Extension of a class of decomposable measures via generalized pseudo-metrics

JID:FSS AID:7105 /FLA

[m3SC+; v1.237; Prn:18/10/2016; 12:47] P.1 (1-14)

Available online at www.sciencedirect.com

ScienceDirect Fuzzy Sets and Systems ••• (••••) •••–••• www.elsevier.com/locate/fss

Extension of a class of decomposable measures via generalized pseudo-metrics ✩ Jialiang Xie a,b , Qingguo Li a,∗ , Chongxia Lu a , Lankun Guo c , Shuili Chen d a College of Mathematics and Econometrics, Hunan University, Changsha, Hunan, 410082, PR China b College of Science, Jimei University, Xiamen, Fujian, 361021, PR China c College of Mathematics and Computer Science, Key Laboratory of High Performance Computing and Stochastic Information Processing, Hunan

Normal University, Changsha, Hunan, 410012, PR China d Chengyi University College, Jimei University, Xiamen, Fujian, 361021, PR China

Received 10 January 2016; received in revised form 30 September 2016; accepted 6 October 2016

Abstract This study considers the application of generalized pseudo-metrics to the extension of decomposable measures. We prove that the extension of a non-strict Archimedean t-conorm-based σ -decomposable measure can be formulated as the closure of a subset of a certain generalized pseudo-metric space. We show that the extension via generalized pseudo-metrics is equivalent to the completion of t-conorm-based σ -decomposable measures and the well-known Carathéodory extension. © 2016 Elsevier B.V. All rights reserved.

Keywords: Decomposable measure; Generalized pseudo-metric space; T-conorm; T-norm

1. Introduction In the past 40 years, a substantial theory of nonadditive measures (also called fuzzy measures) has been developed. One class of fuzzy measures comprises triangular conorm decomposable measures (⊥-decomposable measures) [6,25,48], which include the λ-additive measure, probability measure, and possibility measure [33,47] as special cases. Many elegant results have been obtained regarding ⊥-decomposable measures and related integrals [1,4,13,17,19, 22–24,27,34,37,38,42,50]. Based on these measures and related integrals, pseudo-analysis [36] was developed as a generalization of the classical analysis. In addition, the tools of pseudo-analysis have many applications in various fields such as decision theory, optimization problems, nonlinear partial differential equations, hybrid utility, pattern recognition, and aggregation analysis [2,7,8,10,13,20,29,30,32,33,35,36,43,46].

✩

Fully documented templates are available in the elsarticle package on CTAN.

* Corresponding author.

E-mail addresses: [email protected] (J. Xie), [email protected] (Q. Li). http://dx.doi.org/10.1016/j.fss.2016.10.003 0165-0114/© 2016 Elsevier B.V. All rights reserved.

JID:FSS AID:7105 /FLA

2

[m3SC+; v1.237; Prn:18/10/2016; 12:47] P.2 (1-14)

J. Xie et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

It is well known that probabilistic metric spaces [44] and fuzzy metric spaces [9,11,21] constructed via a triangular norm are generalizations of the classic metric spaces. The connections among ⊥-decomposable measures, various probabilistic metrics, and fuzzy metrics have been studied previously [12,15,16,28,39–41,45]. It should be noted that Qiu et al. [41] developed an approach for constructing a KM fuzzy pseudo-metric from a given ⊥-decomposable measure. Using the KM fuzzy pseudo-metric, a class of decomposable measures can be extended to a σ -algebra. Subsequently, we constructed a pseudo-metric (in the sense of Pap) based on the measurable sets of a given σ -⊥-decomposable measure, and we also analyzed the connection between the σ -⊥-decomposable measure and induced pseudo-metric [49]. In the present study, we apply the pseudo-metric constructed in [49] to the extension of σ -⊥-decomposable measures. The approach employed for extending fuzzy measures is an interesting topic in the theory of fuzzy measures. However, as noted by Wang and Klir [47], although it is impossible to provide a unified extension theorem for all types of fuzzy measures, several extension theorems can be established for some special classes of fuzzy measures. The necessary and sufficient conditions for extending a submeasure (a special null-additive monotone set function) from a ring to the generated σ -ring were first obtained by Dobrakov [5]. Similarly, Pap [26] gave the necessary and sufficient conditions for the extension of ⊥-decomposable measures to monotone order continuous ⊥-subdecomposable set functions. Thus, a theorem based on extending null-additive set functions from a ring to the algebra generated by the ring was proved by Pap [31]. [41] asked whether a σ -⊥-decomposable measure on an algebra A can be extended to a unique σ -⊥-decomposable measure on the generated σ -algebra S(A). As shown by [41], if μ is a (NSA)-type σ -⊥-decomposable measure, then by using Carathéodory’s extension method, μ can be uniquely extended to the σ -algebra S(A). It is rather difficult to intuitively understand Carathéodory’s extension method [14], so we present a pseudo-metric approach for the extension of the decomposable measures. Our results suggest that extending a σ -⊥-decomposable measure from an algebra to a σ -algebra can be reduced to finding the closure of a subset of a generalized pseudo-metric space. The remainder of this paper is organized as follows. In Section 2, we provide some basic notions and auxiliary results that are needed later. In Section 3, we prove that the extension of a (NSA)-type σ -⊥-decomposable measure can be formulated as the closure of a subset of a certain generalized pseudo-metric space. In Section 4, we discuss the completion of σ -⊥-decomposable measures, as well as establishing the connection between this completion and the particular extension via generalized pseudo-metrics. To illustrate the effectiveness of our method for extension via pseudo-metrics, we compare it with Carathéodory’s extension method in Section 5. In particular, the extension via generalized pseudo-metrics is equivalent to the completion of σ -⊥-decomposable measures and the well-known Carathéodory extension. 2. Preliminaries Throughout this study, the notations A, S(A), and P(X) denote an algebra of subsets of the given nonempty set X, the σ -algebra generated by A, and the power set of X, respectively. First, we recall the concepts of a triangular norm and triangular conorm from [7,18,44,48]. Definition 2.1. (i) A function  : [0, 1]2 → [0, 1] is called a triangular norm (t -norm) if it satisfies the following conditions for all x, y, z ∈ [0, 1]: (T1) x1 = x, (boundary condition) (T2) xy ≤ xz whenever y ≤ z, (monotonicity) (T3) xy = yx, (commutativity) (T4) x(yz) = (xy)z. (associativity) (ii) A function ⊥ : [0, 1]2 → [0, 1] is called a triangular conorm (t -conorm) if for all x, y, z ∈ [0, 1] it satisfies (T2)–(T4) and (S1) x⊥0 = x. (boundary condition) (iii) For any t-conorm ⊥, the t-norm ⊥∗ defined by (TCO) x⊥∗ y = 1 − (1 − x)⊥(1 − y), is called the dual t -norm of ⊥.

