Let 𝐵𝑘 = ⋃ 𝐸𝑛 and 𝐸=⋂ ⋃ 𝐸𝑛 = ⋂ 𝐵𝑘;then numbers) such that, for any natural numbers 𝑛,, 𝑚 there exist 𝑖=𝑘 𝑖 𝑘=2 𝑖=𝑘 𝑖 𝑘=1 ̃ ̃ 𝐵̃ ↘ 𝐸̃ 𝜇(̃ 𝐵̃ )=𝜇(̃ ⋃∞ 𝐸̃ )≤𝛿 ,(𝑘≥1) {𝐸𝑛}, {𝐹𝑛}⊂F such that 𝑘 , 𝑘 𝑖=𝑘 𝑛𝑖 𝑘−1 . Hence 𝜇(̃ 𝐸)̃ =;thatis, 0 𝜇̃ possesses (SA) property. ̃ ̃ 1 Re 𝜇(̃ 𝐸𝑛)∨Re 𝜇(̃ 𝐹𝑛)< , 𝑛 Conflict of Interests 1 𝜇(̃ 𝐸̃ )∨ 𝜇(̃ 𝐹̃ )< , (14) Im 𝑛 Im 𝑛 𝑚 The authors declare that there is no conflict of interests regarding the publication of this paper. ̃ ̃ 󸀠 ̃ ̃ 󸀠󸀠 Re 𝜇(̃ 𝐸𝑛 ∪ 𝐹𝑛) <𝜀̸ 0, Im 𝜇(̃ 𝐸𝑛 ∪ 𝐹𝑛) <𝜀̸ 0 . Acknowledgments ̃ ̃ ̃ Thus 𝜇(̃ 𝐸𝑛 ∪ 𝐹𝑛) <𝜀̸ 0 and, thus, (𝜌)̃ lim𝑛 Re 𝜇(̃ 𝐸𝑛)= ̃ ̃ This work was supported by the International Science and (𝜌)̃ lim𝑛 Re 𝜇(̃ 𝐹𝑛)=0and (𝜌)̃ lim𝑛 Im 𝜇(̃ 𝐸𝑛)= (𝜌)̃ 𝜇(̃ 𝐹̃ )=0 𝜇̃ Technology Cooperation Foundation of China (Grant no. lim𝑛 Im 𝑛 .Fromupperautocontinuityof , 2012DFA11270) and the National Natural Science Foundation we have of China (Grant no. 11071151). ̃ ̃ ̃ ̃ (𝜌)̃ lim Re 𝜇(̃ 𝐸𝑛 ∪ 𝐹𝑛)=0, (𝜌)̃ lim Im 𝜇(̃ 𝐸𝑛 ∪ 𝐹𝑛)=0. 𝑛 𝑛 References (15) [1] G. Choquet, “Theory of capacities,” Annales de l'Institut Fourier, ̃ ̃ Therefore 𝜇(̃ 𝐸𝑛 ∪ 𝐹𝑛)<𝜀0 for some 𝑛0 ≥1. This conflicts with vol. 5, pp. 131–295, 1953. the hypothesis. [2] M. Sugeno, Theoryoffuzzyintegralsanditsapplications[Ph.D. thesis], Tokyo Institute of Technology, 1974. Theorem 23. Suppose that 𝜇̃ possesses (P.G.P) property. If [3]L.A.Zadeh,“Fuzzysetsasabasisforatheoryofpossibility,” ̃ ̃ Fuzzy Sets and Systems,vol.1,no.1,pp.3–28,1978. {𝐸𝑛}⊂F and (𝜌)̃ lim𝑛𝜇(̃ 𝐸𝑛)=0, then there exists a sequence {𝛿𝑛} of real numbers satisfying 𝛿𝑛 >0(for all n) and 𝛿𝑛 ↘0 [4] H. Hermes, “Calculus of set valued function and control,” ̃ ̃ ∞ ̃ Journal of Mathematics and Mechanics,vol.18,no.1,pp.47–60, and a subsequence {𝐸𝑛 } of {𝐸𝑛} such that 𝜇(⋃̃ 𝐸𝑛 )<𝛿𝑘 𝑘 𝑖=𝑘+1 𝑖 1968. (forall 𝑘≥1). Furthermore, 𝜇̃ possesses (SA) property. [5] N. S. Papageorgiou, “Contributions to the theory of set-valued 󸀠 󸀠󸀠 󸀠 󸀠󸀠 󸀠 functions and set-valued measures,” Transactions of the Ameri- Proof. For any real numbers 𝜀 ,𝜀 ,𝛿1,𝛿1 >0,let𝜀=𝜀+ 󸀠󸀠 󸀠 󸀠󸀠 󸀠 󸀠 󸀠󸀠 󸀠 can Mathematical Society, vol. 304, pp. 245–265, 1982. 