Some complete metrics on spaces of fuzzy subsets

Fuzzy Sets and Systems 130 (2002) 357–365www.elsevier.com/locate/fss

Some complete metrics on spaces of fuzzy subsetsVolker Kr&atschmer∗

Statistik und Okonometrie, Rechts- und Wirtschaftswissenschaftliche Fakultat, Universitat des Saarlandes, Bau 31,Postfach 15 11 50, D-66041 Saarbrucken, Germany

Received 2 August 2001; received in revised form 30 January 2002; accepted 8 March 2002

Abstract

The classes of the Lp;∞- and Lp-metrics play an important role to develop a probability theory in fuzzy sample spaces. Allof these metrics are known to be separable, but not complete. The classes are closely related as for each Lp;∞-metric thereexists some Lp-metric which induces the same topology. This paper deals with the completion of the Lp;∞- and Lp-metrics.We can also show that the relationship between the classes of Lp;∞- and Lp-metrics still holds for the obtained respectiveclasses of their completions. c© 2002 Elsevier Science B.V. All rights reserved.

Keywords: ; Lp;∞-metrics; Lp-metrics; Completion of the Lp;∞-metrics; Completion of the Lp-metrics

0. Introduction

Several metrics on spaces of fuzzy subsets of Rk

have been investigated within fuzzy literature. Theyoften have been used to develop a probability the-ory in fuzzy sample spaces, mostly in the space ofnormal bounded compact fuzzy subsets of Rk , i.e.normal fuzzy subsets with compact positive �-cutsand bounded support. Klement et al. [12] introduceda class of separable metrics, in the following calledLp;∞-metrics. They are not complete, even in thesubspace of normal, bounded, convex, compact fuzzysubsets, i.e. the normal, bounded, compact fuzzy sub-sets of Rk with convex positive �-cuts [7,8]. Colubi etal. [4,5] embedded the space of normal, bounded, com-pact fuzzy subsets into the space consisting of all thecompact-valued cadlag functions on [0; 1]. Endowing

∗ Tel.: +49-681-302-3169; fax: +49-681-302-3551.E-mail address: [email protected]

(V. Kr&atschmer).

the cadlag space with the convenient Skohorod met-ric, the trace metric on the space of fuzzy subsets,also called Skohorod metric, is complete and sepa-rable. Diamond and Kloeden [7,8] suggested anotherimportant class of metrics on the space of normal,bounded, convex, compact fuzzy subsets, the so calledLp-metrics. They showed that they are separable, butnot complete. Furthermore, each of the Lp-metrics isassociated with one of the Lp;∞-metrics in the waythat they induce the same topology.

It has been proved [14,5] that all of these metricsgenerate identical �-algebras on the respective fuzzysample spaces where the metrics work. Moreover, theyinduce a concept of fuzzy random variable which canbe regarded as a uni@cation of the already known ones.Strong Laws of Large Numbers for fuzzy random vari-ables (in the uni@ed sense) have been found with re-spect to the Lp;∞-metrics [12,11,15] as well as withrespect to the Skohorod metric [4] and the Lp-metrics[13,15].

Besides the Strong Law of Large Numbers there ex-ist a Central Limit theorem and a Gliwenko–Cantelli

0165-0114/02/$ - see front matter c© 2002 Elsevier Science B.V. All rights reserved.PII: S 0165 -0114(02)00120 -3

358 V. Kratschmer / Fuzzy Sets and Systems 130 (2002) 357–365

theorem for fuzzy random variables, simultaneouslyformulated with respect to the Lp-metrics [15]. Thismight be a justi@cation to grant the Lp-metrics a promi-nent part within the probability theory in fuzzy samplespaces.

The aim of this paper is to @nd completions of theLp-metrics as well as of the Lp;∞-metrics, restricted tothe space of normal, bounded, convex, compact fuzzysubsets of Rk .

In Section 1 we shall introduce the Lp;∞- andLp-metrics. They base on the HausdorH metric re-spective the so called Lp-metrics on the space ofnon-void convex, compact subsets of Rk . Some topo-logical properties of these metrics and their mutualrelationship will be summarized.