JID:FSS AID:7105 /FLA

[m3SC+; v1.237; Prn:18/10/2016; 12:47] P.3 (1-14)

J. Xie et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

3

A t-norm  (t -conorm ⊥) is called continuous if it is a continuous function; a t -norm  (t -conorm ⊥) is called Archimedean if  (⊥) is continuous and xx < x (x⊥x > x, resp.) for all x ∈ (0, 1). An Archimedean t -norm  (t-conorm ⊥) is called strict if  is strictly increasing in (0, 1)2 . Theorem 2.1 ([48]). (a) A function ⊥ : [0, 1]2 → [0, 1] is an Archimedean t-conorm iff a continuous, strictly increasing function g : [0, 1] → [0, ∞] with g(0) = 0 exists such that   a⊥b = g (−1) g(a) + g(b) , where g (−1) is the pseudo-inverse of g defined by  −1 g (y) if y ∈ [0, g(1)], (−1) g (y) = 1 if y ∈ [g(1), ∞]. Moreover, ⊥ is strict iff g(1) = ∞. (b) A function  : [0, 1]2 → [0, 1] is an Archimedean t-norm iff a continuous, strictly decreasing function f : [0, 1] → [0, ∞] with f (1) = 0 exists such that   ab = f (−1) f (a) + f (b) , where f (−1) is the pseudo-inverse of f defined by  −1 f (y) if y ∈ [0, f (0)], f (−1) (y) = 0 if y ∈ [f (0), ∞]. Moreover,  is strict iff f (0) = ∞. Due to the associative property, the t-conorm ⊥ can be extended by induction to an n-ary operation by setting   n n−1 ⊥ xi = ⊥ xi ⊥xn . i=1

i=1

Due to monotonicity, for each sequence (xi )i∈N of elements in [0, 1], the following limit can be considered: ∞

n

⊥ xi = lim ⊥ xi .

i=1

n→∞ i=1

For Archimedean t-conorms, it follows that N  N (−1) g(xi ) , where N ∈ N ∪ {∞}. ⊥ xi = g i=1

i=1

Analogous statements hold for the t-norm case [48]. Next, we describe the concept of generalized pseudo-metrics based on a t -conorm. This definition is based on Pap’s proposed method [38], which is compatible with pseudo-operations. Definition 2.2. Let X be a non-empty set and ⊥ is a t -conorm. A function d⊥ : X × X → [0, 1] is a generalized pseudo-metric on X if it satisfies the following conditions: (PM1) d⊥ (x, x) = 0 for all x ∈ X, (PM2) d⊥ (x, y) = d⊥ (y, x), for all x, y ∈ X, (PM3) d⊥ (x, z) ≤ d⊥ (x, y)⊥d⊥ (y, z), for all x, y, z ∈ X. Similar to the classical case in pseudo-metric spaces, we introduce the following concepts.

JID:FSS AID:7105 /FLA

[m3SC+; v1.237; Prn:18/10/2016; 12:47] P.4 (1-14)

J. Xie et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

4

Definition 2.3. A sequence (xi )i∈N in a generalized pseudo-metric space (X, d⊥ ) is said to converge to x if lim d⊥ (xi , x) = 0; a sequence (xi )i∈N in a generalized pseudo-metric space (X, d⊥ ) is called a Cauchy sequence i→∞

if lim d⊥ (xi , xj ) = 0; the generalized pseudo-metric space (X, d⊥ ) is called complete if every Cauchy sequence is i,j →∞

convergent. Let (X, d⊥ ) be a generalized pseudo-metric space. For a point x in X and r > 0, the set B(x, r) = {y ∈ X | d⊥ (x, y) < r} is called an open ball with radius r centered at x. A subset A of X is said to be open provided that for every point x ∈ A, there is an open ball centered at x that is contained in A. For a point x ∈ X, an open set that contains x is called a neighborhood of x. For a subset E, a point x ∈ X is called a point of closure of E provided that every neighborhood of x contains a point in E. If E contains all of its points of closure, then the set E is said to be closed. It is obvious that a subset E is closed iff whenever a sequence in E converges to a limit x ∈ X, the limit x belongs to E. In the sequel, we recall some important decomposable measures and related properties. Definition 2.4 ([3,48]). Let ⊥ be a t -conorm. A set function μ : A → [0, 1] with μ(∅) = 0 and μ(X) = 1 is: (i) a ⊥-decomposable measure if μ(A ∪ B) = μ(A)⊥μ(B) holds for each pair (A, B) of disjoint elements of A; (ii) a σ -⊥-decomposable measure if 

∞ μ Ai = ⊥ μ(Ai ) i∈N

i=1

for all pairwise disjoint sets Ai (i ∈ N) such that



Ai ∈ A;

i∈N

(iii) a ⊥-subdecomposable measure if μ(A ∪ B) ≤ μ(A)⊥μ(B) for all A, B ∈ A; (iv) a σ -⊥-subdecomposable measure if 

∞ μ Ai ≤ ⊥ μ(Ai ) i∈N

i=1

for all pairwise disjoint sets Ai (i ∈ N) such that



Ai ∈ A;

i∈N

(v) continuous from below (above, resp.) if lim μ(Ai ) = μ(A) for Ai  A (for Ai A, resp.).

