An embedding theorem for convex fuzzy sets

Pedro TeranFacultad de Ciencias Economicas y Empresariales

Departamento de Metodos Estadısticos

Universidad de Zaragoza

Gran Vıa, 2. E-50005 Zaragoza, Spain

e-mail: teran@unizar.es

AbstractIn this paper we embed the space of upper semicontinuous convex

fuzzy sets on a Banach space into a space of continuous functions on acompact space. The following structures are preserved by the embed-ding: convex cone, metric, sup-semilattice. The indicator function ofthe unit ball is mapped to the constant function 1. Two applicationsare presented: strong laws of large numbers for fuzzy random variablesand Korovkin type approximation theorems.

Keywords: embedding theorem, fuzzy random variable, strong law oflarge numbers, Korovkin type approximation theorem.

1 Introduction

The aim of this paper is to obtain an embedding theorem for a class offuzzy sets on a Banach space. Since that space does not have a linear spacestructure, embedding theorems are an essential tool in order to ‘transfer’mathematical results and techniques from better known spaces. Puri andRalescu were among the first to use this approach, in order to deal withdifferentiability issues [20], prove limit theorems for fuzzy random variables[13] and define other probabilistic notions [22]. The reader will find a briefoverview of other embedding theorems in Section 3.

1

Section 4 is devoted to the main results of the paper. We will first studythe structure of the codomain V of Puri and Ralescu’s Radstrom type embed-ding. They just showed that it is a normed space. Actually, it is a normedvector lattice with order unit (Theorem 4). As a consequence, we are finallyable to embed the space Fc, defined below, into a space of real continuousfunctions on a compact set (Theorem 6). In doing so, all of its structure ispreserved (operations, metric, sup-semilattice structure and order unit). Wewill also compare this embedding to those based on support functions.

In Section 5 we discuss two applications of our result, corresponding towhat we believe are its main areas of applicability: probability theory andcalculus in spaces of fuzzy sets.

2 Preliminaries

Let E be a Banach space with norm ‖ · ‖. We will denote its closed unitball by B. If E = Rd, we denote its unit sphere by Sd−1. The dual spaceof E will be denoted by E∗. Accordingly, B∗ will be its unit ball. The innerproduct in Rd will be denoted by < ·, · >.

We will denote by K the class of non-empty compact subsets of E, en-dowed with the Hausdorff metric

dH(A,C) = maxsupx∈A

infy∈C

‖x− y‖, supy∈C

infx∈A

‖x− y‖

= minε > 0 | A ⊂ C + εB,C ⊂ A + εBand the operations

A + C = x + y | x ∈ A, y ∈ C, λ · A = λx | x ∈ A.K is thus a metric convex cone and the subclass Kc of all convex sets in

K is a closed convex subcone. The convex hull of A ∈ K will be denoted byco A.

We will denote by F the family of positive upper semicontinuous functionson E having compact support and whose maximum value is 1. Observe thatevery level set Uα = x | U(x) ≥ α of U ∈ F is in K, for α ∈ (0, 1]. Theclosed support of U will be denoted by U0.

The subfamily of all U such that Uα is in Kc for all α ∈ (0, 1] will bedenoted by Fc. It is well known that U ∈ Fc if and only if

U(λx + (1− λ)y) ≥ minU(x), U(y)

2

for λ ∈ [0, 1], that is, U is quasiconcave. Whenever E is a concrete space,e.g. E = Rd, we may write F(E) instead of F , and the same applies to Fc,etcetera.

Denote by LU : [0, 1] → K the mapping such that LU(α) = Uα. We definetwo further subfamilies of Fc:

Fcc = U ∈ Fc | LU is dH-continuous, FcL = U ∈ Fc | LU is dH-Lipschitzian.

Every element of K can be identified to its indicator function IK ∈ F .Observe that the pointwise order in F extends the inclusion order in K.Indeed, A ⊂ C if, and only if IA ≤ IC . Following the conventional usage, wewill write U ⊂ V instead of U ≤ V .

We define the following operations in F :

(U + V )(x) = sup α ∈ [0, 1] |x ∈ Uα + Vα= sup

x1+x2=xminU(x1), V (x2),

(λU)(x) =

U(λ−1x), λ 6= 0

I0(x), λ = 0.

They extend those of K, in the sense that

(λU)α = λUα, and (U + V )α = Uα + Vα

for all α ∈ (0, 1] (see [19]).In order to endow F with a metric structure, we will use the Puri-Ralescu

metric

d∞(U, V ) = supα∈[0,1]

dH(Uα, Vα) = infε > 0 | U ⊂ V + εIB, V ⊂ U + εIB

The space (F , d∞) is complete [23] but not separable [13].For each U ∈ F , define co U ∈ Fc by the equalities (co U)α = co Uα for

α ∈ (0, 1]. Thus, co U is the quasiconcave envelope or fuzzy convex hull of U .If E is a metric space, we will denote by D(E) the family of cadlag

functions on [0, 1] taking on values in E. That is, every element of D(E) isright-continuous, continuous at 0 and has left limits at every point.

We will denote by C(Ω, E) the family of continuous mappings from atopological space Ω to E. We will write B and Lip instead of C in order to

3

mean bounded and Lipschitzian functions, respectively. The sup norm in afunction space will be denoted by ‖ · ‖∞.