For the remainder of the paper we shall consider thespaces Fnob

cocp(Rk) consisting of the normal fuzzy sub-sets of Rk with convex, compact positive �-cuts andsupport functions which are integrable of order p. Aswill be shown in Section 2, the Lp;∞- and Lp-metricsare extendable to the respective spaces Fnob

cocp(Rk). Fur-thermore, we shall prove that the space of normal,bounded, convex, compact fuzzy subsets is dense ineach of the spaces Fnob

cocp(Rk) with respect to the ap-propriate extended Lp;∞- and Lp-metric.

Section 3 is devoted to the statement of the main re-sults. We shall propose that every extended Lp;∞- andevery extended Lp-metric is a (separable) completionof the respective Lp;∞- and Lp-metric. Moreover, weshall state that for each extended Lp;∞-metric thereis some extended Lp-metric which induce the sametopology. This generalizes the result by Diamond andKloeden for the Lp;∞- and Lp-metrics.

These main results will be proved separately in theSections 3 and 4.

1. Notations and preliminaries

Within this paper the Euclidean norm ‖·‖ on Rk is@xed, and Sk−1 denotes the unit sphere in Rk withrespect to ‖·‖. The unit sphere will be equipped withthe relative topology Sk−1 of the standard topology onRk , and [0; 1] will be regarded as a topological sub-space of R with topology [0;1]. The set B(Sk−1) con-sists of all Borel-subsets with respect to Sk−1 , and theBorel-subsets with respect to [0;1] are gathered by the�-algebra B([0; 1]). The measure spaces under

consideration are (Sk−1;B(Sk−1); Sk−1) and ([0; 1];

B([0; 1]); 1), where Sk−1indicates the unit Lebesgue

measure on Sk−1 and 1 the Lebesgue measure on[0; 1]. As the symbol for the product measure spaceof ([0; 1];B([0; 1]); 1) and (Sk−1;B(Sk−1); Sk−1

)we shall use ([0; 1]× Sk−1;B([0; 1])⊗B(Sk−1);1 ⊗ Sk−1

).The indicator mapping of some subset A of some

universal set is denoted by 1A. For every fuzzy sub-set A of Rk with membership function �A and support

�−1A

(]0; 1]) we introduce the notation [A]� which de-signs the �-cut of A for � in ]0; 1], and the topolog-ical closure cl(�−1

A(]0; 1])) of the support for �= 0.

Fnococ(Rk) consists of all the fuzzy subsets of Rk with

non-void convex, compact positive �-cuts, called nor-mal convex, compact fuzzy subsets. Those normalconvex, compact fuzzy subsets having bounded sup-port are gathered in Fnob

coc (Rk). The fuzzy sets from thisspace are often called normal convex, upper semicon-tinuous fuzzy subsets with bounded support becausethey are exactly those normal convex fuzzy subsets ofRk which have bounded supports and upper semicon-tinuous membership functions. The fuzzy subset 0 ofRk is de@ned by [0]� = {0} for �∈ ]0; 1]. It is a mem-ber of Fnob

coc (Rk).We denote the space of non-void convex, compact

subsets ofRk byK+co (Rk). This space will be endowed

with the HausdorH metric dH with respect to ‖·‖. Itis already known that (K+

co (Rk); dH) is separable andcomplete (e.g. [6, (2.4), (2.6), (5.6)]).

Besides the HausdorH metric the so-calledLp-metrics are often used to de@ne a topologicalstructure on K+

co (Rk). They make use of the uniquecharacterization of every subset K ∈K+

co (Rk) by itssupport function

sK : Rk → R; x → supy∈K

y′x;

where y′x denotes the standard scalar product of thevectors y; x. Then for p∈ [1;∞[ the Lp-metric �p isde@ned by

�p(K; L) :=(∫

Sk−1|sK − sL|p dSk−1

)1=p

:

If k = 1, then it is easy to check that the inequalities�p6dH62�p hold for p∈ [1;∞[. In the case of k¿2it has been shown by Vitale [17] that one can @nd

V. Kratschmer / Fuzzy Sets and Systems 130 (2002) 357–365 359

for arbitrary p∈ [1;∞[ a positive constant Cp with�p6dH and

dH(K; L)p+k−1 6 Cpp�p(K; L)pdiam(K ∪ L)k−1

(K; L∈K+co (Rk) and diam(K ∪L) diameter of K ∪L).

Especially, @xing k ∈N; p∈ [1;∞[; x∈Rk , thereexists some positive constant Cp such that

dH(K; {x})p 6 Cpp�p(K; {x})p

holds for every K ∈K+co (Rk) because we have

diam(K ∪{x})62dH(K; {x}) for K ∈K+co (Rk).

