Information Sciences 259 (2014) 225–230

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Remarks on monotone interval-valued set multifunctions

0020-0255/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.ins.2013.08.032

E-mail address: gavrilut@uaic.ro

Alina Cristiana Gavrilut�Faculty of Mathematics, ‘‘Al.I. Cuza’’ University, Carol I Bd. 11, Ias�i 700506, Romania

a r t i c l e i n f o

Article history:Received 7 January 2012Received in revised form 17 June 2013Accepted 17 August 2013Available online 28 August 2013

Keywords:Interval-valued set multifunctionMultisubmeasureSubmeasure

a b s t r a c t

In this paper, motivated by the representation of uncertainty, we discuss the problem ofinterval-valued set multifunctions which are monotone with respect to Guo and Zhangorder relation [7]. We prove that a set multifunction l : C ! PkcðRþÞ is a multisubmeasurein this order relation if and only if for every A 2 C; lðAÞ ¼ ½m1ðAÞ;m2ðAÞ�; m1; m2 : C ! Rþ

being submeasures in the sense of Drewnowski [5]. As application, related results concern-ing variation and continuity properties are established.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

Non-additive measures can be used for modelling problems in non-deterministic environment. In the last years, this areahas been widely developed and a wide variety of topics have been investigated. Also, non-additive integrals have very inter-esting properties from a mathematical point of view, which have been studied and applied to various fields (as informationsciences, decision-making problems, monotone expectation, aggregation approach, etc.).

The idea of modelling the behaviour of phenomena at multiple scales has become an useful tool in pure and appliedmathematics. Fractal-based techniques lie at the heart of this area, since fractals are multiscale objects, which often describesuch phenomena better than traditional mathematical models. In Kunze et al. [13] and Wicks [18], certain hyperspace the-ories concerning the Hausdorff metric and the Vietoris topology, as a foundation for self-similarity and fractality are devel-oped. In fact, for many years, topological methods were used in many fields to study the chaotic nature in dynamicalsystems, which seem to be collective phenomenon emerging out of many segregated components. Most of these systemsare collective (set-valued) dynamics of many units of individual systems. It therefore arised the need of a topological treat-ment of such collective dynamics. Recent studies of dynamical systems, in engineering and physical sciences, have revealedthat the underlying dynamics is set-valued (collective), and not of a normal, individual kind, as it was usually studied before.Interesting approaches of topology in psychology can be found in Lewin et al. [14], Brown [3], etc.

Interval-valued set multifunctions are very important tools since they are related to the representation of uncertainty, anecessity coming from economic uncertainty, fuzzy random variables, interval-probability, martingales of multivalued func-tions, interval-valued capacities (Bykzkan and Duan [4], Jang [9–12], Li and Sheng [15], Weichselberger [17] and manyothers).

In [7], Guo and Zhang defined an interesting order relation on the family P0ðRÞ of all nonvoid subsets of R. Using it, Sofi-an-Boca introduced in [16] a type of a PkcðRþÞ-valued set multifunction, also called a multisubmeasure, but not with respectto the usual inclusion of sets (as we proposed in [6]), but with respect to the order relation of Guo and Zhang [7].

We prove that a set multifunction l : C ! PkcðRþÞ is a multisubmeasure in this order relation if and only if for everyA 2 C; lðAÞ ¼ ½m1ðAÞ;m2ðAÞ�, m1;m2 : C ! Rþ being submeasures in the sense of Drewnowski [5]. We apply this result in

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order to obtain results concerning variation and continuity properties such as exhaustivity, order-continuity and regularityof different types in Hausdorff topology.