i→∞

Theorem 2.2 ([48]). (i) μ is a ⊥-decomposable measure ⇒ μ is monotone; (ii) μ is a ⊥-decomposable measure ⇔ μ(A ∪ B)⊥μ(A ∩ B) = μ(A)⊥μ(B) for any A, B ∈ A; (iii) μ is a σ -⊥-decomposable measure ⇔ μ is ⊥-decomposable and continuous from below. Theorem 2.3 ([3]). (i) μ is a ⊥-decomposable measure ⇒ μ is ⊥-subdecomposable; (ii) μ is a σ -⊥-decomposable measure ⇒ μ is σ -⊥-subdecomposable. Definition 2.5 ([48]). A ⊥-decomposable measure μ, where ⊥ is an Archimedean t-conorm with an additive generator g, is said to be a (NSA)-type ⊥-decomposable measure if ⊥ is non-strict and g ◦ μ is a finite additive measure. In the following sections, we focus on the case where μ is a (NSA)-type σ -⊥-decomposable measure.

JID:FSS AID:7105 /FLA

[m3SC+; v1.237; Prn:18/10/2016; 12:47] P.5 (1-14)

J. Xie et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

5

3. Extension of a class of σ -⊥-decomposable measures via a generalized pseudo-metric In this section, we extend μ from A to S(A) using the outer measure induced by μ and the generalized pseudometric. Definition 3.1 ([41]). Let μ be a σ -⊥-decomposable measure on an algebra A ⊂ P(X). The set function μ∗ : P(X) → [0, 1] defined by

∞

∞ ∗ μ (A) = inf ⊥ μ(Ai ) : Ai ∈ A; A ⊂ Ai , A ∈ P(X) i=1

i=1

is called the outer measure induced by μ. Remark 3.1. For any A ⊆ X and any sequence (Ai )i∈N of sets in A, the union of which contains A, let Bi =  Ai ∩ (∪i−1 n=1 An ) for each i ∈ N. Then, (Bi )i∈N is a disjoint sequence of sets in A such that Bi ⊆ Ai and

∞

i=1

Ai =

∞

Bi .

i=1

Thus, every sequence (Ai )i∈N of sets in Definition 3.1 may be replaced by a disjoint sequence (Bi )i∈N with the same property. Lemma 3.1 ([41]). The outer measure μ∗ induced by μ has the following properties: (i) (ii) (iii) (iv) (v) (vi)

μ∗ |A = μ; μ∗ (∅) = 0; μ∗ is monotonous, i.e., μ∗ (A) ≤ μ∗ (B) whenever A ⊂ B; μ∗ is σ -⊥-subdecomposable; μ∗ is ⊥-subdecomposable; for any sets A, B, C ∈ P(X), μ∗ (AC) ≤ μ∗ (AB)⊥μ∗ (BC), where A B denotes the symmetric difference of sets A and B.

Theorem 3.1 ([49]). Let (X, A, μ) be a σ -⊥-decomposable measure space and A/μ is the set of all equivalence classes for the relation “∼”: A ∼ B iff μ(AB) = 0. If we define d⊥ : A/μ × A/μ → [0, 1] by d⊥ ([A], [B]) = μ(A  B) for all A, B ∈ A, then d⊥ is a pseudo-metric on A/μ, where [A]([B]) denotes the equivalence class containing A(B). Theorem 3.2 ([49]). Let (X, A, μ) be a σ -⊥-decomposable measure space and d⊥ is the pseudo-metric on A/μ, as defined in Theorem 3.1. Then, the maps ([A], [B]) → [A] ∨ [B] and ([A], [B]) → [A] ∧ [B] are uniformly continuous from A/μ × A/μ to A/μ. Using Lemma 3.1 and Theorem 3.1, it is not difficult to prove the following result. Theorem 3.3. Let μ be a σ -⊥-decomposable measure on an algebra A ⊂ P(X) and μ∗ is the outer measure induced by μ. If we define d⊥ : P(X) × P(X) → [0, 1] by d⊥ (A, B) = μ∗ (A  B) for all A, B ∈ P(X), then d⊥ is a generalized pseudo-metric on P(X). ¯ In particular, We denote the d⊥ -closure of A as A. A¯ = {A ∈ P(X) : a sequence (Ai )i∈N ⊂ A exists such that lim d⊥ (Ai , A) = 0}. i→∞

JID:FSS AID:7105 /FLA

[m3SC+; v1.237; Prn:18/10/2016; 12:47] P.6 (1-14)

J. Xie et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

6

According to Theorem 3.3, it is obvious that a set A ∈ A¯ iff for any ε > 0, Aε ∈ A exists such that μ∗ (A  Aε ) < ε. Thus, A ∈ A¯ if A can be approximated by the sets from A arbitrarily closely. Lemma 3.2. Let (Ai )i∈N ⊂ A with lim d⊥ (Ai , A) = 0. Then, lim μ(Ai ) = μ∗ (A). i→∞

i→∞

Proof. According to Theorem 3.3, we can obtain the following two inequalities     μ∗ (A) = d⊥ (A, ∅) ≤ lim d⊥ (A, Ai )⊥d⊥ (Ai , ∅) = lim d⊥ (A, Ai )⊥μ(Ai ) = lim μ(Ai ) i→∞

i→∞

i→∞

and     lim μ(Ai ) = lim d⊥ (Ai , ∅) ≤ lim d⊥ (Ai , A)⊥d⊥ (A, ∅) = lim d⊥ (A, Ai )⊥μ∗ (A) = μ∗ (A).

i→∞

i→∞

Thus, lim μ(Ai

) = μ∗ (A).