A partially ordered linear space is a real linear space V endowed with apartial order ≺ such that

(i) x ≺ y, x′ ≺ y′ =⇒ x + x′ ≺ y + y′,

(ii) x ≺ y, a ∈ R+ =⇒ ax ≺ ay,

that is, ≺ is compatible with the operations of V. If ≺ determines a latticestructure, V is called a vector lattice or Riesz space. We will denote supre-mum and infimum by ∨ and ∧, respectively. A sufficient condition for V tobe a vector lattice is the existence of the positive part x+ = x ∨ 0 of everyx ∈ V.

The positive cone V+ is the set of all x ∈ V such that x+ = x. Thenegative part and absolute value of x are defined to be x− = (−x)+ and|x| = x∨ (−x). In every vector lattice the formulae x∨ y = x + (y−x)+ andx ∧ y = y − (y − x)+ hold. Whence, x ∨ y + x ∧ y = x + y.

We will say that V is a normed vector lattice if it is endowed with a normfor which |x| ≺ |y| yields ‖x‖ ≤ ‖y‖. Such a norm is called a Riesz norm. IfV′ is a vector sublattice of V, it will be called order-dense if

x =∨y ∈ V′ | 0 ≺ y ≺ x

for all x ∈ V+.A vector lattice is called Archimedean if

∧ε>0 εx = 0 for every x ∈ V+.

An order unit of V is an element e ∈ V+ such that every x ∈ V is boundedbetween −ne and ne for some n ∈ N. A Banach lattice is a complete normedvector lattice. A space of real continuous functions on a compact Hausdorffspace will be called a C(K) space.

3 Some existing embedding theorems

In this section we review a number of known embedding theorems. We aim toillustrate the various approaches to this topic and underline the relationshipsand differences with ours.

Embeddings into linear spaces are not possible for F . This is so becauseevery linear continuous mapping from F to a topological linear space V

4

identifies each function with its quasiconcave envelope. Indeed, let f : F → Vbe linear continuous, then

f(U) =1

m(f(U) + . . . + f(U)) = f(

1

m(U + . . . + U)) → f(co U)

since d∞( 1m

(U + . . . + U), co U) → 0 by [5, Lemma ??].Most of the known embeddings have been obtained under the additional

assumption of quasiconcavity. We tell apart three main kinds of embeddingtheorems for upper semicontinuous fuzzy sets: Radstrom type embeddings,embeddings via support functions of level sets and embeddings into spacesof cadlag functions.

As to the first kind, Puri and Ralescu [20] used a Radstrom type embed-ding to show that (Fc, d∞, +, ·) embeds into a normed space V . Since V is aquotient of Fc×Fc, it is not immediately recognizable as a ‘concrete’ space.That is the main disadvantage of their embedding. Since our results will bebased on that kind of embedding, details will follow in next section.

An alternative approach uses support functions. In that way, Klement,Puri and Ralescu [13] showed that (FcL(Rd), d∞, +, ·) is embedded into Lip ([0, 1]×Sd−1,R) endowed with its sup-norm, by the mapping U 7→ sU such that

sU(α, r) = supx∈Uα

< r, x > .

One can define sU for any U ∈ Fc and check easily that sU is bounded,therefore (Fc(R

d), d∞, +, ·) embeds into B([0, 1] × Sd−1,R) which is also aBanach space [7]. Ma [17] showed that the codomain of the embedding canbe taken to be D(C(Sd−1,R)). Roman-Flores and Rojas-Medar [25] recentlyproved for separable E that Fcc is the largest subclass of Fc embeddable intoC([0, 1]×B∗,R) via support functions.

The idea of using spaces of cadlag functions, already present in Ma’sresult, has been recently recovered by Colubi et al. [4] and Kim [12] in orderto embed (F , d∞) into D(K) with its uniform metric. To each U ∈ F ,one associates the function t 7→ U1−t. In doing so, F is identified with theincreasing cone of D(K). Observe that D(K) is not a linear space. Howeverspaces of cadlag functions taking on values in a metric space have beenstudied due to their relevance to the theory of stochastic processes [10].

5

4 A new embedding theorem

In this section we present the main results of the paper. We will pursue theidea of Puri and Ralescu’s original Radstrom type embedding. We will showthat, despite its ‘unfamiliar’ appearance, the codomain of the embedding hasa rich structure and many nice properties. In particular, it is an Archimedeanvector lattice with order unit. Those nice properties allow to apply a furtherembedding theorem into a C(K) space.

We have pointed out that only Fcc can be embedded into a space ofcontinuous functions by using support functions. But Fcc is a quite smallsubset of Fc, in the sense that it is separable whereas the latter is not so [13,Proposition 3.3].

The clearest strength of the new embedding theorem is that the targetspace is a classical Banach space of continuous functions which is well under-stood. That simplifies greatly its use and enlarges the toolset for studyingFc. Also, for some applications the

As already mentioned, we take Puri and Ralescu’s result as our departurepoint.

Lemma 1. (Fc, d∞, +, ·) embeds into a normed linear space.

The proof in [20] assumes that E is reflexive and allows level sets to beclosed convex bounded sets. Since the technique in our setting is identical(see Radstrom’s original paper [24]), we will just describe it briefly.

