Characterizations of compact sets in fuzzy set spaces with Lp metric

Accepted Manuscript Characterizations of compact sets in fuzzy set spaces with Lp metric

Huan Huang, Congxin Wu

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S0165-0114(16)30394-3 http://dx.doi.org/10.1016/j.fss.2016.11.007 FSS 7125

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Fuzzy Sets and Systems

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27 August 2015 14 November 2016 23 November 2016

Please cite this article in press as: H. Huang, C. Wu, Characterizations of compact sets in fuzzy set spaces with Lp metric, Fuzzy Sets Syst. (2016), http://dx.doi.org/10.1016/j.fss.2016.11.007

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Characterizations of compact sets in fuzzy set spaces with Lp metric $ Huan Huanga,∗, Congxin Wub a

b

Department of Mathematics, Jimei University, Xiamen 361021, China Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

Abstract Compactness criteria in fuzzy set spaces endowed with the Lp metric have been studied for several decades. Total boundedness is a key feature of compactness in metric spaces. However, comparing existing compactness criteria in fuzzy set spaces endowed with the Lp metric with the Arzel`a-Ascoli theorem, the latter gives compactness criteria by characterizing totally bounded sets while the former does not characterize totally bounded sets. Currently, compactness criteria are only presented for three particular fuzzy set spaces under assumptions of convexity or star-shapedness. General fuzzy sets have become more important in both theory and applications. Therefore, this paper presents characterizations of totally bounded sets, relatively compact sets, and compact sets in general fuzzy set spaces equipped with the Lp metric, but which do not have any assumptions of convexity or star-shapedness. Subsets of these general sets include common fuzzy sets, such as fuzzy numbers, fuzzy star-shaped numbers with respect to the origin, fuzzy star-shaped numbers, and general fuzzy star-shaped numbers. Existing compactness criteria are stated for fuzzy number space, the space of fuzzy star-shaped numbers with respect to the origin, and the space of fuzzy star-shaped numbers endowed with the Lp metric, respectively. Constructing completions of fuzzy set spaces with respect to the Lp metric is a problem closely dependent on characterizing totally bounded sets. Based on characterizations of total $

Project supported by National Natural Science Foundation of China (No. 61103052), and by Natural Science Foundation of Fujian Province of China(No. 2016J01022) ∗ Corresponding author Email addresses: [email protected] (Huan Huang ), [email protected] (Congxin Wu)

Preprint submitted to Elsevier

November 28, 2016

boundedness and relatively compactness and some discussion of the convexity and star-shapedness of fuzzy sets, we show that the completions of fuzzy set spaces studied here can be obtained using the Lp extension. We also clarify relationships among the ten fuzzy set spaces studied here–the five pairs of original spaces and their corresponding completions. We show that the subspaces have parallel characterizations of totally bounded sets, relatively compact sets, and compact sets. Finally, we discuss properties of the Lp metric on fuzzy set space as an application of our results, and review compactness criteria proposed in previous work. Keywords: Fuzzy sets; Compact sets; Totally bounded sets; Lp metric; Completions 1. Introduction Compactness is a fundamental property in both theory and applications [15, 22, 32], and compactness criteria have attracted much attention. The Arzel`a-Ascoli theorem(s) provide compactness criteria in classic analysis and topology. Fuzzy systems have been successfully used to solve many real-world problems [1, 5, 6, 14, 37], and compactness plays an important role in the analysis and application of fuzzy sets and systems [4, 17, 18, 20, 31, 34, 37]. Many important and interesting works, including [7, 8, 10, 12, 13, 19, 26, 30, 31, 36, 38], have characterized compactness in fuzzy set spaces equipped with different topologies. Since Diamond and Kloeden [8] introduced the dp metric, an Lp type metric, it has become one of the most often used convergence structures on fuzzy sets. Consequently, researchers have started to consider characterizations of compactness in fuzzy set spaces with the dp metric. Diamond and Kloeden [8] gave a compactness criterion in fuzzy number space E m with the dp metric, and Ma [26] modified their characterization. Diamond [7] introduced fuzzy star-shaped numbers as an extension of fuzzy numbers, characterizing the compact sets in (S0m , dp ), where S m denotes the set of all fuzzy star-shaped numbers; S0m denotes the set of all the fuzzy starshaped numbers with respect to the origin; and E m and S0m do not include each other and are subsets of S m . Wu and Zhao [36] pointed out that the modified compactness criterion in [26] still has a fault and showed that the compactness criterion in (S0m , dp ) proposed in [7] has the same type of error as the compactness criterion in 2

(E m , dp ) proposed in [8]. They presented correct characterizations of compactness in (S0m , dp ) and (E m , dp ). Based on their results, Zhao and Wu [38] proposed a characterization of compactness in (S m , dp ). The concepts of “p-mean equi-left-continuity” and “uniform p-mean boundedness” proposed by Diamond and Kloeden [8] and Ma [26], respectively, play important roles in establishing and illustrating characterizations of compactness in fuzzy set spaces with the dp metric. Comparing the characterizations in [36, 38] to the Arzel`a-Ascoli theorem, the latter provides compactness criteria by characterizing totally bounded sets, whereas the former does not characterize totally bounded sets. Since total boundedness is a key feature of compactness in metric space, it is a natural and important problem to consider how to characterize totally bounded sets in fuzzy set spaces with the dp metric. Existing compactness criteria in [36, 38] are stated for three particular fuzzy set spaces with the dp metric. Fuzzy sets in these spaces have assumptions of convexity or star-shapedness. However, general fuzzy sets, which have no assumptions of convexity or star-shapedness, have attracted attention in both theory and applications [2, 24, 37], which has raised the basic and important problem of how to characterize totally bounded sets, relatively compact sets, and compact sets in general fuzzy set spaces endowed with the dp metric. Clearly, many other types of particular fuzzy sets exist. For example, Qiu et al.[28] introduced the set of all general fuzzy star-shaped numbers, denoted by Sm , where S m is a subset of Sm , which is a subset of a kind of general fuzzy set FB (Rm ). If we characterize totally bounded sets, relatively compact sets, and compact sets in general fuzzy set spaces, then we can obtain parallel characterizations for particular fuzzy set spaces immediately because the latter are subspaces of the former. (E m , dp ), (S0m , dp ), (S m , dp ) and (Sm , dp ) are not complete. Kr¨atschmer [25] presented the completion of (E m , dp ), described by the support functions of fuzzy numbers. This raises the basic problem of what are the completions of the rest of the spaces. This issue should be replaced by the more general question of how to generate the completions of fuzzy set spaces with respect to the dp metric. These problems are closely related to the problem of characterizing totally bounded sets and relatively compact sets in fuzzy set spaces equipped with the dp metric. This paper proposes answers to all the above problems or questions. The

3

questions are closely related; in some ways they are different aspects of a same problem. To put this in a more general setting which does not have any assumptions of convexity or star-shapedness, we consider FB (Rm ), the set of all normal, upper semi-continuous, compact-support fuzzy sets on Rm , and we introduce the Lp extension of a fuzzy set space, where Lp extensions of (S0m , dp ), (E m , dp ), (S m , dp ), (Sm , dp ) and (FB (Rm ), dp ) are denoted by (S0m,p , dp ), (E m,p , dp ), (S m,p , dp ), (Sm,p , dp ), and (FB (Rm )p , dp ), respectively. All the fuzzy set spaces discussed here are subspaces of (FB (Rm )p , dp ). We provide characterizations of totally bounded sets, relatively compact sets, and compact sets in (FB (Rm ), dp ) and (FB (Rm )p , dp ), and prove that each Lp extension discussed here is exactly the completion of its original space. We then show that the subspaces of (FB (Rm ), dp ) and (FB (Rm )p , dp ) have parallel characterizations of totally bounded sets, relatively compact sets, and compact sets. Finally, as applications of our results, we consider properties of the dp metric and review compactness characterizations proposed in previous work. The remainder of this paper is organized as follows. Section 2 introduces and discusses some properties of the Hausdorff metric and Section 3 recalls and introduces some relevant concepts and results of fuzzy sets. Section 4 presents characterizations of relatively compact sets, totally bounded sets, and compact sets in (FB (Rm )p , dp ), and Section 5 shows that FB (Rm )p is the completion of FB (Rm ) according to the dp metric. We then provide characterizations of relatively compact sets, totally bounded sets, and compact sets in (FB (Rm ), dp ). Based on the outcomes in Sections 4 and 5 and discussions on convexity and star-shapedness of fuzzy sets, Section 6 shows that (S0m,p , dp ), (E m,p , dp ), (S m,p , dp ), and (Sm,p , dp ) are exactly the completions of (S0m , dp ), (E m , dp ), (S m , dp ), and (Sm , dp ), respectively. We also clarify relationships among the ten fuzzy set spaces discussed, and provide characterizations of totally bounded sets, relatively compact sets, and compact sets in these spaces. Section 7 provides some applications of the results, and finally we provide our conclusions in Section 8. 2. Hausdorff metric Let N be the set of all natural numbers, Q be the set of all rational numbers, Rm be m-dimensional Euclidean space, KC (Rm ) be the set of all the nonempty compact and convex sets in Rm , K(Rm ) be the set of all 4

nonempty compact sets in Rm , and C(Rm ) be the set of all nonempty closed sets in Rm . The well-known Hausdorff metric on C(Rm ) is defined as H(U, V ) = max{H ∗ (U, V ), H ∗ (V, U )} for arbitrary U, V ∈ C(Rm ), where H ∗ (U, V ) = sup d (u, V ) = sup inf d (u, v). u∈U v∈V

u∈U

Proposition 2.1. [9] (C(Rm ), H) is a complete metric space in which K(Rm ) and KC (Rm ) are closed subsets. Hence, K(Rm ) and KC (Rm ) are also complete metric spaces. Proposition 2.2. [9, 30] A nonempty subset U of (K(Rm ), H) is compact if and only if it is closed and bounded in (K(Rm ), H). m Proposition +∞2.3. [9] Letm{un } ⊂ K(R ) satisfy u1 ⊇ u2 ⊇ . . . ⊇ un ⊇ . . .. Then u = n=1 un ∈ K(R ) and H(un , u) → 0 as n → ∞.  On the other hand, if u1 ⊆ u2 ⊆ . . . ⊆ un ⊆ . . . and u = +∞ n=1 un ∈ K(Rm ), then H(un , u) → 0 as n → ∞.