Moreover, Vitale proved that for every k¿2 and forevery p∈ [1;∞[ the metrics �p; dH induce the sametopology, and the metric space (K+

co (Rk); �p) is com-plete and separable. All of these results still hold forevery p∈ [1;∞[ in the case of k = 1. This is an easyconsequence of the inequalities �p6dH62�p for k = 1and p∈ [1;∞[.

The HausdorH metric dH and the Lp-metrics �p in-duce the metrics dp and �p (p∈ [1;∞[) on Fnob

coc (Rk),de@ned by

dp(A; B) :=

(∫ 1

0dH([A]�; [B]�)p d�

)1=p

and

�p(A; B) :=

(∫ 1

0�p([A]�; [B]�)p d�

)1=p

(cf. [12,7,8]). We borrow from Diamond andKloeden [7,8] the names Lp;∞-metric for dp and Lp-metric for �p. For every p∈ [1;∞[ the metrics dp; �pinduce the same topology on Fnob

coc (Rk) with �p6dp,and the metric spaces (Fnob

coc (Rk); dp); (Fnobcoc (Rk); �p)

are separable but not complete (cf. [7] or [8, Sections7.1–7.3]). Our aim is to @nd completions of them.

2. Fuzzy subsets with integrable support functions

As the main tool to @nd completions of the metricspaces (Fnob

coc (Rk); dp) and (Fnobcoc (Rk); �p) we extend

the notion of support functions to fuzzy subsets fromFnococ(Rk). For every A∈Fno

coc(Rk) we can de@ne themapping

sA : ]0; 1] × Rk → R; (�; x) → s[A]�(x);

which is called the support function of A. The mappingsA : [0; 1]×Rk →R, de@ned by

sA(�; x) :=

{sA(�; x) � ¿ 0;

0 � = 0

will be named the trivial extension of sA.

Lemma 2.1. Let A∈Fnococ(Rk), and let B(R) denote

the �-algebra on R generated by the standard topol-ogy. Then we can state:

1. sA|[0; 1]× Sk−1 is B([0; 1])⊗B(Sk−1) − B(R)-measurable.

2. sA|[0; 1]× Sk−1 is 1 ⊗ Sk−1-integrable of order p

for every p∈ [1;∞[ if A is a member of Fnobcoc (Rk).

Proof. Obviously, we have

(1) sA(�; ·) is continuous for every �∈ [0; 1](2) sA(·; x) is nonincreasing on ]0; 1] for x∈Rk

Furthermore, it is known that

|sA(�; x) − sA(�; x)|6 dH([A]�; [A]�)

holds for �; �∈ ]0; 1] and x∈ Sk−1, which implies

(3) sA(·; x) is left continuous on ]0; 1] for everyx∈ Sk−1

(cf. e.g. [14, Lemma 3.4]).De@ning IQ := ]0; 1]∩Q, we can conclude from (2)

and (3) for arbitrary t ∈R

(]0; 1] × Sk−1) ∩ s−1A

(] −∞; t[)

=⋃�∈IQ

[�; 1] × ((s(�; ·))−1(] −∞; t[) ∩ Sk−1)

and

({0} × Sk−1) ∩ s−1A

(] −∞; t[)

=

{{0} × Sk−1 t ¿ 0;

∅ t 6 0:

Hence statement 1 follows from (1).If A∈Fnob

coc (Rk), then we have

|sA(�; x)|p 6 dH([A]�; {0})p 6 dH([A]0; {0})p


for �∈ [0; 1] and x∈ Sk−1. This proves, taking state-ment 1 into consideration, statement 2 since 1 ⊗ Sk−1

is @nite.

In accordance with Lemma 2.1 and Fubini’s theo-rem the metrics �p can be rewritten by

�p(A; B)

=

(∫ 1

0

∫Sk−1

|sA − sB|p dSk−1d1

)1=p

: (1)

Let us de@ne for @xed p∈ [1;∞[ the space Fnobcocp(Rk)

to consist of all those fuzzy subsets A∈Fnococ(Rk) with

|sA|[0; 1]× Sk−1|p being 1 ⊗ Sk−1-integrable. Then

each metric �p can be extended to Fnobcocp(Rk) easily.