Define the equivalence relation ∼ in Fc ×Fc such that

(U, V ) ∼ (U ′, V ′) ⇐⇒ U + V ′ = V + U ′.

Then one proves that the quotient space Fc × Fc/ ∼ is a real linear spaceV . We will denote those equivalence classes by [U, V ], and set 0 = [I0, I0]since it is the neutral element of V . Endowing V with the norm

‖[U, V ]‖ = d∞(U, V ),

the metric convex cone (Fc, d∞, +, ·) is embedded into (V , ‖ · ‖, +, ·) by thenatural identification U 7→ [U, I0].

Notice that the inclusion relation⊂ plays no explicit role in [20]. However,it will be essential to our approach. To this end, we define the ordering

[U, V ] ≺ [U ′, V ′] ⇐⇒ U + V ′ ⊂ V + U ′

6

which extends of inclusion to V . When defined between pairs of elements ofFc, it is a preordering naturally associated to the equivalence relation ∼.

We begin by proving the following curious identity. As a consequence, ananalogous identity holds for d∞.

Lemma 2. Let A,C ∈ Kc. Then,

dH(A,C) = dH(2 co(A ∪ C), A + C).

Proof. Recall that

dH(A,C) = minη > 0 | A ⊂ C + ηB,C ⊂ A + ηB.Set ε = dH(A,C). Notice that

2(A ∪ C) ⊂ A ∪ C + A ∪ C = (A + A) ∪ (A + C) ∪ (C + A) ∪ (C + C),

and each term in the right hand side union is included in A + C + εB, sothat

2 co(A ∪ C) ⊂ co(A + C + εB) = A + C + εB.

Moreover A ⊂ co(A ∪ C) and C ⊂ co(A ∪ C), whence

A + C ⊂ co(A ∪ C) + co(A ∪ C) = 2 co(A ∪ C) ⊂ 2 co(A ∪ C) + εB.

We have proven that

dH(2 co(A ∪ C), A + C) ≤ ε = dH(A,C).

Let us check now the converse inequality. For that purpose, denote ε′ =dH(2 co(A ∪ C), A + C). By the definition of the Hausdorff metric, 2 co(A ∪C) ⊂ A + C + ε′B. Thus,

A + A ⊂ co(A ∪ C) + co(A ∪ C) = 2 co(A ∪ C) ⊂ A + C + ε′B.

Since A + A ⊂ A + C + ε′B and all the involved sets are closed, convex andbounded, we deduce that A ⊂ C + ε′B (see [24, Lemma 1]). Analogouslyone proves that C ⊂ A + ε′B, so that

dH(A,C) ≤ ε′ = dH(2 co(A ∪ C), A + C)

and the proof is complete.

7

Corollary 3. Let U, V ∈ Fc. Then,

d∞(U, V ) = d∞(2 co maxU, V , U + V ).

Proof. It suffices to apply Lemma 2, taking into account that d∞(U, V ) =supα∈[0,1] dH(Uα, Vα) and (maxU, V )α = Uα ∪ Vα.

Notice that maxU, V ∈ F if U, V ∈ F but it need not be quasiconcaveif U and V are so.

Now we will establish the embedding of Fc with its full structure into V ,as well as some of the good properties of the space V .

Theorem 4. The partially ordered metric convex cone (Fc, d∞, +, ·,⊂) isisomorphic to a closed convex subcone of (V , ‖ · ‖, +, ·,≺). Moreover, V is avector lattice such that

(i) [U, V ]+ = [co maxU, V , V ],

(ii) V+ = [U, V ] | V ⊂ U,(iii) [U, V ] ∨ [U ′, V ′] = [co maxU ′ + V, U + V ′, V + V ′],

(iv) [U, V ] ∧ [U ′, V ′] = [U + U ′, co maxU ′ + V, U + V ′],(v) |[U, V ]| = [2 co maxU, V , U + V ],

(vi) ‖ · ‖ is a Riesz norm,

(vii) [IB, I0] is an order unit.

Proof. We know, from Lemma 1 and the definition of ≺, that Fc embeds intoV . Since Fc is complete, its image is closed in V .

The first thing to check is that addition and product by scalars are com-patible with the ordering ≺ in V . This is routine and we leave it to thereader.

Let us show now that V is not just a partially ordered linear space buta vector lattice. We have to check that each [U, V ] has a positive part in V ,we will indeed show that [U, V ]+ = [co maxU, V , V ].

First, U and V are included in co maxU, V ∈ Fc, so that U + V ⊂V + co maxU, V , too. As a consequence,

[U, V ] ≺ [co maxU, V , V ] and 0 ≺ [co maxU, V , V ].

8

That is, [co maxU, V , V ] is an upper bound of [U, V ] and 0. It remains toprove that it is the least one.

Let [U ′, V ′] be such that [U, V ] ≺ [U ′, V ′] and 0 ≺ [U ′, V ′]. By thedefinition of ≺, both U + V ′ ⊂ V + U ′ and V ′ ⊂ U ′ hold. It follows that

maxU, V + V ′ = maxU + V ′, V + V ′ ⊂ V + U ′,

and thus

co maxU, V + V ′ = co(maxU, V + V ′) ⊂ co(V + U ′) = V + U ′

since V + U ′ ∈ Fc.That is, [co maxU, V , V ] ≺ [U ′, V ′] and V is a vector lattice.Let us identify now the positive cone V+. It is formed by all the elements

which equal their positive part, or equivalently whose negative part is 0.Since

[U, V ]− = (−[U, V ])+ = [V, U ]+ = [co maxU, V , U ],

it will equal 0 if, and only if co maxU, V = U . From that identity, V ⊂ Ufollows. Conversely, if the latter holds, also co maxU, V = co U = U .Therefore, (ii) has been proven.