A set K ∈ K(Rm ) is said to be star-shaped relative to a point x ∈ K if, for each y ∈ K, the line xy joining x to y is contained in K. The kernel ker K of K is the set of all points x ∈ K such that xy ⊂ K for each y ∈ K. KS (Rm ) denotes all star-shaped sets in Rm . Then, KC (Rm )  KS (Rm ), and ker K ∈ KC (Rm ) for all K ∈ KS (Rm ). We say that a sequence of sets {Cn } converges to C, in the sense of Kuratowski, if C = lim inf Cn = lim sup Cn , n→∞

n→∞

Where lim inf Cn = {x ∈ X : x = lim xn , xn ∈ Cn }, n→∞

n→∞

lim sup Cn = {x ∈ X : x = lim xnj , xnj ∈ Cnj } = n→∞

j→∞

∞  

Cm .

n=1 m≥n

In this case, we’ll write simply C = limn→∞ Cn (K). The following two known propositions discuss the relationship between the notion of convergence induced by the Hausdorff metric and convergence in the sense of Kuratowski (see [11] for details). 5

Proposition 2.4. Suppose that u, un , n = 1, 2, . . ., are nonempty compact sets in Rm . Then H(un , u) → 0 as n → ∞ implies that u = limn→∞ un (K). Proposition 2.5. Suppose that u, un , n = 1, 2, . . ., are nonempty compact sets in Rm , and that un , n = 1, 2, . . . are connected sets. If u = limn→∞ un (K), then H(un , u) → 0 as n → ∞. Theorem 2.1. KS (Rm ) is a closed set in (K(Rm ), H). Proof. Suppose that {un } ⊂ KS (Rm ), u ∈ K(Rm ), and H(un , u) → 0 as n → ∞. Then we need to prove that u ∈ KS (Rm ). Choose xn ∈ Ker un , n = 1, 2, . . ., then there exists an N such that xn ∈ U for all n ≥ N , where U := {y : d(y, u) ≤ 1}. Since U is a compact set, we know there is a subsequence {xni } of {xn } such that limi→∞ xni = x0 . Therefore. x0 ∈ lim supn→∞ un , and it follows from Proposition 2.4 that x0 ∈ u. To show that u is star-shaped and x0 ∈ ker u, it suffices to show that λx0 + (1 − λ)z ∈ u for all z ∈ u and λ ∈ [0, 1]. Given z ∈ u, since u = lim inf n→∞ un , there is a sequence {zn : zn ∈ un } such that limn→∞ zn = z. Hence, for each λ ∈ [0, 1], λx0 + (1 − λ)z = lim λxni + (1 − λ)zni ∈ lim sup un , i→∞

n→∞

and thus, from Proposition 2.4, λx0 + (1 − λ)z ∈ u. Corollary 2.1. Let u, un be star-shaped sets, n = 1, 2, . . .. If H(un , u) → 0, then lim supn→∞ ker un ⊂ ker u. Proof. The proof follows directly from the proof of Theorem 2.1. 3. Fuzzy set spaces We recall and introduce various fuzzy set spaces, including fuzzy numbers space, fuzzy star-shaped numbers space, and general fuzzy star-shaped numbers space. Some basic properties of these spaces are discussed. Let F (Rm ) represent all fuzzy subsets on Rm , i.e., functions from Rm to m [0, 1] (see [9, 35] for details). 2R := {S : S ⊆ Rm } can be embedded in

6

F (Rm ), as any S ⊂ Rm can be seen as its characteristic function, i.e., the fuzzy set  1, x ∈ S,  S(x) = 0, x ∈ / S. For u ∈ F (Rm ), let [u]α denote the α-cut of u, i.e., {x ∈ Rm : u(x) ≥ α}, [u]α = supp u = {x ∈ Rm : u(x) > 0},

α ∈ (0, 1], α = 0.

For u ∈ F (Rm ), we suppose that (i) u is normal, i.e., there exists at least one x0 ∈ Rm with u(x0 ) = 1; (ii) u is upper semi-continuous; (iii-1) u is fuzzy convex, i.e., u(λx+(1−λ)y) ≥ min{u(x), u(y)} for x, y ∈ Rm and λ ∈ [0, 1]; (iii-20 ) u is fuzzy star-shaped with respect to the origin, i.e., u(λy) ≥ u(y) for all y ∈ Rm and λ ∈ [0, 1]. (iii-2) u is fuzzy star-shaped, i.e., there exists x ∈ Rm such that u is fuzzy star-shaped with respect to x, i.e., u(λy + (1 − λ)x) ≥ u(y) for all y ∈ Rm and λ ∈ [0, 1]; (iii-3) Given α ∈ (0, 1], then there exists xα ∈ [u]α such that xα y ∈ [u]α for all y ∈ [u]α ; (iv-1) [u]0 is a bounded set in Rm ; 1/p

 1 p H([u] , {0}) dα < +∞, where p ≥ 1, and 0 denotes the (iv-2) α 0 m origin of R ; (iv-3) [u]α is a bounded set in Rm when α > 0. • If u satisfies (i), (ii), (iii-1), and (iv-1), then u is a fuzzy number. The set of all fuzzy numbers is denoted by E m . • If u satisfies (i), (ii), (iii-20 ), and (iv-1), then u is a fuzzy star-shaped number with respect to the origin. The set of all fuzzy star-shaped numbers with respect to the origin is denoted by S0m . • If u satisfies (i), (ii), (iii-2), and (iv-1), then u is a fuzzy star-shaped number. The set of all fuzzy star-shaped numbers is denoted by S m . • If u satisfies (i), (ii), (iii-3), and (iv-1), then u is a general fuzzy starshaped number. The set of all general fuzzy star-shaped numbers is denoted by Sm . 7

The definitions of fuzzy star-shaped numbers and general fuzzy starshaped numbers were given by Diamond [9] and Qiu et al. [28], respectively. Rm can be embedded in E m , as any r ∈ Rm can be viewed as the fuzzy number  1, x = r, r(x) = 0, x = r.  E m  S0m and S0m  E m , and if u ∈ S m then α∈(0,1] ker [u]α = ∅. However this inequality may not hold when u ∈ Sm . Clearly S0m = {u ∈ S m : 0 ∈  m m m  Sm . α∈(0,1] ker [u]α }. So E , S0  S To illustrate and prove the conclusions in this paper, we need to introduce Lp type noncompact fuzzy sets. Let u ∈ F (Rm ). • If u satisfies (i), (ii), (iii-1), and (iv-2), then u is an Lp type noncompact fuzzy number. The collection of all such fuzzy sets is denoted by E m,p . • If u satisfies (i), (ii), (iii-20 ), and (iv-2), then u is an Lp type noncompact fuzzy star-shaped number with respect to the origin. The collection of all such fuzzy sets is denoted by S0m,p . • If u satisfies (i), (ii), (iii-2), and (iv-2), then u is an Lp type noncompact fuzzy star-shaped number. The collection of all such fuzzy sets is denoted by S m,p . • If u satisfies (i), (ii), (iii-3), and (iv-2), then u is an Lp type noncompact general fuzzy star-shaped number. The collection of all such fuzzy sets is denoted by Sm,p . Consequently, E m  E m,p , S0m  S0m,p , S m  S m,p and Sm  Sm,p . Thus, a kind of Lp type noncompact fuzzy set may be obtained using the weaker assumption (iv-2) instead of (iv-1) on the corresponding kind of compact fuzzy set. Therefore, the latter is a subset of the former. We call this process Lp extension, and call each kind of Lp type noncompact fuzzy set Lp extension of the corresponding kind of compact fuzzy set. The eight kinds of fuzzy sets above have assumptions of convexity or star-shapedness. However, there is an increasing amount of attention being given to general fuzzy sets for theoretical and real-world applications. For example, the study of fuzzy differential equations [2, 24] considers fuzzy sets with no assumptions of convexity or star-shapedness. Thus, we want to 8

move to considering general fuzzy sets, with no assumptions of convexity or star-shapedness. So we introduce the following kinds of general fuzzy sets. Suppose u ∈ F (Rm ). • If u satisfies (i), (ii), and (iv-1), then u is a normal upper semi-continuous compact-support fuzzy set on Rm . The collection of all such fuzzy sets is denoted by FB (Rm ). • If u satisfies (i), (ii), and (iv-2), then u is a normal upper semi-continuous Lp type noncompact-support fuzzy set on Rm . The collection of all such fuzzy sets is denoted by FB (Rm )p . • If u satisfies (i), (ii), and (iv-3), then u is a normal upper semi-continuous noncompact-support fuzzy set on Rm . The collection of all such fuzzy sets is denoted by FGB (Rm ). FB (Rm )p is the Lp extension of FB (Rm ), and E m , S0m  S m  Sm  FB (Rm ), E m,p , S0m,p  S m,p  Sm,p  FB (Rm )p , FB (Rm )  FB (Rm )p  FGB (Rm ).