Proposition and De!nition 2.2. For every p∈[1;∞[the mapping

�p : Fnobcocp(Rk) × F

nobcocp(Rk) → R

de9ned by

�p(A; B)

:=

(∫ 1

0

∫Sk−1

|sA − sB|p dSk−1d1

)1=p

is a metric which extends �p.

Proof. Symmetry of �p is obvious, the triangle in-equality is provided by the Minkowski inequality,and �p(A; A) = 0 holds for every A∈ Fnob

cocp(Rk).Let A; B∈ Fnob

cocp(Rk) with �p(A; B) = 0. Then thereexists some 1-null set N with �p([A]�; [B]�) = 0for �∈ ]0; 1]\N , i.e. [A]� coincide with [B]� for�∈ ]0; 1]\N . Since ]0; 1]\N is a dense subset of[0; 1], we can @nd for every �∈ ]0; 1]∩N an isotonesequence (�n)n in ]0; �[\N converging to �. Hence

[A]� =∞⋂n=1

[A]�n =∞⋂n=1

[B]�n = [B]�

and A= B. Therefore �p satis@es all the conditions fora metric. Moreover, we can easily see from Lemma 2.1and (1) that �p extends �p. Now the proof is complete.

Applying Fubini’s theorem we immediately obtainan auxiliary result which we shall use later severaltimes.

Lemma 2.3. For each p∈ [1;∞[ and every A; B∈Fnob

cocp(Rk) the mapping AB : [0; 1]→R, de9ned by

AB(�) :=

{�p([A]�; [B]�) � ¿ 0;

0 � = 0

is 1-integrable of order p, and we have

�p(A; B)p =∫ 1

0( AB(�))

p d�:

Using the relationships between the HausdorH met-ric dH and the Lp-metrics �p on K+

co (Rk) we canalso extend the Lp;∞-metrics to the respective spacesFnob

cocp(Rk). But since the supports of fuzzy subsetsA; B∈ Fnob

cocp(Rk) are not bounded necessarily, we haveto modify the description of the Lp;∞-metrics slightly.Every pair (A; B) of fuzzy subsets A; B∈Fno

coc(Rk)will be associated with the mapping hAB : [0; 1]→R,de@ned by

hAB(�) :=

{dH([A]�; [B]�) � ¿ 0;

0 � = 0:

Then we have

dp(A; B) =

(∫ 1

0(hAB(�))

p d�

)1=p

;

which shows the direction for the extension.

Proposition and De!nition 2.4. For every p∈[1;∞[the mapping

dp : Fnobcocp(Rk) × F

nobcocp(Rk) → R

with

dp(A; B) :=

(∫ 1

0(hAB(�))

p d�

)1=p

;

is well de9ned and satis9es the properties of a met-ric. Furthermore dp extends the metric dp, and wehave �p6dp.


Proof. Let Cp be some positive constant with

dH(K; {0})p 6 Cpp�p(K; {0})p

for K ∈K+co (Rk). Then we have hp

0A6C

pp p

0Afor

A∈Fnobcocp(Rk), with 0A being de@ned as in Lemma 2.3.

Let us @x A; B∈ Fnobcocp(Rk). The mapping hAB is left

continuous (cf. [14, Lemma 3.4]). Therefore, de@ningIQ := ]0; 1]∩Q, we get for every t ∈R

]0; 1] ∩ h−1AB

(] −∞; t[) =∞⋃m=1

∞⋃r=1

∞⋂s=r⋃

q∈h−1AB

(]−∞;t−1=m[)∩IQ

[q; q + 1=s[∩]0; 1]:

This means that hAB is B([0; 1]) −B(R)-measurable,where B(R) denotes the �-algebra on R which is gen-erated by the standard topology. Since we also have

hpAB6 C

pp( 0A + 0B)

p;

the mapping hAB is 1-integrable of order p due toLemma 2.3, and dp is well de@ned. Obviously, dp is anextension of dp, and �p6dp. The triangle inequality isprovided by Minkowski’s inequality. If dp(A; B) = 0,then �p(A; B) = 0 and A= B. The remaining conditionsof a metric are clearly ful@lled.

Fnobcoc (Rk) is a dense subset of Fnob

cocp(Rk) with respectto the Lp;∞- and the Lp-metric. This can be concludedfrom the following proposition.