Besides, the image of Fc is closed by the supremum operation, since

[U, I0] ∨ [V, I0] = [co maxU, V , I0].Now we will prove (iii):

[U, V ] ∨ [U ′, V ′] = [U, V ] + ([U ′, V ′]− [U, V ])+

= [U, V ] + ([U ′, V ′] + [V, U ])+ = [U, V ] + [U ′ + V, V ′ + U ]+

= [U, V ] + [co maxU ′ + V, V ′ + U, V ′ + U ]

= [U + co maxU ′ + V, V ′ + U, V + V ′ + U ]

= [co maxU ′ + V, V ′ + U, V + V ′].

The proof of (iv) is analogous. Besides,

|[U, V ]| = [U, V ] ∨ (−[U, V ]) = [U, V ] ∨ [V, U ]

= [co maxV + V, U + U, V + U ] = [2 co maxU, V , U + V ],

so that we also have (v).

9

Observe that if [U, V ] ≺ [U ′, V ′] and both are in V+, then V ⊂ U , V ′ ⊂ U ′

and U + V ′ ⊂ V + U ′. Let ε > ‖[U ′, V ′]‖ = d∞(U ′, V ′). Then, U ′ ⊂ V ′ + εBand we deduce that

U + V ′ ⊂ V + U ′ ⊂ V + V ′ + εB.

Since V ′ ∈ Fc, the cancellation law [24, Lemma 1] ensures that U ⊂ V + εB,and that, together with V ⊂ U , implies that ‖[U, V ]‖ = d∞(U, V ) ≤ ε. Thatholds for all ε > ‖[U ′, V ′]‖, therefore

[U, V ] ≺ [U ′, V ′] ∈ V+ =⇒ ‖[U, V ]‖ ≤ ‖[U ′, V ′]‖.

Let us prove now (vi), i.e. that its norm makes V a normed vector lattice.Assume that |[U, V ]| ≺ |[U ′, V ′]|. By the implication above, ‖ |[U, V ]| ‖ ≺‖ |[U ′, V ′]| ‖, that is,

d∞(2 co maxU, V , U + V ) ≤ d∞(2 co maxU ′, V ′, U ′ + V ′).

As shown by Corollary 3, that is equivalent to saying that d∞(U, V ) ≤d∞(U ′, V ′) i.e. ‖[U, V ]‖ ≤ ‖[U ′, V ′]‖.

It just remains to check that (vii) holds. Let [U, V ] ∈ V . We want toshow that there exists n ∈ N such that

−n[IB, I0] ⊂ [U, V ] ⊂ n[IB, I0],

that is, V ⊂ U + nIB and U ⊂ V + nIB. It suffices to take any n >d∞(U, V ).

Remark 1. If Fc is endowed with the supremum with respect to the ordering⊂, the embedding U 7→ [U, I0] makes Fc isomorphic to a sub sup-semilatticeof V . But the same is not true for the infimum, since

[U, I0] ∧ [V, I0] = [U + V, co maxU, V ]

which actually may not have the form [W, I0]: just take U = I0, V = I1above. That is explained by the fact that Fc is not an inf-semilattice (theminimum of two elements of Fc may not reach the top value 1). ¤

In the following, we show that V can, in its turn, be embedded into aC(K) space, in a way which preserves all of its structure.

10

Proposition 5. There exist a compact Hausdorff topological space Γ and amapping j : V → C(Γ,R) such that:

(i) j is an isometry with respect to ‖ · ‖ and ‖ · ‖∞,

(ii) j is a homomorphism of vector lattices,

(iii) j(V) is dense and order-dense in C(Γ,R),

(iv) j([IB, I0]) = 1.

Proof. We know that V is a vector lattice and that ‖ · ‖ is a Riesz norm.Then, V is Archimedean by [8, Lemma 354B]. Moreover, we have proventhat [IB, I0] is a order unit of V . To each order unit e, we can associate theRiesz norm ‖ · ‖e defined to be

‖x‖e = minε > 0 | |x| ≤ ε · e

(see [8, Lemma 354F]).Let us take e = [IB, I0], then by [8, Theorem 354K] there exist Γ and j

as in the statement of the theorem such that j is a linear isometry between‖ · ‖e and ‖ · ‖∞, and (ii) through (iv) hold.

It just remains to prove that actually ‖[U, V ]‖ = ‖[U, V ]‖e for all [U, V ] ∈V . But

‖[U, V ]‖e = minε > 0 | [2 co maxU, V , U + V ] ≺ ε[IB, I0]

= minε > 0 | 2 co maxU, V ⊂ U + V + εIB.Since U + V ⊂ 2 co maxU, V , the right hand side is actually

minε > 0 | 2 co maxU, V ⊂ U + V + εIB, U + V ⊂ 2 co maxU, V + εIB

= d∞(2 co maxU, V , U + V ) = d∞(U, V ) = ‖[U, V ]‖.