Diamond and Kloeden [9] introduced the dp distance (1 ≤ p < ∞) on S , which is defined as m

 dp (u, v) =

0

1

p

H([u]α , [v]α ) dα

1/p (1)

for all u, v ∈ S m . Since u ∈ F (Rm ) satisfies assumption (ii) is equivalent to [u]α ∈ C(Rm ) for all α ∈ (0, 1], dp distance (1 ≤ p < ∞) can be defined on FGB (Rm ). However, dp distance is not a metric on FGB (Rm ) because dp (u, v) may equal +∞ for some u, v ∈ FGB (Rm ). The dp distance, p ≥ 1, is a metric on FB (Rm )p . All fuzzy set spaces discussed here are subspaces of (FB (Rm )p , dp ). An Lp type noncompact fuzzy set space endowed with the dp metric is called an Lp extension of the corresponding compact fuzzy set space with dp metric. Therefore, fuzzy set spaces (E m,p , dp ), (S0m,p , dp ), (S m,p , dp ), (Sm,p , dp ), and (FB (Rm )p , dp ) are the Lp extensions of the fuzzy set spaces (E m , dp ), (S0m , dp ), (S m , dp ), (Sm , dp ), and (FB (Rm ), dp ), respectively. 9

Diamond and Kloeden [9] showed that (E m , dp ) is not a complete space. Their analysis indicates that the four spaces (S0m , dp ), (S m , dp ), (Sm , dp ), and (FB (Rm ), dp ) are also not complete. Kr¨atschmer [25] provided the completion of (E m , dp ), using support functions of fuzzy numbers. In subsequent sections, we will show that the completion of every incomplete fuzzy set space discussed here is exactly its Lp extension, i.e., their completions can be obtained from the Lp extension. Remark 3.1. For each u ∈ FB (Rm )p , [u]α is a compact set when α ∈ (0, 1], and [u]0 , the 0 cut, is the only possible unbounded cut set. Therefore, [u]α ∈ KC (Rm ) for all u ∈ E m,p and α ∈ (0, 1], and [u]α ∈ KS (Rm ) for all u ∈ Sm,p and α ∈ (0, 1].  Denote ker u := α∈(0,1] ker [u]α for u ∈ Sm,p (see [9, 28]). Given u ∈ Sm,p , then u ∈ S m,p if and only if ker u = ∅. The following representation theorem is used widely in the theory of fuzzy numbers. Proposition 3.1. [27] Given u ∈ E m , then (i) [u]λ ∈ KC (Rm ) for all λ ∈ [0, 1]; (ii) [u]λ = γ<λ [u]γ for all λ ∈ (0, 1];  (iii) [u]0 = γ>0 [u]γ . Moreover, if the family of sets {vα : α ∈ [0, 1]} satisfy conditions (i)–(iii), then there exists a unique u ∈ E m such that [u]λ = vλ for each λ ∈ [0, 1]. Similarly, we can obtain representation theorems for S0m , S m , Sm , FB (Rm ), E m,p , S0m,p , S m,p , Sm,p , and FB (Rm )p , as listed below and used subsequently. Theorem 3.1. Given u ∈ S0m , then  (i) [u]λ ∈ KS (Rm ) for all λ ∈ [0, 1], and 0 ∈ α∈(0,1] ker [u]α ;  (ii) [u]λ = γ<λ [u]γ for all λ ∈ (0, 1];  (iii) [u]0 = γ>0 [u]γ . Moreover, if the family of sets {vα : α ∈ [0, 1]} satisfy conditions (i)–(iii), then there exists a unique u ∈ S0m such that [u]λ = vλ for each λ ∈ [0, 1]. Theorem 3.2. Given u ∈ S m , then  (i) [u]λ ∈ KS (Rm ) for all λ ∈ [0, 1], and α∈(0,1] ker [u]α = ∅;  (ii) [u]λ = γ<λ [u]γ for all λ ∈ (0, 1];  (iii) [u]0 = γ>0 [u]γ . 10

Moreover, if the family of sets {vα : α ∈ [0, 1]} satisfy conditions (i)–(iii), then there exists a unique u ∈ S m such that [u]λ = vλ for each λ ∈ [0, 1]. Theorem 3.3. Given u ∈ Sm , then (i) [u]λ ∈ KS (Rm ) for all λ ∈ [0, 1]; (ii) [u]λ = γ<λ [u]γ for all λ ∈ (0, 1];  (iii) [u]0 = γ>0 [u]γ . Moreover, if the family of sets {vα : α ∈ [0, 1]} satisfy conditions (i)–(iii), then there exists a unique u ∈ Sm such that [u]λ = vλ for each λ ∈ [0, 1]. Theorem 3.4. Given u ∈ FB (Rm ), then m (i) [u]λ ∈ K(R  ) for all λ ∈ [0, 1]; (ii) [u]λ = γ<λ [u]γ for all λ ∈ (0, 1];  (iii) [u]0 = γ>0 [u]γ . Moreover, if the family of sets {vα : α ∈ [0, 1]} satisfy conditions (i)–(iii), then there exists a unique u ∈ FB (Rm ) such that [u]λ = vλ for each λ ∈ [0, 1]. Theorem 3.5. Given u ∈ E m,p , then (i) [u]λ ∈ KC (Rm ) for all λ ∈ (0, 1]; (ii) [u]λ = γ<λ [u]γ for all λ ∈ (0, 1];  (iii) [u]0 = γ>0 [u]γ ; 1/p

 1 p H([u] , {0}) dα < +∞. (iv) α 0 Moreover, if the family of sets {vα : α ∈ [0, 1]} satisfy conditions (i)–(iv), then there exists a unique u ∈ E m,p such that [u]λ = vλ for each λ ∈ [0, 1]. Theorem 3.6. Given u ∈ S0m,p , then  (i) [u]λ ∈ KS (Rm ) for all λ ∈ (0, 1] and 0 ∈ α∈(0,1] ker [u]α ;  (ii) [u]λ = γ<λ [u]γ for all λ ∈ (0, 1];  (iii) [u]0 = γ>0 [u]γ ; 1/p

 1 p H([u] , {0}) dα < +∞. (iv) α 0 Moreover, if the family of sets {vα : α ∈ [0, 1]} satisfy conditions (i)–(iv), then there exists a unique u ∈ S0m,p such that [u]λ = vλ for each λ ∈ [0, 1]. Theorem 3.7. Given u ∈ S m,p , then  (i) [u]λ ∈ KS (Rm ) for all λ ∈ (0, 1] and α∈(0,1] ker [u]α = ∅;  (ii) [u]λ = γ<λ [u]γ for all λ ∈ (0, 1]; 11

 (iii) [u]0 = γ>0 [u]γ ; 1/p

 1 p H([u] , {0}) dα < +∞. (iv) α 0 Moreover, if the family of sets {vα : α ∈ [0, 1]} then there exists a unique u ∈ S m,p such that [u]λ Theorem 3.8. Given u ∈ Sm,p , then (i) [u]λ ∈ KS (Rm ) for all λ ∈ (0, 1]; (ii) [u]λ = γ<λ [u]γ for all λ ∈ (0, 1];  (iii) [u]0 = γ>0 [u]γ ; 1/p

 1 p H([u] , {0}) dα < +∞. (iv) α 0 Moreover, if the family of sets {vα : α ∈ [0, 1]} then there exists a unique u ∈ Sm,p such that [u]λ

satisfy conditions (i)–(iv), = vλ for each λ ∈ [0, 1].

satisfy conditions (i)–(iv), = vλ for each λ ∈ [0, 1].

Theorem 3.9. Given u ∈ FB (Rm )p , then m (i) [u]λ ∈ K(R  ) for all λ ∈ (0, 1]; (ii) [u]λ = γ<λ [u]γ for all λ ∈ (0, 1];  (iii) [u]0 = γ>0 [u]γ ; 1/p

 1 p H([u] , {0}) dα < +∞. (iv) α 0 Moreover, if the family of sets {vα : α ∈ [0, 1]} satisfy conditions (i)– (iv), then there exists a unique u ∈ FB (Rm )p such that [u]λ = vλ for each λ ∈ [0, 1]. 4. Characterizations of relatively compact sets, totally bounded sets, and compact sets in (FB (Rm )p , dp ) We present a characterization of relatively compact sets in the fuzzy set space (FB (Rm )p , dp ), and based on this, we present characterizations of totally bounded sets and compact sets in (FB (Rm )p , dp ), respectively. Compactness characterizations of fuzzy set spaces with the dp metric have been studied for many years. The following two concepts were introduced by Diamond and Kloeden [8] and Ma [26], and are important to establish and illustrate compactness characterizations. Definition 4.1. [8] Let u ∈ FB (Rm )p . If, for a given ε > 0, there is a δ(u, ε) > 0 such that for all 0 ≤ h < δ