Proposition 2.5. Let A∈ Fnobcocp(Rk). For n∈N the

membership function of the fuzzy subset An of Rk isde9ned by �An

:= 1[A](1=n)�A.Then An belongs to Fnob

coc (Rk) for n∈N, andlimn→∞ �p(A; An) = limn→∞ dp(A; An) = 0.

Proof. An ∈Fnobcoc (Rk) for n∈N because the family

{[An]� | �∈ [0; 1]} is a subset of the set {[A]� | �∈[1=n; 1]}.

Let Bn denote the fuzzy subset in Fnobcocp(Rk) with

positive �-cuts being identical with [A]1=n. Then

dp(A; An)p =∫ 1

01[0;1=n](�)(hABn

(�))p d�:

Since [Bn]� = [A]1=n is a subset of [A]� for every�∈ ]0; 1=n], we obtain

|1[0;1=n] hABn|p 6 (2h0A)

p:

Furthermore,

limn→∞ 1[0;1=n](�) (hABn

(�))p = 0

holds for �∈ ]0; 1]. Hence, applying Lebesgue’s the-orem of dominated convergence, we can conclude

0 6 limn→∞ �p(A; An) 6 lim

n→∞ dp(A; An) = 0:

Diamond and Kloeden proposed that the topologicalspaces (Fnob

coc (Rk); dp); (Fnobcoc (Rk); �p) induced by the

metrics dp; �p (p∈ [1;∞[) are locally compact ([7,p. 71; 8, Corollary 8.3.4]). But, however, in the viewof Proposition 2.5 this result is not true.

Corollary 2.6. Let dp ; �p be the topologies onFnobcoc (Rk) induced by the metrics dp respective �p,and let dp ; �p be the topologies on Fnob

cocp(Rk) induced

by the metrics dp respective �p for 9xed p∈ [1;∞[.Then Fnob

coc (Rk) �∈ dp ⊇ �p , especially the topologi-cal spaces (Fnob

coc (Rk); dp); (Fnobcoc (Rk); �p) are not

locally compact.

Proof. Let the membership function �A of the fuzzysubset A of Rk be de@ned by

�A(x) :=

{1

‖x‖2p ‖x‖¿ 1;

1 ‖x‖ ¡ 1:

We have [A]� = {x∈Rk | ‖x‖2p61=�} and sA(�; x)p =

1=√� for �∈ ]0; 1]; x∈ Sk−1. Then it is easy

to check that A∈ Fnobcocp(Rk)\Fnob

coc (Rk). Setting�An

(x) := �A(nx); x∈Rk , we have de@ned a sequence(An)m in Fnob

cocp(Rk)\Fnobcoc (Rk) which converges to

0 with respect to dp. (Observe sAn= (1=n)sA and

dp(An; 0) = (1=n)dp(A; 0).) This implies that 0 is notin the topological interior of Fnob

coc (Rk) with respect todp . Additionally, �p ⊆ dp because of �p6dp. Hence

Fnobcoc (Rk) �∈ dp ⊇ �p .

On the other hand, Fnobcoc (Rk) is a dense sub-

set of Fnobcocp(Rk) with respect to dp and �p due to

Proposition 2.5.


Combining the results, the topological spaces(Fnob

coc (Rk); dp); (Fnobcoc (Rk); �p) are not locally com-

pact (cf. [9, Theorem 3.3.9]).

3. Statement of the main results

As one of their most important results Diamondand Kloeden claimed that each pair (dp; �p) inducesthe same topology on Fnob

coc (Rk). Since we have�p6dp, the only interesting part is to show that�p-convergence implies dp-convergence. For k = 1this is easy to prove. Let k¿2 and let (An)n denotea sequence in Fnob

coc (Rk) which converges to someA∈Fnob

coc (Rk) with respect to �p. If [A]0 is a singleton,say x, we can @nd some positive constant Cp with

dH([An]�; {x})p 6 Cpp�p([An]�; {x})p

for n∈N; �∈ [0; 1]. Then limn→∞ dp(A; An) = 0 fol-lows immediately.

In their book [8] as well as in their article [7] Dia-mond and Kloeden use the following argument if [A]0

is not a singleton:First, they established the following inequalities

dp(An; A) 6 dp+k−1(An; A);

and(PCppdp+k−1(A; An)p+k−1

diam([A]0 ∪ [An]0)k−1

)1=p

6 �p(A; An)

for some positive constant PCp, chosen independentlyof n∈N.