i→∞

i→∞

i→∞

2

Now, we state the first main theorem of this study. Theorem 3.4. Let μ be a σ -⊥-decomposable measure on an algebra A. Then, (i) A¯ is a σ -algebra containing A; (ii) μ∗ |A¯ is a σ -⊥-decomposable measure. Proof. (i) We split the proof into a series of claims. Claim 1. A¯ is an algebra containing A. ¯ In order to verify that A¯ is an algebra, it is sufficient to We already noted that A ⊂ A¯ from the definition of A. ¯ For any A, B ∈ A, ¯ convergent sequences (Ai )i∈N and ¯ then both A ∪ B and A belong to A. show that if A, B ∈ A, (Bi )i∈N in A exist that converge to A and B, respectively. By Theorem 3.2, we have   0 = lim d⊥ (Ai , A)⊥d⊥ (Bi , B) ≥ lim d⊥ (Ai ∪ Bi , A ∪ B) ≥ 0, i→∞

i→∞

which implies that (Ai ∪ Bi )i∈N is a convergent sequence in A, and lim d⊥ (Ai ∪ Bi , A ∪ B) = 0. Thus, A ∪ B belongs i→∞

¯ In addition, since to A.

lim d⊥ (Ai , A ) = lim μ∗ (Ai A ) = lim μ∗ (Ai A) = lim d⊥ (Ai , A) = 0,

i→∞

i→∞

i→∞

i→∞

¯ Thus, A¯ is an algebra containing A. then we have A ∈ A. Claim 2. μ∗ |A¯ is a ⊥-decomposable measure. Let A, B ∈ A¯ be disjoint. Then, the sequences (Ai )i∈N and (Bi )i∈N in A exist that converge to A and B, respectively. According to the proof of Claim 1, (Ai ∪ Bi )i∈N is a sequence in A that converges to A ∪ B. By the inclusion Ai ∩ Bi ⊂ (Ai A) ∪ (Bi B), we have 0 ≤ lim μ(Ai ∩ Bi ) i→∞

= lim μ∗ (Ai ∩ Bi ) i→∞   ≤ lim μ∗ (Ai A) ∪ (Bi B) i→∞

JID:FSS AID:7105 /FLA

[m3SC+; v1.237; Prn:18/10/2016; 12:47] P.7 (1-14)

J. Xie et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

7

≤ lim μ∗ (Ai A)⊥μ∗ (Bi B) i→∞   = lim d⊥ (Ai , A)⊥d⊥ (Bi , B) = 0, i→∞

which implies that lim μ(Ai ∩ Bi ) = 0. Thus, by the equality (Theorem 2.2 (ii)) i→∞

μ(Ai ∪ Bi )⊥μ(Ai ∩ Bi ) = μ(Ai )⊥μ(Bi ), and the continuity of ⊥, we have μ∗ (A ∪ B) = lim μ(Ai ∪ Bi )⊥ lim μ(Ai ∩ Bi ) i→∞ i→∞   = lim μ(Ai ∪ Bi )⊥μ(Ai ∩ Bi ) i→∞   = lim μ(Ai )⊥μ(Bi ) i→∞

= lim μ(Ai )⊥ lim μ(Bi ) i→∞

i→∞

= μ∗ (A)⊥μ∗ (B), which completes the whole proof. Claim 3. A¯ is a σ -algebra.

¯ i ∈ N be pairwise disjoint. Then, the sequences (Aij )j ∈N in A exist that converge to Ai for every Let Ai ∈ A, i ∈ N. Using the same argument employed in the proof of Claim 2, we can find that lim μ(A1j ∩ A2j ) = 0. Thus, j →∞

we have the following equality μ∗ (A1 ) = lim μ(A1j ) j →∞   = lim μ (A1j ∩ A2j ) ∪ (A1j ∩ A2j ) j →∞

= lim μ(A1j ∩ A2j )⊥ lim μ(A1j ∩ A2j ) j →∞

= lim

j →∞

j →∞

μ(A1j ∩ A2j ).

Similarly, we have μ∗ (A2 ) = lim μ(A2j ∩ A1j ). For any fixed n ∈ N, in general, we have μ∗ (Ai ) = lim μ(Bij(n) ) j →∞

for every i ∈ {1, 2, ...n}, where ⎛ 

(n) ⎝ Bij = Aij

j →∞

⎞ Akj ⎠ .

k∈{1,2,...n}−{i}

  (n) Let g be an additive generator of ⊥. By the continuity of g, we have g μ∗ (Ai ) = lim g(μ(Bij )) for every i ∈ j →∞

{1, 2, ...n}. Consequently, for any fixed n ∈ N, we have n n        (n)  g μ∗ (Ai ) = lim g μ(Bij ) ≤ g μ(X) = g(1), i=1

j →∞

i=1

where the inequality holds because for any fixed j ∈ N, measure. Hence, ∞    g μ∗ (Ai ) ≤ g(1), i=1

and we can find n such that

n

(n) i=1 Bij

is a disjoint union and g ◦ μ is a finite additive

JID:FSS AID:7105 /FLA

[m3SC+; v1.237; Prn:18/10/2016; 12:47] P.8 (1-14)

J. Xie et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

8

⎛ lim ⎝

n,m→∞

m 

⎞ ⎞ ⎛ ∞   ∗   ∗  g μ (Ai ) ⎠ = 0 and lim ⎝ g μ (Ai ) ⎠ = 0. i→∞

i=n+1

i=n+1

μ∗ ,

Therefore, by the σ -⊥-subdecomposability of we have ⎛ ⎞  n ∞ ∞



lim d⊥ Ai , Ai = lim μ∗ ⎝ Ai ⎠ n→∞

i=1

n→∞

i=1

i=n+1

 ≤ lim

n→∞

∞

⊥ μ∗ (Ai )

i=n+1

⎛

= g (−1) ⎝ lim

∞  

n→∞



⎞   g μ∗ (Ai ) ⎠ = 0, 

i=n+1

 ¯ d⊥ ) and it which shows that ( ni=1 Ai ) n∈N is a convergent sequence in the generalized pseudo-metric space (A, ∞ ¯ ¯ A . Thus, A ∈ A. Therefore, A is a σ -algebra. converges to ∞ i=1 i i=1 i ¯ i ∈ N be pairwise disjoint. By the proof of (i), we have (ii) Let Ai ∈ A, ⎛ ⎞ ∞  n ∞