 1 1/p p H([u]α , [u]α−h ) dα < ε, h

12

where 1 ≤ p < +∞, then we say u is p-mean left-continuous. Suppose that U is a nonempty set in FB (Rm )p . If the above inequality holds uniformly for all u ∈ U , then we say U is p-mean equi-left-continuous. Definition 4.2. [26] We say a set U ⊂ FB (Rm )p is uniformly p-mean bounded if there is a constant M > 0 such that dp (u,  0) ≤ M for all u ∈ U . If U is uniformly p-mean bounded, this is equivalent to U being a bounded set in (FB (Rm )p , dp ). Diamond [7, 8] characterized the compact sets in (E m , dp ) and (S0m , dp ). Proposition 4.1. [8] A closed set U of (E m , dp ) is compact if and only if: (i) {[u]0 : u ∈ U } is bounded in (K(Rm ), H); (ii) U is p-mean equi-left-continuous. Proposition 4.2. [7] A closed set U of (S0m , dp ) is compact if and only if: (i) {[u]0 : u ∈ U } is bounded in (K(Rm ), H); (ii) U is p-mean equi-left-continuous. Ma [26] found an error in Proposition 4.1, and modified it to the following Proposition 4.3. For notational convenience, let u ∈ F (Rm ), Ma used the symbol u(α) to denote the fuzzy set  u(x), if u(x) ≥ α, (α) u (x) := 0, if u(x) < α. Proposition 4.3. [26] A closed set U of (E m , dp ) is compact if and only if: (i) U is uniformly p-mean bounded; (ii) U is p-mean equi-left-continuous; (h) (iii) For {uk } ⊂ U , if {uk : k = 1, 2, . . .} converges to u(h) ∈ E m in dp (h) metric for any h > 0, then there exists a u0 ∈ E m such that u0 = u(h). Wu and Zhao [36] pointed out that there exists a fault in the compactness characterization of Ma [26] and showed that the compactness characterization of Diamond [7] has the same type of error as the compactness characterization of Diamond [8]. So Propositions 4.1, 4.2, and 4.3 were all wrong. They gave correct compactness criteria in (E m , dp ) and (S0m , dp ), respectively. Proposition 4.4. [36] A closed set U of (E m , dp ) is compact if and only if: (i) U is uniformly p-mean bounded; (ii) U is p-mean equi-left-continuous; 13

(iii) Let ri be a decreasing sequence in (0, 1] converging to zero. For {uk } ⊂ (r ) U , if {uk i : k = 1, 2, . . .} converges to u(ri ) ∈ E m in dp metric, then there (r ) exists a u0 ∈ E m such that [u0 i ]α = [u(ri )]α when ri < α ≤ 1. Proposition 4.5. [36] A closed set U of (S0m , dp ) is compact if and only if: (i) U is uniformly p-mean bounded; (ii) U is p-mean equi-left-continuous; (iii) Let ri be a decreasing sequence in (0, 1] converging to zero. For {uk } ⊂ (r ) U , if {uk i : k = 1, 2, . . .} converges to u(ri ) ∈ S0m in dp metric, then there (r ) exists a u0 ∈ S0m such that [u0 i ]α = [u(ri )]α when ri < α ≤ 1. From Proposition 4.5, Zhao and Wu [38] proposed further a compactness criterion in (S m , dp ). Proposition 4.6. [38] A closed set U of (S m , dp ) is compact if and only if: (i) U is uniformly p-mean bounded; (ii) U is p-mean equi-left-continuous; (iii) Let ri be a decreasing sequence in (0, 1] converging to zero. For {uk } ⊂ (r ) U , if {uk i : k = 1, 2, . . .} converges to u(ri ) ∈ S m in dp metric, then there (r ) exists a u0 ∈ S m such that [u0 i ]α = [u(ri )]α when ri < α ≤ 1. Comparing Propositions 4.4, 4.5, and 4.6 with the Arzel`a-Ascoli theorem, the latter provides compactness criteria by characterizing the totally bounded set whereas the former does not. Since total boundedness is a key feature of compactness, it is a basic and important problem to consider characterizations of totally bounded sets in fuzzy set spaces. To obtain a characterization of totally bounded sets in (FB (Rm )p , dp ), we first characterize relatively compact sets in (FB (Rm )p , dp ). Some fundamental conclusions and concepts in classic analysis and topology are listed below, which are useful in this paper (see [3] for details). • Lebesgue’s dominated convergence theorem. Let {fn } be a sequence of integrable functions that converges almost everywhere to a function f , and suppose that {fn } is dominated by an integrable function g. Then f is integrable, and f = limn→∞ fn . • Fatou’s Lemma. If {fn } is a sequence of nonnegative integrable functions that converges almost everywhere to a function f , and if the sequence  { fn } is bounded above, then f is integrable and f ≤ lim inf fn . 14

• Absolute continuity of Lebesgue integral. Suppose that f is Lebesgue integrable on E, then for arbitrary ε > 0, there is a δ > 0  such that A f dx < ε whenever A ⊆ E and m(A) < δ. • Minkowski’s inequality. Let p ≥ 1, and let f, g be measurable functions on R such that |f |p and |g|p are integrable. Then |f + g|p is integrable, and Minkowski’s inequality

 |f + g|

p

1/p

 ≤

|f |

p

1/p

 +

|g|

p

1/p

holds. • A relatively compact subset Y of a topological space X is a subset with compact closure. In the case of a metric topology, the criterion for relative compactness becomes that any sequence in Y has a subsequence convergent in X. • Let (X, d) be a metric space. A set U in X is totally bounded if and only if, for each ε > 0, it contains a finite ε approximation, where an ε approximation to U is a subset S of U such that ρ(x, S) < ε for each x ∈ U. • Let (X, d) be a metric space. Then if a set U in X is relatively compact then it is totally bounded. For subsets of a complete metric space these two meanings coincide. Thus (X, d) is a compact space if and only if X is totally bounded and complete. We start with some lemmas that discuss properties of bounded subsets and elements of (FB (Rm )p , dp ). Lemma 4.1. Suppose that U is a bounded set in (FB (Rm )p , dp ), then {[u]α : u ∈ U } is a bounded set in (K(Rm ), H) for each α > 0. Proof. If there exists an α0 > 0 such that {[u]α0 : u ∈ U } is not a bounded set in (K(Rm ), H), then there is a u ∈ U such that [u]α0 ∈ / K(Rm ) or {[u]α0 : u ∈ U } is a unbounded set in K(Rm ). For both cases, there exist un ∈ U such that H([un ]α0 , {0}) > n · ( α10 )1/p when n = 1, 2, . . ., and hence 1/p

 1 p H([u ] , {0}) dα > n, which contradicts the boundedness of U . n α 0 15

Lemma 4.2. Suppose that u ∈ FB (Rm )p and α ∈ (0, 1], then H([u]α , [u]β ) → 0 as β → α−. Proof. The proof follows directly from Theorem 3.9 and Proposition 2.3. Lemma 4.3. If u ∈ FB (Rm )p , then u is p−mean left-continuous. Proof. Given ε > 0, then since u ∈ FB (Rm )p , from the absolute continuity of Lebesgue integral, we know there exists an h1 > 0 such that



h1

0

1/p p

H([u]α , {0}) dα

≤ ε/3.

(2)

Since [u] h1 ∈ K(Rm ), then there exists an M > 0 such that H([u] h1 , {0}) < 2 2 M . Thus, H([u]α , [u]α−h ) ≤ H([u]α , {0}) + H([u]α−h , {0}) ≤ 2M for all α ≥ h1 and h ≤ h1 /2. From Lemma 4.2, H([u]α , [u]α−h ) → 0 when h → 0+. Lebesgue’s dominated convergence theorem and (3),



1

h1

(3)

Thus, from

1/p

p

H([u]α , [u]α−h ) dα

→0

when h → 0+. Therefore, there exists an h2 > 0 such that



1 h1

1/p

p

H([u]α , [u]α−h ) dα

< ε/3

(4)

for all 0 ≤ h ≤ min{h1 /2, h2 }. Combining (2) and (4), for all h ≤ h3 = min{h1 /2, h2 },

 h

1

H([u]α , [u]α−h ) dα

 ≤

 ≤

1/p

p

1/p

h1

H([u]α , [u]α−h ) dα

h h1 h

p

H([u]α , {0}) dα

1

+

1/p p



h1



h1

+ h

16

p

1/p

H([u]α , [u]α−h ) dα 1/p p

H([u]α−h , {0}) dα

+ ε/3

 ≤

h1 0

1/p p

H([u]α , {0}) dα



h1

+ 0

1/p p

H([u]α , {0}) dα

+ ε/3

≤ ε/3 + ε/3 + ε/3 = ε.

(5)

From the arbitrariness of ε, we know that u is p−mean left-continuous. Finally, we arrive at the main results of this section. The following theorem characterizes the relatively compactness of (FB (Rm )p , dp ). Theorem 4.1. U is a relatively compact set in (FB (Rm )p , dp ) if and only if (i) U is uniformly p-mean bounded; (ii) U is p-mean equi-left-continuous. Proof. Necessary. If U is a relatively compact set in (FB (Rm )p , dp ), then since (FB (Rm )p , dp ) is a metric space, it follows that U is a bounded set in (FB (Rm )p , dp ), i.e., U is uniformly p-mean bounded. Now we prove that U is p−mean equi-left-continuous. Given ε > 0, since U is a relatively compact set, there exists an ε/3 net {u1 , u2 , . . . un } of U . From Lemma 4.3, {uk : k = 1, 2, . . . n} is p-mean equi-left-continuous. Hence there exists δ > 0 such that

 1 1/p p H([uk ]α , [uk ]α−h ) dα ≤ ε/3. (6) h

for all h ∈ [0, δ) and k = 1, 2, . . . , n. Given u ∈ U , there is an uk such that dp (u, uk ) ≤ ε/3, and thus, by (6), we know that for all h ∈ [0, δ),

 h

1

H([u]α , [u]α−h ) dα

 ≤

1/p

p

1 h

p

H([u]α , [uk ]α ) dα

1/p



1

+ h

p

H([uk ]α , [uk ]α−h ) dα

≤ ε/3 + ε/3 + ε/3 = ε.

1/p



1

+ h

p

H([uk ]α−h , [u]α−h ) dα (7)

From the arbitrariness of ε and u ∈ U , U is p-mean equi-left-continuous. Sufficient. If U satisfies (i) and (ii), then to show U is a relatively compact set, it suffices to find a convergent subsequence of an arbitrarily given sequence in U . 17

1/p

Let {un } be a sequence in U . To find a subsequence {vn } of {un } that converges to v ∈ FB (Rm )p according to dp metric, we split the proof into three steps. Step 1. Find a subsequence {vn } of {un } and v ∈ FGB (Rm ) such that a.e. H([vn ]α , [v]α ) → 0 ([0, 1]).