The @rst one follows from Jensen’s inequality, thesecond one from

diam([A]� ∪ [An]�) 6 diam([A]0 ∪ [An]0)

and the relationship between the HausdorH metric dH

and the Lp-metric �p. Without any further explana-tion they conclude that the desired convergence withrespect to the metric dp follows from these inequal-ities. The argument is hard to see, in particular if(diam([A]0 ∪ [An]0))n is unbounded. But modifyingthe line of reasoning, we can show that each pair(dp; �p) induces the same topology on Fnob

cocp(Rk). Thisveri@es the result of Diamond and Kloeden as a by-product.

Theorem 3.1. The metrics dp; �p induce the sametopology on Fnob

cocp(Rk) for p∈ [1;∞[.

The metric spaces (Fnobcocp(Rk); �p) are indeed com-

pletions of the respective metric spaces (Fnobcoc (Rk); �p).

Theorem 3.2. (Fnobcocp(Rk); �p) is complete, separable

and a completion of (Fnobcoc (Rk); �p) for p∈ [1;∞[.

As an immediate consequence of �p6dp (forp∈ [1;∞[), Proposition 2.5 and Theorems 3.1, 3.2we obtain

Corollary 3.3. (Fnobcocp(Rk); dp) is complete, separable

and a completion of (Fnobcoc (Rk); dp) for p∈ [1;∞[.

4. Proof of Theorem 3.1

We @rstly want to recall the notion of the uniformintegrability:

Let (&;F; ') be some @nite measure space andlet Lp(&;F; ') be the space consisting of all real-valued mappings on & which are '-integrable of orderp (p∈ [1;∞[). A subset M ⊆L1(&;F; ') is said tobe uniformly '-integrable if the following conditionsare ful@lled:

1. supf∈M

∫ |f| d'¡∞.2. For every positive constant * there is some positive

constant � such that

'(A) 6 � ⇒ supf∈M

∫A|f| d'6 *

holds for A∈F.

Uniform integrability plays an important role to linkconvergence in p-mean with convergence in measure(e.g. [1, Satz 21.4; 10, p. 195]):

A sequence (fn)n inLp(&;F; ') converges to somef∈Lp(&;F; ') in p-mean if and only if (fn)n con-verges to f in '-measure and the set {|fn|p | n∈N}is uniformly '-integrable.

Applying to the situation of Theorem 3.1, we haveto show necessarily that {|hAAn

|p | n∈N} is uniformly1-integrable if (An)n denotes a sequence in Fnob

cocp(Rk)


which converges to some A∈ Fnobcocp(Rk) with respect

to �p.

Lemma 4.1. Let (An)n be a sequence in Fnobcocp(Rk)

which converges to some A∈ Fnobcocp(Rk) with respect

to �p (p∈ [1;∞[).Then the set {|hAAn

|p | n∈N} is uniformly 1-integrable.

Proof. Let the mappings AB (A; B∈ Fnobcocp(Rk))

be de@ned as in Lemma 2.3. Next we can choosesome positive constant Cp such that the inequal-ity dH(K; {0})6Cp�p(K; {0}) holds for everyK ∈K+

co (Rk).On the one hand we @nd

|hAAn|p6 |Cp( 0A + An0)|p

6 |Cp(2 0A + AnA)|p

and

supn∈N

∫A|hAAn

|p d1

6 supn∈N

∫A|Cp(2 0A + AnA)|p d1 (2)

for every A∈B([0; 1]).On the other hand, limn→∞ �p(An; A) = 0 im-

plies that ( AnA)n converges to 0 in p-mean due toLemma 2.3. Hence (Cp(2 0A + AnA))n convergesto 2Cp 0A in p-mean, and (|Cp(2 0A + AnA)|p)n isuniformly 1-integrable. The statement of the lemmafollows then immediately from inequality (2).

Next we deal with the problem whether �p-convergence implies dp-convergence.

Lemma 4.2. Let (An)n be a sequence in Fnobcocp(Rk)

which converges to some A∈ Fnobcocp(Rk)with respect to

�p (p∈ [1;∞[). Then there exists some subsequence(A (n))n which converges to A with respect to dp.