μ∗ Ai = μ∗ Ai ⊥μ∗ ⎝ Ai ⎠ i=1

i=1

 = and

⎛

∞

μ∗ ⎝

i=n+1

⎛ ⎞  ∞

n Ai ⎠ ⊥ μ∗ (Ai ) ⊥μ∗ ⎝

i=1

⎞

i=n+1

⎛

Ai ⎠ = g (−1) ⎝

i=n+1

∞ 

⎞ g(μ∗ (Ai ))⎠ → 0 as n → ∞,

i=n+1

which implies that ⎞ ⎛ ∞ ∞  n  



μ∗ Ai = lim ⎝ ⊥ μ∗ (Ai ) ⊥μ∗ Ai ⎠ n→∞

i=1

⎛

 =

i=1

i=n+1

⎞  ∞ 

 lim ⊥ μ∗ (Ai ) ⊥ ⎝ lim μ∗ Ai ⎠ n

n→∞ i=1

n→∞

i=n+1

∞

= ⊥ μ∗ (Ai ). i=1

Thus,

μ∗ |

A¯

is a σ -⊥-decomposable measure. 2

By combining Lemma 3.2 and Theorem 3.4, it is not difficult to prove the following extension theorem. Theorem 3.5 ([41]). Let μ be a σ -⊥-decomposable measure of (NSA)-type on an algebra A. Then, μ can be uniquely extended to a σ -⊥-decomposable measure on S(A). Remark 3.2. In [5] (and in [27], Section 5.3), the authors described the necessary and sufficient conditions for extending a submeasure from a ring to the generated σ -ring. Similarly, Pap [26] studied the extensions of the ⊥-decomposable measure μ with respect to a continuous t -conorm ⊥ from a ring R to the σ -ring  generated by R. In particular, let μ : R → [0, 1] be an order continuous (for any sequence (Ei )i∈N ⊆ R, Ei ∅, lim μ(Ei ) = 0) i→∞

⊥-decomposable measure with respect to a continuous t -conorm ⊥ and  is the σ -ring generated by R. Then, μ can

JID:FSS AID:7105 /FLA

[m3SC+; v1.237; Prn:18/10/2016; 12:47] P.9 (1-14)

J. Xie et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

9

be extended to a unique monotone order continuous ⊥-subadditive set function, μ ¯ :  → [0, 1], iff the following conditions hold: (i) if (Ai )i∈N is a sequence from R such that μ(An Am ) → 0 as n, m → ∞, then the limit of the sequence (μ(Am ))m∈N exists; (ii) lim μ(Ei ) = 0 for any sequence (Ei )i∈N of pairwise disjoints sets from R. i→∞

By comparing these results, we can discuss the extension problem for ⊥-decomposable measures from different aspects. 4. The completeness of the σ -⊥-decomposable measure μ∗ |A¯ In this section, we discuss the completion of σ -⊥-decomposable measures, as well as establishing the connection between this completion and the particular extension via generalized pseudo-metrics. Definition 4.1. A σ -⊥-decomposable measure μ on a σ -algebra M is called complete if M contains all the subsets of every set in M with μ-measure zero. It is clear from the definition of the outer measure that if A ⊂ B ∈ A¯ and μ∗ (B) = 0, then A ∈ A¯ and μ∗ (A) = 0. Hence, μ∗ |A¯ is a complete σ -⊥-decomposable measure. Theorem 4.1. Let μ be a σ -⊥-decomposable measure on σ -algebra M. Set Aμ = {A ⊂ X|E, F ∈ M, s.t., E ⊂ A ⊂ F and μ(F \ E) = 0}.   Now, we define a set function μ ¯ : Aμ → [0, 1] as μ(A) ¯ = μ(E) or μ(F ) . Then, (i) Aμ is a σ -algebra on X that includes M; (ii) μ¯ is a complete σ -⊥-decomposable measure on Aμ , where its restriction to M is μ. Proof. (i) It is clear that M ⊂ Aμ and hence X ∈ Aμ . Note that the relations E ⊂ A ⊂ F and μ(F \ E) = 0 imply the relations F  ⊂ A ⊂ E  and μ(E  \ F  ) = μ(E  ∩ F ) = μ(F \ E) = 0. Thus, Aμ is closed under complement. Next, suppose that (Ai )i∈N is a sequence of sets in Aμ . For each i, choose the sets Ei and Fi in M such that Ei ⊂ Ai ⊂ Fi ∞ ∞ ∞ ∞ ∞      and μ(Fi \ Ei ) = 0. Then, Ei and Fi belong to M with Ei ⊂ Ai ⊂ Fi , and i=1

μ

∞ 

Fi \

i=1

Thus,

∞ 

∞

i=1

i=1

i=1

i=1

∞  ∞  

Ei ≤ μ (Fi \ Ei ) ≤ ⊥ μ(Fi \ Ei ) = 0.

i=1

i=1

i=1

Ai belongs to Aμ . Consequently, Aμ is a σ -algebra on X that includes M.

i=1

¯ = μ(A) by letting E and F equal to A, and thus μ¯ is an extension of μ. (ii) For any A in M, we have μ(A) It is clear that μ(∅) ¯ = 0. Let A1 , A2 ∈ Aμ be disjoint sets. By the definition of Aμ , E1 , E2 , F1 , F2 exist such that E1 ⊂ A1 ⊂ F1 and E2 ⊂ A2 ⊂ F2 . Then, μ(A ¯ 1 ∪ A2 ) = μ(E1 ∪ E2 ) = μ(E1 )⊥μ(E2 ) = μ(A ¯ 1 )⊥μ(A ¯ 2 ). Thus, μ¯ is a ⊥-decomposable measure on Aμ . Let (Ai )i∈N be an increasing sequence of sets in Aμ . For each i, choose the sets Ei and Fi in M such that Ei ⊂ Ai ⊂ Fi and μ(Fi \ Ei ) = 0. If we let Eiδ