Remark 2. Neither j(Fc) nor V+ contains the other. Indeed, if 0 6∈ U1, then[U, I0] 6∈ V+. On the other hand, if V ⊂ U , there does not necessarilyexist W ∈ Fc such that U = V + W or, equivalently, [U, V ] = [W, I0]. Theintersection j(Fc) ∩ V+ is the set of the images of all elements of Fc whose1-level set contains 0. ¤

11

By composing the former results, we obtain the sought embedding theo-rem.

Theorem 6. Let Γ be as in Proposition 5. Then, there exists a mappingh : Fc → C(Γ,R) which has the following properties:

(i) h is a linear isometry with respect to d∞ and ‖ · ‖∞,

(ii) h is a homomorphism of sup-semilattices (and so is positive),

(iii) h(IB) = 1,

(iv) h(Fc)− h(Fc) is dense and order-dense in C(Γ,R).

Proof. This follows from the properties of the embeddings of Fc into V andof V into C(Γ,R). For (iv), notice that h(Fc)− h(Fc) = j(V).

As a further consequence, we retrieve the following classical result.

Corollary 7. Every Banach space E embeds isometrically into a C(K) space.

Proof. Notice that E is embedded into Fc by the mapping x 7→ Ix, thenapply Theorem 6.

Let us identify explicitly the compact set Γ in the statement of the em-bedding theorem. Obviously Γ is only determined up to homeomorphism.

Proposition 8. In Theorems 5 and 6, Γ can be taken to be the set of allmappings T : Fc → R such that

(i) T (aU + bV ) = aT (U) + bT (V ) for all a, b ≥ 0, U, V ∈ Fc, i.e. T islinear,

(ii) T (co maxU, V ) = maxTU, TV , i.e. T is a sup-semilattice homo-morphism,

(iii) T (IB) = 1,

endowed with the pointwise convergence topology (the relative product topol-ogy inherited from RFc .)

12

Proof. By [8, Corollary 354L], Γ can be identified to the set of Riesz homo-morphisms T from V to R such that T ([IB, I0]) = 1, endowed with thepointwise convergence topology.

For each T under those conditions, we define T : Fc → R by TU =T ([U, I0]).

Clearly, T satisfies (iii). By [8, Proposition 352G], T is a Riesz homo-morphism if, and only if it is linear and preserves positive parts. From thelinearity, (i) follows, moreover

T ([U, V ]) = T ([U, I0])− T ([V, I0]) = T (U)− T (V )

for all U, V ∈ Fc.The preservation of positive parts means that T ([U, V ]+) = T ([U, V ])+

for all U, V ∈ Fc. Thus,

T ([U, V ]+) = T ([co maxU, V , V ]) = T (co maxU, V )− T (V )

and T ([U, V ]) = T (U)− T (V ), whence

T (co maxU, V )− T (V ) = [T (U)− T (V )]+ = maxT (U), T (V ) − T (V ).

This yields (ii).We have shown that each T in the statement of the proposition corre-

sponds to some Riesz homomorphism T , and obviously this correspondenceis injective. Let us prove that one can also pass from T to T .

Take T as in the statement and define T : V → R by T ([U, V ]) = T (U)−T (V ). Now T is well defined, since [U, V ] = [U ′, V ′] implies U + V ′ =V + U ′ and by the linearity of T we obtain T (U) + T (V ′) = T (V ) + T (U ′)i.e. T (U)− T (V ) = T (U ′)− T (V ′).

Notice that T (−[U, V ]) = T ([V, U ]) = T (V )− T (U) = −T ([U, V ]). Thenit is routine to show that T is linear. In order to prove (ii), one just has toreverse the argument concerning positive part preservation. Finally, (iii) isobvious.

It only remains to prove that the topologies are homeomorphic. Butsince they are sets of real-valued functions with the pointwise convergencetopology, it is obvious that the topological structures are the same as soonas the domains of the functions are bijective.

Remark 3. Notice that Γ contains all the support functionals s(x∗, α, ·) :Fc → R since they satisfy properties (i) to (iii). It is also interesting to

13

point out that the pointwise convergence s(x∗n, αn, ·) → s(x∗, α, ·) impliesconvergence on the subclass Ixx∈E, but

s(x∗n, αn, Ix) = x∗n(x), s(x∗, α, Ix) = x∗(x)

and x∗n(x) → x∗(x) is just the weak∗ convergence in E∗.Recall that Hormander’s embedding theorem, in the modified version of

Gine and Hahn [9] allows to identify compact convex sets with their supportfunctions, seen as continuous functions on B∗ with the weak∗ topology. Thisrelates both embedding theorems. ¤

5 Two applications

In this section, two applications of Theorem 6 are shown. They are oneanalytical (convergence of linear positive operators), the other probabilistic(convergence of fuzzy random variables). For space reasons, they are treatedin less detail than the rest of the paper. Our aim is just to give the reader asuggestion of what can be comfortably done with the help of Theorem 6.

5.1 Strong laws of large numbers for fuzzy randomvariables

A large field of forseeable applications of the embedding theorem is proba-bility theory in spaces of upper semicontinuous functions on Banach spaces.