(8)

Since U is uniformly p-mean bounded, by Lemma 4.1, {[u]α : u ∈ U } is a bounded set in K(Rm ) for each α ∈ (0, 1]. Thus, from Proposition 2.2, for each α > 0, {[u]α : u ∈ U } is a relatively compact set in (K(Rm ), H). Arrange all rational numbers in (0, 1] into a sequence q1 , q2 , . . . , qn , . . .. (1) (1) Then {un } has a subsequence {un } such that {[un ]q1 } converges to uq1 ∈ (1) (1) (k) K(Rm ), i.e. H([un ]q1 , uq1 ) → 0. If {un }, . . . , {un } have been chosen, we (k+1) (k) (k+1) can choose a subsequence {un } of {un } such that {[un ]qk+1 } converges m to uqk+1 ∈ K(R ). Thus we obtain nonempty compact sets uqk , k = 1, 2, . . .. with uqm ⊆ uql whenever qm > ql . (n) Put vn = {un } for n = 1, 2, . . .. Then {vn } is a subsequence of {un } and H([vn ]qk , uqk ) → 0 as n → ∞

(9)

for k = 1, 2, . . .. Define {vα : α ∈ [0, 1]} as  u , α ∈ (0, 1]; qk <α qk vα = α = 0. α∈(0,1] vα , Then vα , α ∈ [0, 1], have the following properties: (i) vλ ∈ K(Rm ) for all λ ∈ (0, 1];  (ii) vλ = γ<λ vγ for all λ ∈ (0, 1]; (iii) v0 =

 γ>0

vγ .

From Proposition 2.3, obtain that vα ∈ K(Rm ) for all α ∈ (0, 1]. Thus, property (i) is proved, and properties (ii) and (iii) follow immediately from the definition of vα . Define a function v : Rm → [0, 1] as    x ∈ λ>0 vλ , x∈vλ λ, v(x) = 0, otherwise. 18

Then v is a fuzzy set on Rm . From properties (i), (ii), and (iii) of vα , we know that [v]α = vα . a.e. Therefore, v ∈ FGB (Rm ). If (I) and (II) are true, then (8), i.e., H([vn ]α , [v]α ) → 0 ([0, 1]). (I) P (v) is at most countable, where P (v) = {α ∈ (0, 1) : {v > α}  [v]α },  where {v > α} := β>α [v]β . (II) If α ∈ (0, 1)\P (v), then H([vn ]α , [v]α ) → 0 as n → ∞.

(10)

First, we show that (I). Let D(v) := {α ∈ (0, 1) : [v]α  {v > α} }, then P (v) ⊆ D(v) (Indeed, P (v) = D(v)). Following the appendix of [21], D(v) is at most countable. Therefore, P (v) is at most countable. Second, we show that (II). Suppose that α ∈ (0, 1)\P (v), then from Proposition 2.3, H([v]β , [v]α ) → 0 as β → α. Thus, given ε > 0, we can find a δ > 0 such that H(uq , vα ) < ε for all q ∈ Q with |q − α| < δ. So H ∗ ([vn ]α , vα ) ≤ H ∗ ([vn ]q1 , vα ) ≤ H ∗ ([vn ]q1 , uq1 ) + ε for q1 ∈ Q ∩ (α − δ, α). Hence, from (9) and the arbitrariness of ε, H ∗ ([vn ]α , vα ) → 0 (n → ∞).

(11)

On the other hand, H ∗ (vα , [vn ]α ) ≤ H ∗ (vα , [vn ]q2 ) ≤ H ∗ (uq2 , [vn ]q2 ) + ε for q2 ∈ Q ∩ (α, α + δ). Hence, from (9) and the arbitrariness of ε, H ∗ (vα , [vn ]α ) → 0 (n → ∞).

(12)

Combined with (11) and (12), we thus obtain (10). Step 2. Prove that

 0

1

p

H([vn ]α , [v]α ) dα 19

1/p → 0.

(13)

Given ε > 0, then, for all h < 1/2, 1/p

 h p H([vn ]α , [v]α ) dα

 ≤



0

≤

p

H([vn ]α , [vn ]α+h ) dα

0

p

1 h

0



H([vn ]β−h , [vn ]β ) dβ

p

H([vn ]α+h , [v]α+h ) dα

 + h

p

H([vn ]β , [v]β ) dβ

h 1

p

H([vn ]β , [v]β ) dβ

 0

 h



H([v]α+h , [v]α ) dα

h

p

H([v]β , [v]β−h ) dβ (14)

Since U is p-mean equi-left-continuous, there exists an h ∈ (0, 1/2) such that

 1 1/p p H([vn ]β−h , [vn ]β ) dβ < ε/4 (15) h

for all n = 1, 2, . . .. From (8), if n → ∞ then H([vn ]β−h , [vn ]β ) → H([v]β−h , [v]β ) a.e. on β ∈ [h, 1]. Therefore, from Fatou’s Lemma, we have

 1

 1 1/p 1/p p p H([v]β−h , [v]β ) dβ ≤ lim inf H([vn ]β−h , [vn ]β ) dβ ≤ ε/4, n

h

h

(16) Since [vn ]h and [v]h are contained in {[u]α : u ∈ U }, which is compact, it follows from Lebesgue’s dominated convergence theorem and (8) that

 1 1/p p H([vn ]α , [v]α ) dα →0 h

as n → ∞. Hence, there exists an N (h, ε) such that 1/p

 1 p H([vn ]α , [v]α ) dα ≤ ε/4 h

for all n ≥ N . Combined with (14), (15), (16), and (17), 1/p

 1 p H([vn ]α , [v]α ) dα 0

20

1/p p

H([v]β , [v]β−h ) dβ

1

+

p

2h

+

1/p

1/p

h

+

1/p

2h

+

1/p

p

1/p

h

+

H([vn ]β−h , [vn ]β ) dβ

h



1/p

2h

=



1/p

h

(17)

1/p .

 ≤

1/p

h 0

p

H([vn ]α , [v]α ) dα



≤ ε/4 +

1 h



1

+ h 1/p

p

H([vn ]α , [v]α ) dα

p

1/p

H([vn ]α , [v]α ) dα



+ ε/4 + h

1

p

1/p

H([vn ]α , [v]α ) dα

≤ε for all n ≥ N . Thus we obtain (13) from the arbitrariness of ε. Step 3. Show that v ∈ FB (Rm )p . By (13), we know that there is an N such that



1 0

and then

 0

1

p

H([v]α , [vN ]α ) dα

< 1,

1/p

H([v]α , {0}) dα

 ≤

1/p

p

1 0

≤1+

p

1/p

H([v]α , [vN ]α ) dα

 0

1

p



1

+ 1/p

H([vN ]α , {0}) dα

0

p

1/p

H([vN ]α , {0}) dα

< +∞.

By properties (i),(ii), and (iii) of vα and Theorem 3.9, v ∈ FB (Rm )p . From steps 1–3, for any arbitrary sequence {un } of U , there exists a subsequence {vn } of {un } that converges to v ∈ FB (Rm )p . Thus, U is a relatively compact set in (FB (Rm )p , dp ). Using the characterization of relatively compact sets in (FB (Rm )p , dp ) given in Theorem 4.1, we can subsequently derive characterizations of totally bounded sets and compact sets as follows. Theorem 4.2. U is a totally bounded set in (FB (Rm )p , dp ) if and only if (i) U is uniformly p-mean bounded; (ii) U is p-mean equi-left-continuous. Proof. This only uses the total boundedness of U to show the necessary part of Theorem 4.1. Thus, a proof follows immediately from Theorem 4.1. 21

Theorem 4.3. Let U be a subset of (FB (Rm )p , dp ), then U is compact in (FB (Rm )p , dp ) if and only if (i) U is uniformly p-mean bounded; (ii) U is p-mean equi-left-continuous; (iii) U is a closed set in (FB (Rm )p , dp ). Proof. The proof follows immediately from Theorem 4.1. 5. The relationship between (FB (Rm ), dp ) and (FB (Rm )p , dp ) and properties of (FB (Rm ), dp ) We show that (FB (Rm )p , dp ) is the completion of (FB (Rm ), dp ), and then present characterizations of totally bounded sets, relatively compact sets, and compact sets in (FB (Rm ), dp ). Diamond and Kloeden [9] showed that (E m , dp ) is not a complete space. Ma [26] presented the following example to show this. Let  −x e , if 0 ≤ x ≤ n, n = 1, 2, . . . . un (x) = 0, otherwise, Then {un } is a Cauchy sequence in (E 1 , d1 ) ∩ (S01 , d1 ). Put  −x e , if 0 ≤ x < +∞, u(x) = 0, otherwise, then u ∈ FB (R1 )1 \FB (R1 ), and un converges to u in the d1 metric. Since E 1 , S01 ⊂ S 1 ⊂ S1 ⊂ FB (R1 ), none of (E 1 , d1 ), (S01 , d1 ), (S 1 , d1 ), (S1 , d1 ), and (FB (R1 ), d1 ) are complete. Similarly, we can show that none of (E m , dp ), (S0m , dp ), (S m , dp ), (Sm , dp ), and (FB (Rm ), dp ) are complete. First, we show that the completion of (FB (Rm ), dp ) is exactly its Lp extension (FB (Rm )p , dp ). Two facts are first proved, as shown in Theorems 5.1 and 5.2. Theorem 5.1. (FB (Rm )p , dp ) is a complete space. Proof. It suffices to prove that each Cauchy sequence has a limit in (FB (Rm )p , dp ). Let {un : n ∈ N} be a Cauchy sequence in (FB (Rm )p , dp ), we assert that {un : n ∈ N} is a relatively compact set in (FB (Rm )p , dp ).