=

∞  j =i

Ej ,

Fiσ

=

∞  j =i

Fj ,

JID:FSS AID:7105 /FLA

[m3SC+; v1.237; Prn:18/10/2016; 12:47] P.10 (1-14)

J. Xie et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

10

then both (Eiδ )i∈N and (Fiσ )i∈N are increasing sequences of sets in M with Eiδ ⊂ Ai ⊂ Fiσ , and ∞ ∞ ∞ 

 



  μ(Fiσ \ Eiδ ) = μ (Eiδ ) \ (Fiσ ) = μ ( Ej ) \ ( Fj ) ≤ μ (Ej \ Fj ) = 0. j =i

j =i

j =i

     ∞ ∞ ∞ ∞ ∞ ∞     Since Eiδ ⊂ Ai ⊂ Fiσ and μ ( Fiσ ) \ ( Eiδ ) ≤ μ (Fiσ \ Eiδ ) = 0, then we have i=1

μ( ¯

i=1

∞

i=1

Ai ) = μ(

i=1

∞

i=1

i=1

i=1

i=1

Eiδ ) = lim μ(Eiδ ) = lim μ(A ¯ i ). i→∞

i→∞

Hence, μ¯ is continuous from below. Consequently, μ ¯ is a σ -⊥-decomposable measure on Aμ by part (iii) of Theorem 2.2. The completeness of μ¯ is obvious by the definition of μ. ¯ Finally, if μ¯ ∗ is an arbitrary complete σ -⊥-decomposable measure extension of μ defined on a σ -algebra M ∗ , then M ∗ must include the subsets of μ-measure zero sets and M. Hence, M ∗ ⊃ Aμ , i.e., μ¯ is the minimal complete σ -⊥-decomposable measure extension of μ. Thus, our claim is proved. 2 The following theorem establishes the connection between this completion and the particular extension via generalized pseudo-metrics. Theorem 4.2. Let μ be a σ -⊥-decomposable measure on an algebra A and μ∗ is the outer measure induced by μ. ¯ Then, the completion of μ to S(A) is identical to μ∗ on A. ¯ μ¯ and μ∗ coincide on Aμ . To complete Proof. Since μ∗ on A¯ is a complete measure, then it follows that Aμ ⊂ A, ¯ the proof, we only need to prove that A ⊂ Aμ . ¯ g be an additive generator of ⊥. Then, for any ε > 0, a set Aε ∈ S(A) exists such that A ⊂ Aε and Let A ∈ A, ∞  μ∗ (Aε \ A) < ε. Indeed, by the definition of μ∗ , Ai ∈ A with A ⊂ Ai exists and i=1 ∞      g μ(Ai ) ≤ g μ∗ (A) + g(ε). i=1

Set Aε =

∞ 

¯ and by the σ -⊥-decomposability of μ∗ on A, ¯ we have Ai . It is clear that A ⊂ Aε , Aε ∈ S(A) ⊂ A,

i=1 ∞      g μ∗ (Aε ) ≤ g μ(Ai ) . i=1

Then,     g μ∗ (Aε ) ≤ g μ∗ (A) + g(ε), which shows that μ∗ (Aε \ A) < ε. Set Aσ = μ∗ (Aσ

μ∗ (A

∞  n=1

A 1 . Then, A ⊂ Aσ ∈ S(A) ⊂ A¯ and μ∗ (A) = μ∗ (Aσ ), because n

\ A) ≤ 1 \ A) < 1/n for all n. Now, we can apply this to the complement of A and find a set B ∈ n ¯ S(A) ⊂ A such that A ⊂ B and μ∗ (B \ A ) < ε. Set Aδ = B  . We obtain Aδ ⊂ A and μ∗ (A \ Aδ ) = μ∗ (B \ A ) < 1/n, which is the required relation. This shows that A ∈ Aμ , which completes the proof. 2 5. Comparison of the extension approach via generalized pseudo-metrics with Carathéodory’s strategy In this section, we compare our extension approach via generalized pseudo-metrics with Carathéodory’s strategy.

JID:FSS AID:7105 /FLA

[m3SC+; v1.237; Prn:18/10/2016; 12:47] P.11 (1-14)

J. Xie et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

11

Theorem 5.1. Let μ be a σ -⊥-decomposable measure on an algebra A. Then, a set A belongs to A¯ iff μ∗ (E ∩ A)⊥μ∗ (E \ A) = μ∗ (E) for all sets E ⊂ X. Proof. (⇐). Suppose that A ∈ P(X) and g is the normed additive generator of ⊥. For each ε ∈ (0, 1), ε1 ∈ (0, 1) exists such that ε1 ⊥ε1 ≤ ε. By the definition of μ∗ and Remark 3.1, a corresponding disjoint sequence (Ai )i∈N of sets in A exists such that A⊂

∞

∞ ∞

       Ai and g μ∗ ( Ai ) ≤ g μ(Ai ) ≤ g μ∗ (A) + g(ε1 ).