Assume that E is separable. Let X be a fuzzy random variable, i.e. arandom element of Fc in the sense that each mapping Xα : ω 7→ X(ω)α isa random compact set (see [18]). Then X can be identified with a randomelement of C(Γ,R), namely h X, which need not be Borel (nor Radon) asa consequence of [4].

However it is known that many results related to the strong law of largenumbers (SLLN) in Banach spaces do not require Borel measurability, see[16]. It is indeed enough to define measurability of a random element ξ of aBanach space to be the measurability of all fn(ξ) for a fixed sequence fn ∈ B∗

such that ‖x‖ = supn |fn(x)| for all x ∈ E (i.e. a norming sequence).E being separable, there exists a countable x∗nn ⊂ B∗ which is dense in

B∗ in the weak∗ topology. Define fn,q(U) = s(x∗n, Uq) for all U ∈ Fc, n ∈ N

14

and q ∈ (0, 1] ∩Q. Observe that

d∞(U, V ) = supn,q|fn,q(U)− fn,q(V )|.

Since fn,q : Fc → R are linear and d∞-continuous, they are easily extended toelements of C(Γ,R)∗ by taking into account part (iv) of Theorem 6. Denotethe extensions by fn,q. Then ‖ϕ‖∞ = supn,q |fn,q(ϕ)| for any ϕ ∈ C(Γ,R).

By [2, Theorem III.15], each mapping ω 7→ s(x∗, Xα(ω)) is measurable.All fn,q(h X) are then clearly measurable and h X is measurable in therelaxed sense above. We will refer to this kind of measurability as countablescalar measurability. A countably scalarly measurable mapping will be calleda c.s.m. random element.

Consequently, the bulk of [16, Chapter 7] can be applied to random el-ements of Fc via the embedding h. That is, in order to obtain the SLLNfor a sequence Xnn of c.s.m. random elements of Fc, one just has to ap-ply the corresponding Banach space result to the centered random elementsh(Xn)− h(EXn) (where E is the expectation operator). Notice that

‖ 1

n

n∑i=1

(h(Xi)−h(EXi))‖ = ‖h(n∑

i=1

Xi)−h(n∑

i=1

EXi)‖ = d∞(n∑

i=1

Xi,

n∑i=1

EXi).

Recall that a fuzzy random variable X is called integrably bounded when‖X0‖ is integrable. X is integrably bounded if, and only if there existsEX ∈ F (see [23]).

The following SLLN is new in the context of fuzzy random variables,yet simple enough to serve here as an illustrative example. Chapter 7 in [16]contains more sophisticated results, e.g. Theorem 7.12, that are adapted withthe same technique but whose exposition would need additional notions.

Proposition 9. Let Xnn be a sequence of independent integrably boundedc.s.m. random elements of Fc such that

∑∞i=1 i−rd∞(Xi, EXi)

r < ∞ for some1 ≤ r ≤ 2. Then, the following are equivalent:

(i) d∞( 1n

∑ni=1 Xi,

1n

∑ni=1 EXi) → 0 in probability,

(ii) d∞( 1n

∑ni=1 Xi,

1n

∑ni=1 EXi) → 0 almost surely.

Proof. It follows from [16, Corollary 7.14] and the discussion above.

15

With the former proposition at hand, we aim now at removing the hypoth-esis of quasiconcavity and showing that the Chung type condition

∑∞i=1 i−r

d∞(Xi, EXi)r < ∞ is indeed sufficient to obtain the strong law of large

numbers.Our first step is to adapt Puri and Ralescu’s version [21] of the Shapley-

Folkman inequality. Recall that the inner radius of a compact set A ∈ K isthe quantity

r(A) = supa∈co A

infR ≥ 0 | ∃a1, . . . , as ∈ A, a ∈ coaisi=1, ‖a− ai‖ ≤ R.

We define the levelwise inner radius of U ∈ F to be r(U) = supα∈[0,1] r(Uα).Note that r(IA) = r(A) and necessarily r(U) ≤ 2 · ‖U0‖ < ∞.

Lemma 10. Assume that E has type p > 1. Then, for every U1, . . . , Un ∈ F ,

d∞(n∑

i=1

Ui,

n∑i=1

co Ui) ≤ c1/p · (n∑

i=1

r(Ui)p)1/p,

where c is the type-p constant of E.

The proof is easy, taking into account Puri and Ralescu’s inequality. Thenwe obtain the following.

Theorem 11. Let E be a separable Banach space with type p > 1. LetXnn be a sequence of independent integrably bounded fuzzy random vari-ables such that

∑∞i=1 i−rd∞(Xi, EXi)

r < ∞ for some 1 ≤ r ≤ 2. Then,

d∞(1

n

n∑i=1

Xi,1

n

n∑i=1

E co Xi) → 0

Proof. Notice that

d∞(1

n

n∑i=1

Xi,1

n

n∑i=1

E co Xi) ≤ d∞(1

n

n∑i=1

Xi,1

n

n∑i=1

co Xi)

+d∞(1

n

n∑i=1

co Xi,1

n

n∑i=1

E co Xi) = (I) + (II).

16

By Lemma 10,

(I) ≤ c1/p · (n∑

i=1

r(1

nXi)

p)1/p ≤ 2c1/p · 1

n(

n∑i=1

‖(Xi)0‖p)1/p.