22

To show this, by Theorem 4.1 it is equivalent to prove that {un : n ∈ N} is a bounded set in (FB (Rm )p , dp ) and that {un : n ∈ N} is p-mean equileft-continuous. The former follows immediately from {un : n ∈ N} being a Cauchy sequence. To prove the latter, suppose ε > 0, then since {un : n ∈ N} is a Cauchy sequence, there exists an N ∈ N that satisfies dp (un , um ) ≤ ε/3 for all n, m ≥ N . From Lemma 4.3, {uk : 1 ≤ k ≤ N } is p-mean equi-left-continuous, hence we can find an h > 0 such that 1/p

 1 p H([uk ]α−h , [uk ]α ) dα ≤ ε/3 (18) h

for all 1 ≤ k ≤ N . If k > N , then



 ≤

1

h

H([uk ]α−h , [uk ]α ) dα

1 h

1/p

p

p

H([uk ]α−h , [uN ]α−h ) dα

1/p



1

+ h

p

1/p

H([uN ]α−h , [uN ]α ) dα

≤ ε/3 + ε/3 + ε/3 = ε.



1

+ h

H([uN ]α , [uk ]α ) dα

(19)

From the arbitrariness of ε and ineqs. (18) and (19), {un : n ∈ N} is p-mean equi-left-continuous. From the relatively compactness of {un : n ∈ N} in (FB (Rm )p , dp ), there exists a subsequence {unk : k = 1, 2, . . .} of {un : n ∈ N} such that limk→∞ unk = u ∈ FB (Rm )p . Since {un : n ∈ N} is a Cauchy sequence, un , n = 1, 2, . . . also converges to u in (FB (Rm )p , dp ). This completes the proof. Remark 5.1. From Theorems 4.1 and 4.2, a set U in (FB (Rm )p , dp ) is totally bounded if and only if it is relatively compact. This fact alone can ensure that (FB (Rm )p , dp ) is complete. Theorem 5.2. FB (Rm ) is a dense set in (FB (Rm )p , dp ). Proof. Given u ∈ FB (Rm )p . Put un = u(1/n) , n = 1, 2, . . .. Then  if α ≥ 1/n, [u]α , [un ]α = [u]1/n , if α ≤ 1/n, 23

p

1/p

for all α ∈ [0, 1]. Since [un ]0 = [u]1/n ∈ K(Rm ), then, from Theorems 3.4 and 3.9, un ∈ FB (Rm ) for n = 1, 2, . . ..

 1/p 1 p H([u] , {0}) dα < +∞. Thus, from the Since u ∈ FB (Rm )p , α 0 absolute continuity of the Lebesgue’s integral, it holds that, for each ε > 0, there is a δ(ε) > 0 such that

 0

δ

1/p p

H([u]α , {0}) dα

< ε.

(20)

Note that  dp (un , u) =

1/n 0

 ≤

1/n 0

 ≤2

1/p H([u]1/n , [u]α )p dα 1/p H([u]α , {0})p dα

1/n 0



1/n

+ 0

1/p H([u]α , {0})p dα

1/p H([u]1/n , {0})p dα

,

then, from ineq. (20), dp (un , u) → 0 as n → ∞. Therefore, for each u ∈ FB (Rm )p , we can find a sequence {un } ⊂ FB (Rm ) such that un converges to u, i.e., FB (Rm ) is dense in FB (Rm )p . Remark 5.2. From the proof of Theorem 5.2, given u ∈ FB (Rm )p , then u(1/n) ∈ FB (Rm ) for each n ∈ N, and dp (u(1/n) , u) → 0 as n → ∞. Now, we obtain the relationship between (FB (Rm )p , dp ) and (FB (Rm ), dp ). Theorem 5.3. (FB (Rm )p , dp ) is the completion of (FB (Rm ), dp ). Proof. The proof follows immediately from Theorems 5.1 and 5.2. Next, we show some characterizations of totally bounded sets, relatively compact sets, and compact sets in (FB (Rm ), dp ) that follow from the conclusions in Section 4. Theorem 5.4. Let U ⊂ FB (Rm ), then U is totally bounded in (FB (Rm ), dp ) if and only if (i) U is uniformly p-mean bounded; (ii) U is p-mean equi-left-continuous. 24

Proof. Since FB (Rm ) ⊂ FB (Rm )p , U ⊂ FB (Rm ) is a totally bounded set in (FB (Rm ), dp ) if and only if U is a totally bounded set in (FB (Rm )p , dp ). Therefore, the proof follows immediately from Theorem 4.2. Given U ∈ FB (Rm )p , for notational convenience, let U denote the closure of U in (FB (Rm )p , dp ). Theorem 5.5. Let U ⊂ FB (Rm ), then U is relatively compact in (FB (Rm ), dp ) if and only if (i) U is uniformly p-mean bounded; (ii) U is p-mean equi-left-continuous; (iii) U ⊂ FB (Rm ). Proof. The proof follows immediately from Theorem 4.1 and FB (Rm ) ⊂ FB (Rm )p . Theorem 5.6. Let U ⊂ FB (Rm ), then U is compact in (FB (Rm ), dp ) if and only if (i) U is uniformly p-mean bounded; (ii) U is p-mean equi-left-continuous; (iii) U = U . Proof. The proof follows immediately from Theorem 4.3. Condition (iii) in Theorem 5.6 involves the closure of U in the completion space (FB (Rm )p , dp ). Finally, we derive another characterization of compactness that depends only on U itself, for which we need the following concept. Let Br := {x ∈ Rm : x ≤ r}, where r is a positive real number, then r is the characteristic function of Br . Given u ∈ FB (Rm ), then u ∨ B r ∈ B m FB (R ). Define r

|u| :=

 0

1

r ]α , [B r ]α )p dα H([u ∨ B

1/p .

Then, for u ∈ FB (Rm ), |u|r = 0 if and only if [u]0 ⊆ Br . Since r ]α , [v ∨ B r ]α ) ≤ H([u]α , [v]α ), H([u ∨ B it thus holds that

dp (u, v) ≥ ||u|r − |v|r |. 25

(21)

Theorem 5.7. Let U ⊂ FB (Rm ), then U is relatively compact in (FB (Rm ), dp ) if and only if U satisfies conditions (i) and (ii) in Theorem 4.1, and the following condition (iii ). (iii ) Given {un : n = 1, 2, . . .} ⊂ U , there exists a r > 0 and subsequence {vn } of {un } such that limn→∞ |vn |r = 0. Proof. Suppose that U is a relatively compact set but does not satisfy condition (iii ). Take r = 1, then there exists ε1 > 0 and a subsequence (1) (1) {un : n = 1, 2, . . .} of {un : n = 1, 2, . . .} such that |un |1 > ε1 for all (1) (k) n = 1, 2, . . .. If {un }, . . . , {un } and positive numbers ε1 , . . . , εk have been (k+1) (k) chosen, we can find a subsequence {un } of {un } and εk+1 > 0 such that (k+1) (n) |un |k+1 > εk+1 for all n = 1, 2, . . .. Put vn = un for n = 1, 2, . . .. Then {vn } is a subsequence of {un } and lim inf |vn |k ≥ εk n→∞

(22)

for k = 1, 2, . . .. Let v ∈ FB (Rm )p be an accumulation point of {vn }, then from (21) and (22), |v|k ≥ εk > 0 for all k = 1, 2, . . .. Thus, v ∈ / FB (Rm ), which contradicts the fact that U is a relatively compact set in (FB (Rm ), dp ). Suppose that U ⊂ (FB (Rm ), dp ) satisfies condition (iii ). Given a sequence {un } in U with limn→∞ un = u ∈ FB (Rm )p , then, from (21), there exists a r > 0 such that limn→∞ |un |r = |u|r = 0. Hence, [u]0 ⊆ Br , i.e., u ∈ FB (Rm ). Therefore, from Theorem 4.1, if U meets conditions (i), (ii), and (iii ), then U is a relatively compact set in (FB (Rm ), dp ). Theorem 5.8. Let U be a set in FB (Rm ), then U is compact in (FB (Rm ), dp ) if and only if U is closed in (FB (Rm ), dp ) and satisfies conditions (i), (ii), and (iii ) in Theorem 5.7. Proof. The proof follows immediately from Theorem 5.7. 6. Subspaces of (FB (Rm )p , dp ) We have already shown that (FB (Rm )p , dp ) is the completion of (FB (Rm ), dp ). Therefore, we might expect that (E m,p , dp ), (S0m,p , dp ), (S m,p , dp ), and (Sm,p , dp ) 26

are the completions of (E m , dp ), (S0m , dp ), (S m , dp ), and (Sm , dp ), respectively. We prove that this is true and discuss the relationship among all these fuzzy set spaces, summarizing the conclusions in a figure. Combining these conclusions and the results of Sections 4 and 5, we give characterizations of total boundedness, relative compactness, and compactness of all subspaces of (FB (Rm )p , dp ) discussed here. First, we give a series of conclusions on relationships among subspaces of (FB (Rm )p , dp ), which will be summarized in a figure. Theorem 6.1. Sm,p is a closed set in (FB (Rm )p , dp ). Proof. It suffices to show that each accumulation point of Sm,p belongs to itself. Given a sequence {un } in Sm,p with lim un = u ∈ FB (Rm )p , then a.e. H([un ]α , [u]α ) → 0 ([0, 1]). Suppose that α ∈ (0, 1]. If H([un ]α , [u]α ) → 0, then by Theorem 2.1, [u]α ∈ KS (Rm ). If H([un ]α , [u]α ) → 0, thenthere exists a sequence βn → α− such that [u]βn ∈ KS (Rm ). Since [u]α = n [u]βn , this implies that [u]α ∈ KS (Rm ). Therefore, u ∈ Sm,p . Theorem 6.2. Sm is a dense set in (Sm,p , dp ). Proof. Given u ∈ Sm,p , let un = u(1/n) , n = 1, 2, . . .. Then {un } ⊂ Sm . From Remark 5.2, dp (un , u) → 0 as n → ∞. Therefore, Sm is dense in (Sm,p , dp ). Theorem 6.3. S m,p is a closed subset of (Sm,p , dp ). Proof. Let {un } be a sequence in S m,p which converges to u ∈ Sm,p , then we only need to prove that u ∈ S m,p .