i=1 n

 lim g μ∗ (

n→∞

i=1

i=1

  Ai ) = lim g μ( n→∞

i=1

n

n ∞        Ai ) = lim g μ(Ai ) = g μ(Ai ) , n→∞

i=1

so a positive integer n exists such that if A0 =

n 

i=1

i=1

Ai ∈ A, then

i=1 ∞ ∞

       g μ∗ ( Ai ) ≤ g μ(Ai ) ≤ g μ∗ (A0 ) + g(ε1 ). i=1

i=1

According to the assumption, we have μ∗ (

∞ 

Ai ) = μ∗ (

i=1

∞ 

Ai \ A0 )⊥μ∗ (A0 ), and thus

i=1

∞ ∞



      g μ∗ ( Ai \ A0 ) = g μ∗ ( Ai ) − g μ∗ (A0 ) ≤ g(ε1 ). i=1

i=1

Thus, ∞

    g μ∗ (A \ A0 ) ≤ g μ∗ ( Ai \ A0 ) ≤ g(ε1 ). i=1

In addition, by the assumption, we can obtain μ∗(

∞ 

Ai ) = μ∗ (A)⊥μ∗ (

i=1

∞ 

Ai \ A), so

i=1

∞ ∞



      g μ∗ ( Ai \ A) = g μ∗ ( Ai ) − g μ∗ (A) ≤ g(ε1 ). i=1

i=1

Therefore, ∞

    g μ∗ (A0 \ A) ≤ g μ∗ ( Ai \ A) ≤ g(ε1 ). i=1

Hence,     g μ∗ (A \ A0 ) + g μ∗ (A0 \ A) ≤ g(ε1 ) + g(ε1 ). Since g is uniformly continuous and ⊥ is continuous, then we obtain μ∗ (AA0 ) = μ∗ (A \ A0 )⊥μ∗ (A0 \ A) ≤ ε1 ⊥ε1 ≤ ε, ¯ which implies that for any ε > 0, A ⊂ X, A0 ∈ A exists such that μ∗ (AA0 ) < ε, i.e., A ∈ A. ∗ ∗ ∗ (⇒). By the σ -⊥-subdecomposability of μ , it is sufficient to verify that μ (E ∩ A)⊥μ (E \ A) ≤ μ∗ (E) for any ¯ Let ε > 0 and g is the normed additive generator of ⊥. Suppose that a set sequence (Ai )i∈N ⊆ A E ⊂ X and A ∈ A.

JID:FSS AID:7105 /FLA

[m3SC+; v1.237; Prn:18/10/2016; 12:47] P.12 (1-14)

J. Xie et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

12

satisfies E ⊂

∞ 

Ai and

i=1 ∞ ∞

       g μ∗ ( Ai ) ≤ g μ(Ai ) ≤ g μ∗ (E) + g(ε). i=1

Then, E ∩ A ⊂

i=1 ∞ 

Ai ∩ A and E \ A ⊂

i=1

∞ 

Ai \ A, from which we obtain

i=1

∞ ∞



        g μ∗ (E ∩ A) + g μ∗ (E \ A) ≤ g μ∗ ( Ai ∩ A) + g μ∗ ( Ai \ A) i=1



i=1

 ∞  ≤ g ⊥ μ∗ (Ai ∩ A) + g ⊥ μ∗ (Ai \ A) ∞

i=1

i=1



   n = g lim ⊥ μ∗ (Ai ∩ A) + g lim ⊥ μ∗ (Ai \ A) n

n→∞ i=1



n→∞ i=1

  n  = lim g ⊥ μ∗ (Ai ∩ A) + lim g ⊥ μ∗ (Ai \ A) n→∞

n→∞

=

i=1

n 

= lim ∞ 

n

n→∞

 gμ∗ (Ai ∩ A) + lim

i=1

n→∞

i=1

n 

 gμ∗ (Ai \ A)

i=1



 gμ∗ (Ai ∩ A) + gμ∗ (Ai \ A)

i=1

=

∞    g μ∗ (Ai ∩ A)⊥μ∗ (Ai \ A) i=1

=

∞ 

gμ∗ (Ai )

i=1

=

∞ 

gμ(Ai )

i=1

  ≤ g μ∗ (E) + g(ε). ε is arbitrary, so we have μ∗ (E ∩ A)⊥μ∗ (E \ A) ≤ μ∗ (E), which completes the proof of theorem. 2 6. Conclusion In this study, we presented an intuitive interpretation of the extension of (NSA)-type σ -⊥-decomposable measures. According to this interpretation, finding the closure of a subset of a generalized pseudo-metric space is an effective method for extending an (NSA)-type σ -⊥-decomposable measure from an algebra to a σ -algebra. Acknowledgements The authors are very grateful to the editors and anonymous reviewers for their valuable comments and suggestions. This study was supported partly by the National Natural Science Foundation of China (Nos. 11371130, 61103052, and 11401195), the Natural Science Foundation of Fujian Province (Nos. 2014H0034 and 2016J01022), projects of the Education Department of Fujian Province (No. JA15280), and Li Shangda Discipline Construction Fund of Jimei University.

JID:FSS AID:7105 /FLA

[m3SC+; v1.237; Prn:18/10/2016; 12:47] P.13 (1-14)

J. Xie et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

13

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46]