The latter goes to 0 almost surely since the Marcinkiewicz-Zygmund SLLN(e.g. [3]) ensures that 1

np (∑n

i=1 ‖(Xi)0‖p) → 0 for p > 1.As regards (II), the condition

∑∞i=1 i−rd∞(Xi, EXi)

r < ∞ implies thatthe weak law of large numbers holds for the sequence. This can be shownas in the proof of [16, Theorem 9.17] (that theorem is stated for a separableBanach space, but it can be checked that separability is not necessary inorder to prove the claim above.)

Almost sure convergence of (II) to 0 follows then from Proposition 9,taking into account that co Xnn are c.s.m. random elements of Fc.

Theorem 11 can be compared to [6, Corollary 1] and [15, Theorem 8.2](in both papers E = Rd.) In the former paper, the additional hypothesisthat E co Xnn is relatively d∞-compact is imposed. That condition is verystrong, since it implies but is not implied by the condition that ‖E(Xn)0‖n

is bounded. In [15], convergence is obtained in weaker separable metricswhich are shown to be embeddable using support functions into spaces oftype p > 1. Therefore, the proof scheme breaks down for d∞ since we haveshown that (loosely speaking) the linear span of Fc is dense in a type-1 space.We would like to remark that Theorem 11 does not subsume [15, Theorem8.2] either since the space Fc is a bit smaller than the spaces in the latterpaper.

5.2 Korovkin type approximation theorems

This subsection is devoted to discussing Korovkin type theorems in spaces ofFc-valued continuous functions. Korovkin’s original theorem [14] states thesurprising fact that Tnf → f for every f ∈ C([0, 1],R), for a sequence Tn oflinear positive operators, as soon as convergence holds for the three functions1, x, x2. The family 1, x, x2 is then called a Korovkin system for the spaceC([0, 1],R). The main object of Korovkin type approximation theory is thestudy of Korovkin systems in more general spaces.

A vast amount of research has been produced since Korovkin’s discovery,and the reader is referred to [1] for an extensive account of the theory and its

17

applications. That book considers C0(Ω,R) spaces, i.e. spaces of continuousfunctions which vanish at infinity on a locally compact Hausdorff space Ω.That generality is sufficient for our purposes, since obviously C(Γ,R) is sucha space.

An operator S : Fc → Fc is called linear if S(aU + bV ) = aS(U) + bS(V )for a, b ≥ 0, and positive if S(U) ⊂ S(V ) whenever U ⊂ V . The properties ofthe embedding h allow to extend naturally and uniquely each linear positiveoperator S to the whole C(Γ,R). In this way, all the results in [1] canbe applied to operators in Fc. Here, the fact that the target space of theembedding is one of continuous functions is essential.

We refer the reader directly to [1] instead of quoting a more or less random(and more or less fortunate) selection of results. The reader interested inKorovkin type results for F valued mappings is also referred to [28, 27].

Operators in C(Ω,Fc) or C0(Ω,Fc) can be studied in an equally simpleway. Observe that C(Ω,Fc) embeds into C(Ω, C(Γ,R)), which in its turnembeds naturally into C(Ω×Γ,R), again a space of real continuous functionsto which the classical Korovkin theory can be applied (C0(Ω,Fc) is managedanalogously.)

That device is similar to the one used by Keimel and Roth in their paper[11]. There, in order to obtain Korovkin theorems for mappings whose valuesare in Kc(R

d), the authors mapped a set to its support function (which is anelement of C(Sd−1,R)) and then passed to the space C(Ω× Sd−1,R).

In [28, Theorem 3.1], we have proven a Korovkin type theorem in C(F)(that is, without assuming quasiconcavity). The above-mentioned embeddinginto C(Ω × Γ,R) and an application of a real valued analogous result [1,Theorem 4.2.10] yield a very simplified proof of it in the case of C(Fc). SinceC(F) does not support a similar embedding, that simple proof does not allowto subsume the theorem in full generality. We state here, for the reader’sconvenience, a slightly improved version of that result [27, Theorem 1].

Theorem 12. Let Ω be a compact Hausdorff topological space, and let φ :Ω2 −→ R be a continuous mapping such that φ(x, x) = 0 for all x ∈ Ω andφ(x, y) > 0 for all x, y ∈ Ω with x 6= y. Let u ∈ Fc be such that aIB ⊂ ufor some a > 0. Let Ln,λ : C(Ω,F) −→ B(Ω,F), n ∈ N, λ ∈ Λ (Λ being anindex set), be positive linear operators, such that there exists n0 ∈ N with

supλ∈Λ,n≥n0

‖Ln,λ(u)‖C < ∞.

18

Let T : C(Ω,F) −→ C(Ω,F) be an F -operator.Then, the following conditions are equivalent:

i) D∞(Ln,λ(X), T (X)) −→ 0 uniformly in λ ∈ Λ for all X ∈ C(F),

ii) D∞(Ln,λ(φ(x, ·)u), T (φ(x, ·)u)) −→ 0 uniformly in λ ∈ Λ for all x ∈ Ω,

D∞(Ln,λ(A), T (A)) −→ 0 uniformly in λ ∈ Λ for all A ∈ F ,

iii) supx∈Ω ‖(Ln,λ(φ(x, ·)IB))0‖(x) −→ 0 uniformly in λ ∈ Λ,

D∞(Lnλ(A), T (A)) −→ 0 uniformly in λ ∈ Λ for all A ∈ F .