 1/p 1 → 0, it holds that Since dp (un , u) = 0 H([un ]α , [u]α )p dα H([un ]α , [u]α ) → 0 a.e. on [0, 1].

(23)

Hence, {ker un : n = 1, 2, . . .} is a bounded set in (K(Rm ), H), and it follows that lim sup ker un = ∅. (24) n→∞

We assert that lim sup ker un ⊂ ker [u]α for all α ∈ (0, 1]. n→∞

27

(25)

Therefore, from (24) and (25), 

∅ = lim sup ker un ⊂ n→∞

ker [u]α = ker u.

α∈(0,1]

Thus, from Theorem 3.7, u ∈ S m,p . Now we prove (25). The proof is divided into two cases. Case 1. α ∈ (0, 1] satisfies the condition H([un ]α , [u]α ) → 0. From Corollary 2.1, lim sup ker un ⊂ lim sup ker [un ]α ⊂ ker [u]α . n→∞

n→∞

Case 2. α ∈ (0, 1] satisfies the condition H([un ]α , [u]α ) → 0. From (23), there is a sequence αn ∈ (0, 1], n = 1, 2, . . ., such that αn → α− and H([un ]αn , [u]αn ) → 0. From case 1, lim sup ker un ⊂ n→∞

+∞ 

ker [u]αn .

(26)

n=1

Since H([u]α , [u]αn ) → 0, from Corollary 2.1, lim sup ker [u]αn ⊂ ker [u]α . n→∞

(27)

Combined with (26) and (27), lim sup ker un ⊂ ker [u]α . n→∞

Corollary 6.1. Suppose {un } is a sequence in (S m,p , dp ) and u ∈ Sm,p . If dp (un , u) → 0, then u ∈ S m,p and lim supn→∞ ker un ⊂ ker u. Proof. The proof follows immediately from the proof of Theorem 6.3. Corollary 6.2. (S m , dp ) is a closed subspace of (Sm , dp ). Proof. The proof follows immediately from Corollary 6.1. Theorem 6.4. S m is a dense set in (S m,p , dp ). Proof. The proof is similar to the proof of Theorem 5.2. 28

Figure 1: Relationship among various subspaces of (F (Rm )p , dp ). A ≺ B denotes that A is a closed subspace of B and A −→ B means that B is the completion of A.

Theorem 6.5. (E m,p , dp ) is a closed subspace of (S m,p , dp ). Proof. From Proposition 2.1, (KC (Rm ), H) is a closed set in (KS (Rm ), H). Thus, the proof proceeds similarly to the proof of Theorem 6.3. Corollary 6.3. (E m , dp ) is a closed subspace of (S m , dp ). Proof. The proof follows immediately from Theorem 6.5. Theorem 6.6. E m is a dense set in (E m,p , dp ). Proof. The proof is similar to the proof of Theorem 6.4. Theorem 6.7. (Sm,p , dp ) is the completion of (Sm , dp ). Proof. The proof follows from Theorems 5.1, 6.1, 6.2. Theorem 6.8. (S m,p , dp ) is the completion of (S m , dp ). Proof. The proof follows from Theorems 6.3, 6.4, 6.7. Theorem 6.9. (E m,p , dp ) is the completion of (E m , dp ). Proof. The proof follows from Theorems 6.5, 6.6, 6.8. From Corollary 6.1 and Remark 5.2, we can obtain the following two theorems. Theorem 6.10. (S0m,p , dp ) is the completion of (S0m , dp ). Theorem 6.11. (S0m , dp ) is a closed subspace of (S m , dp ).

29

Figure 1 summarizes the results from this section. Kr¨atschmer [25] presented the completion of (E m , dp ), generated and described via the support functions of fuzzy numbers. This paper shows that the completions of (E m , dp ), (S0m , dp ), (S m , dp ), (Sm , dp ), and (FB (Rm ), dp ) can be obtained from Lp extensions. The completions illustrated in this form are somewhat more concise and clear, and simpler to apply for theoretical research and practical applications. Based on the relationships among subspaces of (F (Rm )p , dp ), and characterizations of total boundedness, relative compactness, and compactness given in Sections 4 and 5, we derive characterizations of totally bounded sets, relatively compact sets, and compact sets in subspaces of (F (Rm )p , dp ). Theorem 6.12. Let U ⊂ Sm,p (U ⊂ E m,p , U ⊂ S0m,p , U ⊂ S m,p ), then U is totally bounded if and only if it is relatively compact in (Sm,p , dp ) ((E m,p , dp ), (S0m,p , dp ), (S m,p , dp )), which is equivalent to (i) U is uniformly p-mean bounded, and (ii) U is p-mean equi-left-continuous. Proof. In a complete space, a set is totally bounded if and only if it is relatively compact. Therefore, the proof follows from Theorems 4.1 and the completeness of (Sm,p , dp ), (E m,p , dp ), (S0m,p , dp ) and (S m,p , dp ). Let U ⊂ Sm,p (U ⊂ E m,p , U ⊂ S0m,p , U ⊂ S m,p ), then U is exactly the closure of U in Sm,p (E m,p , S0m,p , S m,p ). Theorem 6.13. Let U be a set in (Sm,p , dp ) ((E m,p , dp ), (S0m,p , dp ), (S m,p , dp )), then U is compact in (Sm,p , dp ) ((E m,p , dp ), (S0m,p , dp ), (S m,p , dp )) if and only if (i) U is uniformly p-mean bounded; (ii) U is p-mean equi-left-continuous; (iii) U = U . Proof. The proof follows immediately from Theorem 6.12. Theorem 6.14. Let U ⊂ Sm (U ⊂ E m , U ⊂ S0m , U ⊂ S m ), then U is totally bounded in (Sm , dp ) ((E m , dp ), (S0m , dp ), (S m , dp )) if and only if (i) U is uniformly p-mean bounded; (ii) U is p-mean equi-left-continuous. Proof. The proof follows immediately from Theorem 4.2. 30

Theorem 6.15. Let U ⊂ Sm (U ⊂ E m , U ⊂ S0m , U ⊂ S m ), then U is relatively compact in (Sm , dp ) ((E m , dp ), (S0m , dp ), (S m , dp )) if and only if (i) U is uniformly p-mean bounded; (ii) U is p-mean equi-left-continuous; (iii) U ⊂ Sm (U ⊂ E m , U ⊂ S0m , U ⊂ S m ). Proof. The proof follows immediately from Theorem 6.12. Theorem 6.16. Let U ⊂ Sm (U ⊂ E m , U ⊂ S0m , U ⊂ S m ), then U is compact in (Sm , dp ) ((E m , dp ), (S0m , dp ), (S m , dp )) if and only if (i) U is uniformly p-mean bounded; (ii) U is p-mean equi-left-continuous; (iii) U = U . Proof. The proof follows from Theorem 4.3 and the completeness of (Sm,p , dp ), (E m,p , dp ), (S0m,p , dp ) and (S m,p , dp ). Theorem 6.17. Let U ⊂ Sm (U ⊂ E m , U ⊂ S0m , U ⊂ S m ), then U is relatively compact in (Sm , dp ) ((E m , dp ), (S0m , dp ), (S m , dp )) if and only if (i) U is uniformly p-mean bounded; (ii) U is p-mean equi-left-continuous; (iii ) Given {un : n = 1, 2, . . .} ⊂ U , there exists a r > 0 and a subsequence {vn } of {un } such that limn→∞ |vn |r = 0. Proof. Since FB (Rm ) ∩ Sm,p = Sm , FB (Rm ) ∩ E m,p = E m , FB (Rm ) ∩ S0m,p = S0m , and FB (Rm ) ∩ S m,p = S m , the proof follows from Theorems 5.7 and 6.12. Theorem 6.18. Let U be a set in Sm (E m , S0m , S m ), then U is compact in (Sm , dp ) ((E m , dp ), (S0m , dp ), (S m , dp )) if and only if U is closed in (Sm , dp ) ((E m , dp ), (S0m , dp ), (S m , dp )) and satisfies conditions (i), (ii), and (iii ) in Theorem 6.17. Proof. The proof follows immediately from Theorem 6.17. 7. Applications of the results of this paper Using results in this paper, we consider some properties of dp convergence on fuzzy set spaces, and review compactness characterizations presented in [36, 38]. 31

Let us discuss properties of dp convergence from the point of view of level sets of fuzzy sets. Let {uk , k = 1, 2, . . .} ⊂ FGB (Rm ) and u ∈ FGB (Rm ). For notational a.e. convenience, we use H([uk ]α , [u]α ) → 0 (P ) to denote H([uk ]α , [u]α ) → 0 as k → ∞ holds almost everywhere on α ∈ P . Lemma 7.1. Suppose that {uk , k = 1, 2, . . .} ⊂ FGB (Rm ), u ∈ FGB (Rm ) (α) and α ∈ (0, 1], then dp (uk , u) → 0 is equivalent to a.e. (i) H([uk ]β , [u]β ) → 0 ([α, 1]). (ii) H([uk ]α , [u]α ) → 0 and [u]γ ≡ [u]α when γ ∈ [0, α]. (α)