2. The problem of multisubmeasures in Guo and Zhang’s order relation [7]

Let T be an abstract nonvoid space, C a ring of subsets of T, X a Banach space, P0ðXÞ the family of all nonvoid subsets ofX; Pf ðXÞ the family of all nonvoid, closed subsets of X, Pbf ðXÞ the family of all nonvoid, closed, bounded subsets of X; PkcðXÞthe family of all nonvoid, compact, convex subsets of X and h the Hausdorff pseudometric on Pf ðXÞ, which becomes a metricon Pbf ðXÞ [8].

h induces Hausdorff topology on Pf ðXÞ and ðPbf ðXÞ;hÞ is a complete metric space.We denote jMj = h(M, {0}), for every M 2 Pf ðXÞ, where 0 is the origin of X.Some properties of this ‘‘norm’’ and the continuity of several operations with subsets with respect to the Hausdorff topol-

ogy are studied in [1,2].On P0ðXÞ we introduce the Minkowski addition þ

�defined by:

Mþ�

N ¼ M þ N; for every M;N 2 P0ðXÞ;

where M + N = {x + y; x 2M,y 2 N} and M þ N is the closure of M + N with respect to the topology induced by the norm of X.Let us first recall the following notions:

Definition 2.1. [6] A set multifunction l : C ! Pf ðXÞ, with l(;) = {0} is said to be:

(i) a multisubmeasure if l is monotone with respect to the inclusion of sets (i.e., l(A) # l(B), for every A; B 2 C, withA # B) and subadditive with respect to the inclusion and the Minkowski addition of sets (i.e., lðA [ BÞ#lðAÞþ

�lðBÞ,

for every A; B 2 C, with A \ B = ; (or, equivalently, for every A; B 2 C));(II) a finitely additive multimeasure if lðA [ BÞ ¼ lðAÞþ

�lðBÞ, for every A; B 2 C, with A \ B = ;.

Definition 2.2. We call the variation of a set multifunction l : C ! Pf ðXÞ, with l(;) = {0}, the set function �l : C ! Rþdefined for every A 2 C by �lðAÞ ¼ sup

Ppi¼1jlðAiÞj; ðAiÞi¼1;p is a partition of A

n o.

The following example illustrates the practical importance of studying interval-valued set multifunctions:

Example 2.3. Suppose ðX;FÞ is a probability space.In Dempster–Shafer mathematical theory of evidence, belief functions Belief (Bel) and Plausibility (Pl) are defined by a

probability distribution m : PðXÞ ! ½0;1�, with m(;) = 0 andP

A # XmðAÞ ¼ 1.For every A # X; BelðAÞ ¼

PB # AmðBÞ and PlðAÞ ¼

PB;B\A–;mðBÞ.

We recall that:

(i) Bel(A) + Pl(cA) = 1; Bel(A) 6 Pl(A).(ii) Bel(X) = 1, Bel(;) = 0, Pl(X) = 1, Pl(;) = 0.

(iii) Belð[ni¼1AiÞP

P;–S # fA1 ;...;Angð�1ÞjSj�1Belð\Ai2SAiÞ (a general version of super-additivity), for every n 2 N� and every {A1-

, . . . , An} �X.for n = 2, Bel(A1 [ A2) P Bel(A1) + Bel(A2) � Bel(A1 \ A2).

(iv) Plð[ni¼1AiÞ 6

P;–S # fA1 ;...;Angð�1ÞjSj�1Plð\Ai2SAiÞ (a general version of subadditivity).

for n = 2, Pl(A1 [ A2) 6 Pl(A1) + Pl(A2) � Pl(A1 \ A2).Belief and Plausibility non-additive measures identify a family of probability distribution for which they are lower andupper probability measures: for every A # X, P(A) 2 [Bel(A),Pl(A)].The Belief Interval of A is the range defined by the minimum and maximum values which could be assigned to A:[Bel(A),Pl(A)]. This interval probability representation contains the precise probability of a set of interest (in the clas-sical sense). The probability is uniquely determined if Bel(A) = Pl(A). In this case, which corresponds to the classicalprobability, all the probabilities, P(A), are uniquely determined for all subsets of X.