H. Agahi, R. Mesiar, Y. Ouyang, Chebyshev type inequalities for pseudo-integrals, Nonlinear Anal. 72 (2010) 2737–2743. B. Bede, D. O’Regan, The theory of pseudo-linear operators, Knowl.-Based Syst. 38 (2013) 19–26. B. Cavallo, L. D’Apuzzo, M. Squillante, Independence and convergence in non-additive settings, Fuzzy Optim. Decis. Mak. 8 (2009) 29–43. D. Denneberg, Non-additive Measure and Integral, Kluwer Academic Publishers, Dordrecht, 1994. I. Dobrakov, On submeasures I, in: Dissertationes Math., vol. 112, 1974, Waszawa. D. Dubois, M. Prade, A class of fuzzy measures based on triangular norms, Int. J. Gen. Syst. 8 (1982) 43–61. D. Dubois, H. Prade, Aggregation of decomposable measures with application to utility theory, Theory Decis. 41 (1996) 59–95. D. Dubois, E. Pap, H. Prade, Hybrid probabilistic-possibilistic mixtures and utility functions, in: J. Fodor, B. de Baets, P. Perny (Eds.), Preferences and Decisions Under Incomplete Knowledge, in: STUDFUZZ, vol. 51, Spalgebraer, Heidelberg, 2000, pp. 51–73. A. George, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Syst. 64 (1994) 395–399. M. Grabisch, J.L. Marichal, R. Mesiar, E. Pap, Aggregation Functions, Encyclopedia of Mathematics and Its Applications, vol. 127, Cambridge University Press, 2009. V. Gregori, A. Sapena, On fixed point theorems in fuzzy metric spaces, Fuzzy Sets Syst. 125 (2002) 245–253. O. Hadži´c, E. Pap, Fixed Point Theory in Probabilistic Metric Spaces, Kluwer Academic Publishers, Dordrecht, 2001. O. Hadži´c, E. Pap, Probabilistic multi-valued contractions and decomposable measures, Int. J. Uncertain. Fuzziness Knowl.-Based Syst. (2002) 59–74. P.R. Halmos, Measure Theory, Spalgebraer-Verlag, New York, 1974. L. Halˇcinová, O. Hutník, R. Mesiar, On some classes of distance distribution function-valued submeasures, Nonlinear Anal. 74 (2011) 1545–1554. L. Halˇcinová, O. Hutník, R. Mesiar, On distance distribution functions-valued submeasures related to aggregation functions, Fuzzy Sets Syst. 194 (2012) 15–30. E.P. Klement, R. Mesiar, E. Pap, Integration with respect to decomposable measures, based on a conditionally distributive semialgebra on the unit interval, Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 8 (2000) 701–717. E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer Academic Publishers, Dordrecht, 2000. E.P. Klement, R. Mesiar, E. Pap, A universal integral as common frame for Choquet and Sugeno integral, IEEE Trans. Fuzzy Syst. 18 (1) (2010) 178–187. A. Kolesárová, A. Stupˇnanová, J. Beganová, Aggregation-based extensions of fuzzy measures, Fuzzy Sets Syst. 194 (2012) 1–14. I. Kramosil, J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetika 11 (1975) 326–334. R. Mesiar, E. Pap, Idempotent integral as limit of g-integrals, Fuzzy Sets Syst. 102 (1999) 385–392. R. Mesiar, J. Li, E. Pap, The Choquet integral as Lebesgue integral and related inequalities, Kybernetika 46 (2010) 1098–1107. B. Mihailovi´c, E. Pap, Asymmetric general Choquet integrals, Acta Polytech. Hung. 6 (2009) 161–173. E. Pap, Lebesgue and Saks decompositions of ⊥-decomposable measures, Fuzzy Sets Syst. 38 (1990) 345–353. E. Pap, Extension of the continuous t -conorm decomposable measure, Univ. Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 20 (1990) 121–130. E. Pap, Null-Additive Set Functions, Kluwer Academic Publishers, Dordrecht, 1995. E. Pap, O. Hadži´c, R. Mesiar, A fixed point theorem in probabilistic metric spaces and an application, J. Math. Anal. Appl. 202 (1996) 433–469. E. Pap, Pseudo-analysis as a mathematical base for soft computing, Soft Comput. 1 (1997) 61–68. E. Pap, Decomposable measures and nonlinear equations, Fuzzy Sets Syst. 92 (1997) 205–221. E. Pap, Extension of null-additive set functions on algebra of subsets, Novi Sad J. Math. 31 (2001) 2–13. E. Pap, Applications of decomposable measures on nonlinear differential equations, Novi Sad J. Math. 31 (2001) 89–98. E. Pap (Ed.), Handbook of Measure Theory, Elsevier Science, Amsterdam, 2002. E. Pap, M. Štrboja, Generalization of the Jensen inequality for pseudo-integral, Inf. Sci. 180 (2010) 543–548. E. Pap, Theory and Applications of Non-additive Measures and Corresponding Integrals, Modeling Decisions for Artificial Intelligence, Spalgebraer, Berlin Heidelberg, 2013, pp. 1–10. E. Pap, Pseudo-analysis, in: 2013 IEEE 9th International Conference on the Computational Cybernetics, ICCC, IEEE, 2013, pp. 213–218. E. Pap, M. Štrboja, Generalization of the integral inequalities for integral based on nonadditive measures, in: E. Pap (Ed.), Intelligent Systems: Models and Application, Springer, 2013, pp. 3–22. E. Pap, M. Štrboja, I. Rudas, Pseudo-Lp space and convergence, Fuzzy Sets Syst. 238 (2014) 113–128. D. Qiu, W.Q. Zhang, On decomposable measures induced by metrics, J. Appl. Math. 2012 (2012), Article ID 701206, 8 pages. D. Qiu, W.Q. Zhang, C. Li, On decomposable measures constructed by using stationary fuzzy pseudo-ultrametrics, Int. J. Gen. Syst. 42 (2013) 395–404. D. Qiu, W.Q. Zhang, C. Li, Extension of a class of decomposable measures using fuzzy pseudometrics, Fuzzy Sets Syst. 222 (2013) 33–44. D. Qiu, C. Lu, On properties of pseudo-integrals based on pseudo-addition decomposable measures, Abstr. Appl. Anal. 2014 (2014), Article ID 782040, 12 pages. I.J. Rudas, E. Pap, J. Fodor, Information aggregation in intelligent systems, an application oriented approach, Knowl.-Based Syst. 38 (2013) 3–13. B. Schweizer, A. Sklar, Probabilistic Metric Spaces, North-Holland, New York, 1983. Y.H. Shen, On the probabilistic Hausdorff distance and a class of probabilistic decomposable measures, Inf. Sci. 263 (2014) 126–140. A. Stupˇnanová, Special fuzzy measures on infinite countable sets and related aggregation functions, Fuzzy Sets Syst. 167 (2011) 57–64.

JID:FSS AID:7105 /FLA

14

[47] [48] [49] [50]

[m3SC+; v1.237; Prn:18/10/2016; 12:47] P.14 (1-14)

J. Xie et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

Z. Wang, G.J. Klir, Generalized Measure Theory, Spalgebraer Press, New York, 2009. S. Weber, ⊥-decomposable measures and integrals for Archimedean t -conorm, J. Math. Anal. Appl. 101 (1984) 114–138. J.L. Xie, Q.G. Li, S.L. Chen, Y. Gao, On pseudo-metric spaces induced by σ -⊥-decomposable measures, Fuzzy Sets Syst. 289 (2016) 33–42. D. Zhang, C. Guo, Integrals of set-valued functions for ⊥-decomposable measures, Fuzzy Sets Syst. 78 (1996) 341–346.