Proof. According to Lemma 2.3, limn→∞ �p(An; A)= 0 implies that we can @nd some subsequence

(A (n))n and a 1-null set N such that

limn→∞ �p([A (n)]�; [A]�)p = 0

holds for every �∈ ]0; 1[\N (cf. [3, Propositions 3.1.2,3.1.4]). Therefore, we obtain

limn→∞ dH([A (n)]�; [A]�)p = 0

for every �∈ ]0; 1[\N because dH and �p inducethe same topology on K+

co (Rk). In particular(hA (n)A)n converges to 0 in 1-measure. Furthermore,

limn→∞ �p (A (n); A)p = 0, hence {|hA (n)A|p| n∈N}is uniformly 1-integrable due to Lemma 4.1.

Then (hA (n)A)n converges to 0 in p-mean, which

means limn→∞ dp(A (n); A)p = 0.

We are now in the position to prove Theorem 3.1.

Proof of Theorem 3.1. Since �p6dp, it suQces toshow that �p-convergence implies dp-convergence.Let (An)n be a sequence in Fnob

cocp(Rk) which convergesto some A∈ Fnob

cocp(Rk) with respect to �p. This impliesthat every subsequence of (An)n converges to A withrespect to �p. Then, according to Lemma 4.2, everysubsequence of (An)n includes a further subsequencewhich converges to A with respect to dp. This showslimn→∞ dp(An; A) = 0 and completes the proof.

5. Proof of Theorem 3.2

For p∈ [1;∞[ we consider Lp([0; 1]× Sk−1),the Lp-space with respect to the measure space([0; 1]× Sk−1;B([0; 1])⊗B(Sk−1); 1 ⊗ Sk−1

). En-dowed with the usual Lp-norm ‖·‖p, we obtainLp([0; 1]× Sk−1) as a separable Banach Space (cf. [3,Theorem 3.4.1, Proposition 3.4.5]). Real-valued map-pings g1; g2, which are 1 ⊗ Sk−1

-integrable of orderp, will be identi@ed by equivalence class 〈g1〉= 〈g2〉if and only if g1 = g2 a.s. with respect to 1 ⊗ Sk−1

.The space Lp([0; 1]× Sk−1) consists of all theseequivalence classes.

As a direct consequence of its de@nition we canembed Fnob

cocp(Rk) into Lp([0; 1]× Sk−1). Moreover, ac-cording to the de@nitions of �p and ‖·‖p, we can @nd


some embedding which is isometric with respect to �pand ‖·‖p.

Proposition and De!nition 5.1. For every p∈[1;∞[we can de9ne by

jFnobcocp(Rk );p(A) := 〈sA|[0; 1] × Sk−1〉

some mapping jFnobcocp(Rk );p from Fnob

cocp(Rk) to the space

Lp([0; 1]× Sk−1) holding

1. jFnobcocp(Rk );p is injective.

2. ‖jFnobcocp(Rk );p(A) − jFnob

cocp(Rk );p(B)‖p = �p(A; B) for

A; B∈ Fnobcocp(Rk).

jFnobcocp(Rk );p will be called the standard embedding of

Fnobcocp(Rk) in Lp([0; 1]× Sk−1).

The embedding result is an important tool to proveTheorem 3.2

Proof of Theorem 3.2. Proposition 2.5 shows thatFnobcoc (Rk) is a dense subset of Fnob

cocp(Rk) with respectto �p. Since �p is separable, �p is separable too.Therefore, it remains to prove the completeness of �p.

Let (An)n be some Cauchy sequence with re-spect to �p and let jFnob

cocp(Rk );p be the standard em-

bedding of Fnobcocp(Rk) into Lp([0; 1]× Sk−1). Then

(jFnobcocp(Rk );p(An))n is a Cauchy sequence with respect

to ‖·‖p due to Proposition and De@nition 5.1. Since(Lp([0; 1]× Sk−1); ‖·‖p) is a Banach space, thereexists some 〈f〉 ∈Lp([0; 1]× Sk−1) with

limn→∞ ‖jFnob

cocp(Rk );p(An) − 〈f〉‖p = 0:

Applying Fubini’s theorem, we can @nd some 1-nullset N and a subsequence (A (n))n with

limn→∞

∫|sA (n)