In [7] Guo and Zhang introduced an interesting order relation on P0ðRþÞ:

Definition 2.4. If M;N 2 P0ðRþÞ, then M^N if the following two conditions hold:

(i) for every x 2M, there exists yx 2 N so that x 6 yx;(ii) for every y 2 N, there exists xy 2 N so that xy 6 y.

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Using this order relation and inspired by the definition of a Pf ðXÞ-valued multisubmeasure [6], Sofian-Boca proposed in[16] a set multifunction l : C ! PkcðRþÞ also called a multisubmeasure, replacing the inclusion of sets by this order relationand the Minkowski addition by the usual addition (if M; N 2 PkcðRþÞ, then M þ N 2 PkcðRþÞ, so the Minkowski addition isnot needed in this case).

To avoid the confusions, we shall call Sofian-Boca’s multisubmeasure [16] in ‘‘^’’ to be a quasi-multisubmeasure.

As we shall see in the following, this set multifunction has in fact a particular form.

First of all, we remark that l : C ! PkcðRþÞ if and only if there exist two set functions m1; m2 : C ! Rþ so that for everyA 2 C; lðAÞ ¼ ½m1ðAÞ;m2ðAÞ� and m1(A) 6m2(A).

Also, l(;) = {0} if and only if m1(;) = 0 and m2(;) = 0. In what follows, unless stated otherwise, we consider a setmultifunction l : C ! PkcðRþÞ, with l(;) = {0}.

Proposition 2.5.

(i) l is monotone with respect to ‘‘^’’ if and only if both m1, m2 are monotone (in the classical sense);(ii) l is subadditive with respect to ‘‘^’’ if and only if both m1, m2 are subadditive (in the classical sense);

(iii) l is a finitely additive multimeasure if and only if both m1, m2 are finitely additive (in the classical sense).

Proof.

(i) Necessity. For every A; B 2 C, with A � B, we prove that m1(A) 6m1(B) and m2(A) 6m2(B). Indeed, since l(A) ^ l(B),then:

(a) For every x 2 l(A), there exists yx 2 l(B) so that x 6 yx.

Particularly, for x = m2(A), we have m2(A) = x 6 yx 6m2(B) and(b) For every y 2 l(B), there exists xy 2 l(A) so that xy 6 y.

Particularly, for y = m1(B), we have m1(A) 6 xy 6 y = m1(B).

Sufficiency. For every A; B 2 C, with A � B, we prove that l(A)^l(B).Precisely: (a) For every x 2 l(A), there is yx 2 l(B) so that x 6 yx.Indeed, for every x 2 l(A) = [m1(A),m2(A)], there is yx = m2(B) so that x 6m2(A) 6m2(B) = yx.(b) For every y 2 l(B), there is xy 2 l(A) so that xy 6 y.

Indeed, for every y 2 l(B), there is xy = m1(A) 2 l(A) so that xy = m1(A) 6m1(B) 6 y.(ii) Necessity. For every A; B 2 C, we prove that m1(A [ B) 6m1(A) + m1(B) and m2(A [ B) 6m2(A) + m2(B).

Indeed, since l(A [ B)^l(A) + l(B), we have:

� For every x 2 l(A [ B), there exists yx 2 l (A) + l(B) so that x 6 yx, where yx = mx + nx, mx 2 l(A), nx 2 l(B).

Particularly, for x = m2(A [ B) 2 l(A [ B), we have m2(A [ B) = x 6 yx = mx + nx 6m2(A) + m2(B).� For every y 2 l(A) + l(B), there exists xy 2 l (A [ B) so that xy 6 y.

Particularly, for y = m1(A) + m1(B) 2 l(A) + l(B), we have m1(A [ B) 6 xy 6 y = m1(A) + m1(B).Sufficiency. For every A; B 2 C, we prove that l (A [ B)^l(A) + l(B).