𝑖𝜀 ,𝛿1 =𝛿1 +𝑖𝛿1 , 𝛿1 ∈(0,𝜀),and𝛿1 ∈(0,𝜀).Since𝜇̃ 𝛿 ∈(0,𝜀) [6]E.KleinandA.C.Thompson,Theory of Correspondences,John possesses (P.G.P) property, there exists a 1 such that Wiley & Sons, New York, NY, USA, 1984. 𝜇(̃ 𝐸∪̃ 𝐹)̃ <󸀠 𝜀 𝜇(̃ 𝐸∪̃ 𝐹)̃ <󸀠 𝜀 𝜇(̃ 𝐸)̃ ∨ Re and Im whenever Re [7] W. Zhang, “Measure fuzzy numbers,” Chinese Science Bulletin, 𝜇(̃ 𝐹)̃ <󸀠 𝛿 𝜇(̃ 𝐸)̃ ∨ 𝜇(̃ 𝐹)̃ <󸀠󸀠 𝛿 (𝜌)̃ 𝜇(̃ 𝐸̃ )=0 1 and Im 1 .Since lim𝑛 𝑛 , vol. 23, pp. 9–10, 1986. 𝑛 𝜇(̃ 𝐸̃ ∪𝐹)̃ < 𝜀󸀠 𝜇(̃ 𝐸̃ ∪𝐹)̃ < there exists an 1 such that Re 𝑛1 ,Im 𝑛1 [8]C.X.Wu,D.L.Zhang,C.M.Guo,andC.Wu,“Fuzzynumber 𝜀󸀠󸀠 𝜇(̃ 𝐸̃ )<𝛿󸀠 𝜇(̃ 𝐸̃ )<𝛿󸀠󸀠 fuzzy measures and fuzzy integrals. (I). Fuzzy integrals of whenever Re 𝑛1 1 and Im 𝑛1 1 . Therefore ̃ ̃ functions with respect to fuzzy number fuzzy measures,” Fuzzy 𝜇(̃ 𝐸𝑛 ∪𝐹) < 𝜀.Since𝜇̃ possesses (P.G.P) property, there exists 1 Sets and Systems, vol. 98, no. 3, pp. 355–360, 1998. 𝛿 =𝛿󸀠 +𝑖𝛿󸀠󸀠 𝛿󸀠 ∈(0,𝛿󸀠 ∧𝜀󸀠/2) 𝛿󸀠󸀠 ∈(0,𝛿󸀠󸀠 ∧ a 2 2 2 such that 2 1 , 2 1 [9]C.M.Guo,D.L.Zhang,andC.X.Wu,“Fuzzy-valuedfuzzy 󸀠󸀠 ̃ ̃ 󸀠 ̃ ̃ 󸀠󸀠 𝜀 /2),Re𝜇(̃ 𝐸∪𝐹) <1 𝛿 ,andIm𝜇(̃ 𝐸∪𝐹) <2 𝛿 whenever measures and generalized fuzzy integrals,” Fuzzy Sets and ̃ ̃ 󸀠 ̃ ̃ 󸀠󸀠 󸀠 Re(𝜇(̃ 𝐸)∨𝜇(̃ 𝐹)) <2 𝛿 and Im(𝜇(̃ 𝐸)∨𝜇(̃ 𝐹)) <2 𝛿 .As𝛿2 =𝛿2 + Systems, vol. 97, no. 2, pp. 255–260, 1998. 𝑖𝛿󸀠󸀠 >0 𝑛 >𝑛 𝜇(̃ 𝐸̃ )<𝛿󸀠 [10] C. X. Wu, D. L. Zhang, B. Zhang et al., “Fuzzy number fuzzy 2 , there exists an 2 1 such that Re 𝑛2 2 and ̃ 󸀠󸀠 ̃ ̃ 󸀠 ̃ ̃ 󸀠󸀠 measure and fuzzy integrals (I): fuzzy-valued functions with 𝜇(̃ 𝐸𝑛 )<𝛿 𝜇(̃ 𝐸𝑛 ∪𝐹) < 𝛿 𝜇(̃ 𝐸𝑛 ∪𝐹) < 𝛿 Im 2 2 .HenceRe 2 1,Im 2 2 , respect to fuzzy number fuzzy measure on fuzzy sets,” Fuzzy Sets 𝜇(̃ 𝐸̃ ∪𝐸̃ ∪𝐹)̃ < 𝜀󸀠 𝜇(̃ 𝐸̃ ∪𝐸̃ ∪𝐹)̃ < 𝜀󸀠󸀠 and, thus, Re 𝑛1 𝑛2 and Im 𝑛1 𝑛2 . and Systems,vol.107,no.2,pp.219–226,1999. The Scientific World Journal 7

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