(�; ·) − f(�; ·)|p dSk−1= 0

for �∈ ]0; 1]\N (cf. [3, Proposition 3.1.2, Propo-sition 3.1.4]). In particular, using Minkowski’s in-equality, ([A (n)]�)n is a Cauchy sequence with re-spect to the Lp-metric �p for �∈ ]0; 1]\N . Hence,due to the completeness of (K+

co (Rk); �p), thereexists for every �∈ ]0; 1]\N a K� ∈K+

co (Rk) with

limn→∞ �p([A (n)]�; K�) = 0. This implies limn→∞dH([A (n)]�; K�) = 0 for �∈ ]0; 1]\N because dH and�p induce the same topology on K+

co (Rk).Drawing on Theorem II in [2] we obtain for

�; �∈ ]0; 1]\N with �¿�

K� =∞⋂n=1

cl

(⋃m¿n

[A (m)]�)

⊆∞⋂n=1

cl

(⋃m¿n

[A (m)]�)

= K�;

where cl(⋃

m¿n [A (m)]�) denotes the topological clo-sure of

⋃m¿n [A (m)]�, and cl(

⋃m¿n [A (m)]�) the

topological closure of⋃

m¿n [A (m)]�.We de@ne a system (C�)�∈[0;1] of subsets of Rk by

C0 :=Rk ; C� :=⋂

�∈]0;�[\N K� (�∈ ]0; 1]), which sat-is@es C� ⊇C� for �¡�, and C� ∈K+

co (Rk) for �¿0.For the remainder of the proof it suQces to show

(1) There exists some 1-null set PN ⊇N with C� =K�

for �∈ ]0; 1[\ PN .

Then C� =⋂

�∈]0;�[ C� for �∈ ]0; 1] follows imme-diately since ]0; 1[\ PN is a dense subset of [0; 1]. Asa known result of representation-theory, (C�)�∈[0;1]

describes then the �-cuts of some fuzzy subset A ofRk ([16, p. 91]). Obviously, by construction, A∈Fnococ(Rk).In the view of

limn→∞

∫|sA (n)

(�; ·) − f(�; ·)|p dSk−1= 0

and limn→∞ �p([A (n)]�; [A]�) = 0 for �∈ ]0; 1[\ PN wecan conclude for arbitrary �∈ ]0; 1[\ PN∫

|sA(�; ·) − f(�; ·)|p dSk−1= 0;

using Minkowski’s inequality. Application ofTonelli’s theorem leads to∫

|sA − f|p d1 ⊗ Sk−1

=∫ 1

0

∫|sA(�; ·) − f(�; ·)|p dSk−1

d� = 0:


Hence sA|[0; 1]× Sk−1 is 1 ⊗ Sk−1-integrable of

order p with 〈sA|[0; 1]× Sk−1〉= 〈f〉. Particularly,A∈ Fnob

cocp(Rk) and limn→∞ �p(An; A) = 0.Proof of (1).For @xed q∈Qk ∩ Sk−1 the real-valued mapping q

on ]0; 1], de@ned by

q(�) :=

{sK�(q) � ∈]0; 1]\N;

sC�(q) � ∈ N

is nonincreasing. Hence there exists some at mostcountable set Nq ⊆ ]0; 1] such that the restriction q|]0; 1]\Nq of q to ]0; 1]\Nq is continuous. Then,de@ning PN :=N ∪ ⋃q∈Qk ∩ Sk−1 Nq, every mapping q|]0; 1[\ PN (q∈Qk ∩ Sk−1) is continuous.

Let x∈ Sk−1; �∈ ]0; 1[\ PN , and let * be some arbi-trary positive constant. Since support functions of sub-sets K ∈K+

co (Rk) are continuous, we can @nd someq∈Qk ∩ Sk−1 with

max{|sK�(x) − sK�(q)|; |sC�(x) − sC�(q)|} ¡*2:

Moreover, there is some isotone sequence (�n)nin ]0; �[\ PN which converges to �. According toK�n ⊇C� ⊇K� for n∈N this means

q(�n) = sK�n(q) ¿ sC�(q) ¿ sK�(q) = q(�):

Therefore, sK�(q) = sC�(q) due to the continuity of q|]0; 1[\ PN , and

|sK�(x) − sC�(x)|

6 |sK�(x) − sK�(q)| + |sC�(q) − sC�(x)|

¡ *:

Now we can conclude sK�(x) = sC�(x) and �p(K�; C�)= 0. Step (1) is shown, and the proof is complete.

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