Proof. The proof follows from the definition of dp metric and uk . Remark 7.1. Suppose α ∈ (0, 1], then from Lemma 7.1, even if {uk } is (α) a convergent sequence in (FB (Rm )p , dp ), {uk } is not necessarily a convergent sequence in (FB (Rm )p , dp ). The following example is a small change to Example 3.11 in [29] or Example 4.1 in [36], and demonstrates this case. Example 7.1. Consider a sequence {un : n = 1, 2, . . .} in E 1 defined as ⎧ 1 (1 − n1 ) + nx , x ∈ [0, 13 ], ⎪ ⎪ ⎨ 31 , x ∈ [ 13 , 23 ], 3 n = 1, 3, 5, . . . , un (x) = 2x − 1, x ∈ [ 23 , 1], ⎪ ⎪ ⎩ 0, x∈ / [0, 1], ⎧ 1 1 x ⎨ 3 (1 − n ) + 2n , x ∈ [0, 23 ], 2x − 1, x ∈ [ 23 , 1], n = 2, 4, 6, . . . . un (x) = ⎩ 0, x∈ / [0, 1], Then dp (un , u0 ) → 0, where

⎧ 1 x ∈ [0, 23 ], ⎨ 3, 2x − 1, x ∈ [ 23 , 1], u0 (x) = ⎩ 0, x∈ / [0, 1]. (1)

Therefore, {un } is a convergent sequence in (E 1 , dp ). However, {un3 } is a (1)

(1)

3 , v) → 0 and dp (u2n3 , w) → 0, divergent sequence in (E 1 , dp ). In fact, dp (u2n−1 where ⎧ 1 x ∈ [ 13 , 23 ], ⎨ 3, 2x − 1, x ∈ [ 23 , 1], v(x) = ⎩ 0, x∈ / [ 13 , 1],

32

 w(x) =

2x − 1, x ∈ [ 23 , 1], 0, x∈ / [ 23 , 1].

(1)

Since v = w, {un3 } is not a convergent sequence in (E 1 , dp ). Theorem 7.1. Suppose U ⊂ FGB (Rm ) satisfying {[u]α : u ∈ U } is a bounded set in K(Rm ) for each α ∈ (0, 1]. Let {ri , i = 1, 2, . . .} be a sequence in (0, 1], then given {un } ⊂ U , it has a subsequence {unk } satisfying the following two statements. a.e. (i) There exists a u0 ∈ FGB (Rm ) such that H([unk ]α , [u0 ]α ) → 0 ([0, 1]). (ii) There exist uri , i = 1, 2, . . ., in K(Rm ) such that H([unk ]ri , uri ) → 0 for each ri . (r ) Thus dp (unki , u(ri )) → 0 for each ri , where u(ri ) ∈ FB (Rm ), i = 1, 2, . . ., is defined as  [u0 ]α , α ∈ (ri , 1], [u(ri )]α = α ∈ [0, ri ]. uri , Proof. Given {un } ⊂ U , since {[u]α : u ∈ U } is a bounded set in K(Rm ) for each α ∈ (0, 1], then from the proof of Theorem 4.1, {un } has a subsequence a.e. {unj } such that H([unj ]α , [u0 ]α ) → 0 ([0, 1]), where u0 is one of the elements in FGB (Rm ). Since each α > 0, {[u]α : u ∈ U } is a relatively compact set in (K(Rm ), H), using the diagonal method and proceeding according to the proof of Theorem 4.1, we can choose a subsequence {unk } of {unj } such that H([unk ]ri , uri ) → 0 for each ri , where {uri , i = 1, 2, . . .} is a sequence of elements in K(Rm ). Therefore, {unk } is a subsequence of {un } that satisfies statements (i) (r ) and (ii). Thus, by Lemma 7.1, dp (unki , u(ri )) → 0 for all ri . Remark 7.2. Suppose U ⊂ FB (Rm )p satisfies conditions (i) in Theorem 4.1, i.e., U is uniformly p-mean bounded. Let {ri , i = 1, 2, . . .} be a sequence in (0, 1], then given {un } ⊂ U , there is a subsequence {unk } of {un } such that (r ) {unki } converges to u(ri ) ∈ FB (Rm ) in dp metric for each ri . In fact, if U ⊂ FB (Rm )p satisfies conditions (i) in Theorem 4.1, then {[u]α : u ∈ U } is a bounded set in K(Rm ) for each α ∈ (0, 1]. Therefore, the conclusion follows immediately from Theorem 7.1. From Theorem 4.1 and Lemma 7.1, we can obtain two corollaries. Corollary 7.1. Suppose {uk , k = 1, 2, . . .} ⊂ FB (Rm )p and u ∈ FGB (Rm ), then statements (i) and (ii) listed below are equivalent. 33

(i) There exists a decreasing sequence {ri , i = 1, 2, . . .} in (0, 1] converging (r ) to zero such that dp (uk i , u(ri ) ) → 0 for all ri . a.e. (ii) H([uk ]α , [u]α ) → 0 ([0, 1]). Furthermore, if U := {uk , k = 1, 2, . . .} satisfies conditions (i) and (ii) in Theorem 4.1, then statements (i) and (ii) above are equivalent to statement (iii) below. (iii) dp (uk , u) → 0 and u ∈ FB (Rm )p . Corollary 7.2. Suppose that {uk , k = 1, 2, . . .} ⊂ FB (Rm )p , {ri , i = 1, 2, . . .} is a decreasing sequence in (0, 1] that converges to zero, and u(ri ) ∈ FGB (Rm ), i = 1, 2, . . .. Then, the following statements are equivalent. (r ) (i) dp (uk i , u(ri )) → 0 for all ri . a.e. (ii) There exists a unique u0 ∈ FGB such that H([uk ]α , [u0 ]α ) → 0 ([0, 1]) (r ) and [u0 ]α = [u0 i ]α = [u(ri )]α when ri < α ≤ 1. For each ri , H([uk ]ri , [u(ri )]ri ) → 0 and [u(ri )]ri ≡ [u(ri )]θ when θ ∈ [0, ri ]. Furthermore, if U := {uk , k = 1, 2, . . .} satisfies conditions (i) and (ii) in Theorem 4.1, then statements (i) and (ii) above are equivalent to statement (iii) below. (iii) There exists a unique u0 ∈ FB (Rm )p such that dp (uk , u0 ) → 0 and (r ) [u0 ]α = [u0 i ]α = [u(ri )]α when ri < α ≤ 1. For each ri , H([uk ]ri , [u(ri )]ri ) → 0 and [u(ri )]ri ≡ [u(ri )]θ when θ ∈ [0, ri ]. Remark 7.3. [16, 33] describe more studies on this topic, and consider the relationships between dp convergence and other types of convergence on fuzzy set spaces. The results above enable us to review the characterizations of compactness proposed in previous work. Comparing Theorem 6.13 and Proposition 4.4, the latter has an additional condition (iii): Let ri be a decreasing sequence in (0, 1] converging to zero, then for {uk } ⊂ (r ) U , if {uk i : k = 1, 2, . . .} converges to u(ri ) ∈ E m in dp metric, there (r ) exists a u0 ∈ E m such that [u0 i ]α = [u(ri )]α when ri < α ≤ 1. The reason for this is that (E m , dp ) is not complete. The function of condition (iii) in Proposition 4.4 is to guarantee completeness of the closed subspace (U, dp ) of (E m , dp ). 34

If U satisfies conditions (i) and (ii) of Proposition 4.4, then from Remark 7.2, given {uk } ⊆ U , there exists a subsequence {unk } of {uk } such that (r ) {unki : k = 1, 2, . . .} converges to u(ri ) ∈ E m in the dp metric for each ri . Thus, from Corollary 7.2, {unk } is a convergent sequence in (E m,p , dp ), and its limit point u0 ∈ E m,p is defined as (r )

[u0 ]α = [u0 i ]α := [u(ri )]α when ri < α ≤ 1, for each α ∈ (0, 1]. Therefore, if U satisfies conditions (i) and (ii) of Proposition 4.4, then condition (iii) in Proposition 4.4 is equivalent to U ⊂ E m . Thus, Proposition 4.4 can be written in the following form: Proposition 4.4 A closed set U ⊂ (E m , dp ) is compact if and only if: (i) U is uniformly p-mean bounded; (ii) U is p-mean equi-left-continuous; (iii) U ⊂ E m . Similarly, Propositions 4.5 and 4.6 can be written as: Proposition 4.5 A closed set U in (S0m , dp ) is compact if and only if: (i) U is uniformly p-mean bounded; (ii) U is p-mean equi-left-continuous; (iii) U ⊂ S0m . Proposition 4.6 A closed set U in (S m , dp ) is compact if and only if: (i) U is uniformly p-mean bounded; (ii) U is p-mean equi-left-continuous; (iii) U ⊂ S m . 8. Conclusions The key result of this paper is a characterization of total boundedness for fuzzy set spaces endowed with the dp metric. To obtain this characterization, we introduced the Lp extension of a fuzzy set space, then presented a characterization of relatively compact sets in (FB (Rm )p , dp ), which is the Lp extension of (FB (Rm ), dp ). We showed that this characterization is also a characterization of totally bounded sets in (FB (Rm )p , dp ). Thus, this characterization is a characterization of totally bounded sets in any subspace of (FB (Rm )p , dp ) because total boundedness is dependent only on the set itself and the metric dp . Therefore, we can also deduce that (FB (Rm )p , dp ) is the completion of (FB (Rm ), dp ). Using these results, we provided characterizations of relative compactness and compactness in (FB (Rm ), dp ), (FB (Rm )p , dp ) and their subspaces. 35

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