6 Concluding remarks

Provided that E is reflexive, the same technique allows to obtain the em-bedding theorem for the larger space Gc of all non-negative u.s.c. functionswhose maximum value is 1 and whose upper level sets are closed, bounded(instead of compact) and convex subsets of E.

Theorem 6 has been applied in [27] in order to give a construction methodfor Korovkin systems in C(F) from real Korovkin systems via Korovkin sys-tems in C(Fc). It is used in [26] too, in the study of fuzzy valued sub- andsupermartingales.

Note: In [26], references made to Corollary 6 of (a preprint version of) thispaper should be understood as referring to Theorem 6.

Acknowledgements: I thank the area editor for some interesting remarks.Research has been partially funded by the F.P.I. fellowship FP98-71701353 andthe research grant BFM 2002-03263 from the Spanish Ministerio de Ciencia yTecnologıa.

References

[1] F. Altomare, M. Campiti (1994). Korovkin type approximation theory and itsapplications. De Gruyter Studies Math. 17. De Gruyter, Berlin.

19

[2] C. Castaing, M. Valadier (1977). Convex analysis and measurable multifunc-tions. Lecture Notes Math. 580. Springer, Berlin.

[3] K. L. Chung (1974). A course in probability theory. Academic Press, NewYork.

[4] A. Colubi, J. S. Domınguez Menchero, M. Lopez-Dıaz, D. Ralescu (2002).A DE [0, 1] representation of random upper semicontinuous functions.Proc. Amer. Math. Soc. 130, 3237–3242.

[5] A. Colubi, M. Lopez-Dıaz, J. S. Domınguez Menchero, M. A. Gil (1999). Ageneralized strong law of large numbers. Probab. Theory Relat. Fields 114,401–417.

[6] A. Colubi, M. Lopez-Dıaz, J. S. Domınguez Menchero, R. Korner (2001). Amethod to derive strong laws of large numbers for random upper semicontin-uous functions. Statist. Probab. Letters 53, 269–275.

[7] P. Diamond, P. Kloeden (1994). Metric spaces of fuzzy sets. World Scientific,Singapore.

[8] D. H. Fremlin (2002). Measure theory. Vol. 3. Tor-res Fremlin, Essex. [Abridged version available fromhttp://www.essex.ac.uk/maths/staff/fremdh/mt.htm]

[9] E. Gine, M. G. Hahn (1985). Characterization and domains of attraction ofp-stable random compact sets. Ann. Probab. 13 447–468.

[10] A. Jakubowski (1986). On the Skorohod topology. Ann. Inst. H. PoincareProbab. Statist. 22, 263–285.

[11] K. Keimel, W. Roth (1988). A Korovkin type approximation theorem forset–valued functions. Proc. Amer. Math. Soc. 104, 819–824.

[12] Y. K. Kim (2001). Compactness and convexity on the space of fuzzy sets.J. Math. Anal. Appl. 264, 122–132.

[13] E. P. Klement, M. L. Puri, D. A. Ralescu (1986). Limit theorems for fuzzyrandom variables. Proc. Roy. Soc. London Ser. A 407, 171–182.

[14] P. P. Korovkin (1960). Linear operators and approximation theory. Hindustan,Delhi.

20

[15] V. Kratschmer. Integrals of random fuzzy sets. Preprint, Universitat des Saar-landes, 2003.

[16] M. Ledoux, M. Talagrand (1991). Probability theory in Banach spaces.Springer, Berlin.

[17] M. Ma (1993). On embedding problems of fuzzy number space: part 5. FuzzySets and Systems 55, 313–318.

[18] G. Matheron (1975). Random sets and integral geometry. Wiley, New York.

[19] M. L. Puri, D. A. Ralescu (1981). Differentielle d’une fonction floue.C. R. Acad. Sci. Paris Ser. I Math. 293, 237–239.

[20] M. L. Puri, D. A. Ralescu (1983). Differentials of fuzzy functions.J. Math. Anal. Appl. 91, 552–558.

[21] M. L. Puri, D. A. Ralescu (1985). Limit theorems for random compact setsin Banach spaces. Math. Proc. Cambridge Phil. Soc. 97, 151–158.

[22] M. L. Puri, D. A. Ralescu (1985). The concept of normality for fuzzy randomvariables. Ann. Probab. 13, 1373–1379.

[23] M. L. Puri, D. A. Ralescu (1986). Fuzzy random variables. J. Math. Anal.Appl. 114, 409–422.

[24] H. Radstrom (1952). An embedding theorem for spaces of convex sets.Proc. Amer. Math. Soc. 3, 165–169.

[25] H. Roman-Flores, M. A. Rojas-Medar (2002). Embedding of level-continuousfuzzy sets on Banach spaces. Inform. Sci. 144, 227–247.

[26] P. Teran (2004). Cones and decomposition of sub- and supermartingales.Fuzzy Sets and Systems, to appear.

[27] P. Teran. Function valued Korovkin systems without quasiconcavity and setvalued Korovkin systems without convexity. Submitted for publication, 2003.

[28] P. Teran, M. Lopez-Dıaz (2001). Approximation of mappings with valueswhich are upper semicontinuous functions. J. Approx. Theory 113, 245–265.

21