(a) For every x 2 l(A [ B) = [m1(A [ B),m2(A [ B)], there exists yx = m2(A) + m2(B) 2 l(A) + l(B) so thatx 6m2(A [ B) 6m2(A) + m2(B) = yx.

(b) For every y 2 l(A) + l(B), there exists xy = m1(A [ B) 2 l(A [ B) so that xy = m1(A [ B) 6m1(A) + m1(B) 6 y.The statement is straightforward using the properties of the addition of intervals. h

(iii)

By Proposition 2.5 (i) and (ii), we have:

Corollary 2.6. l is a quasi-multisubmeasure if and only if m1, m2 are submeasures in the sense of Drewnowski [5].

By Proposition 2.5 (iii) and Corollary 2.6, we get:

Corollary 2.7. If l is a finitely additive multimeasure, then l is a quasi-multisubmeasure.

Obviously, the converse is not generally valid.

One can also easily verify the following result (which, together with Corollary 2.6 illustrates the difference between amultisubmeasure and a quasi-multisubmeasure, both of them PkcðRþÞ-valued).

Proposition 2.8. l : C ! PkcðRþÞ (where for every A 2 C; lðAÞ ¼ ½m1ðAÞ;m2ðAÞ�Þ is a multisubmeasure in the sense of [6] if andonly if �m1 and m2 are submeasures in the sense of Drewnowski [5].

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Remark 2.9. By the definitions and the above statements, one can easily check that l : C ! PkcðRþÞ (where for everyA 2 C; lðAÞ ¼ ½0;m2ðAÞ�Þ is a multisubmeasure in the sense of [6] if and only if l is a quasi-multisubmeasure.

3. Applications to variation and continuity properties

In this section we stablish results concerning variation and continuity properties as exhaustivity, order-continuity anddifferent types of regularity in Hausdorff topology.

Suppose l : C ! PkcðRþÞ is monotone with respect to Guo and Zhang order relation ‘‘^’’ [7], with l(A) = [m1(A),m2(A)],for every A 2 C, where m1; m2 : C ! Rþ; m1ðAÞ 6 m2ðAÞ, for every A 2 C and m1(;) = m2(;) = 0. Then both m1 and m2 aremonotone in the classical sense.

Proposition 3.1. jl(A)j = m2(A) and �lðAÞ ¼ m2ðAÞ, for every A 2 C.

Proof.

(i) For every A 2 C, we have: eðlðAÞ; f0gÞ ¼ supx2½m1ðAÞ;m2ðAÞ�x ¼ m2ðAÞ and eðf0g;lðAÞÞ ¼ dð0;lðAÞÞ ¼ infy2½m1ðAÞ;m2ðAÞ�y ¼ m1ðAÞ,whence jl(A)j = max{m1(A),m2(A)} = m2(A).

(ii) By (i), �lðAÞ ¼ supPp

i¼1jlðAiÞj; ðAiÞi¼1;p is a partition of An o

¼ supPp

i¼1m2ðAiÞ; ðAiÞi¼1;p is a partition of An o

¼ m2ðAÞ,for every A 2 C. h

We now recall from [6] the following continuity properties:

Definition 3.2. lis said to be (with respect to h):

(i) order continuous if limn?1jl (An)j = 0, for every sequence of sets ðAnÞn2N � C, with An&;;(ii) exhaustive if limn?1jl (An)j = 0, for every sequence of pairwise disjoint sets ðAnÞn2N � C.

By the definitions and Proposition 3.1. (i), we get:

Remark 3.3. l : C ! PkcðRþÞ is exhaustive (order-continuous, respectively) if and only if the same is m2, so m1 does nothave any importance.

In the sequel, let T be, particularly, a locally compact, Hausdorff space, B0 (respectively, B00Þ the Baire d-ring (respectively,r-ring) generated by compact sets, which are Gd (that is, countable intersections of open sets) and B (respectively, B0) theBorel d-ring (respectively, r-ring) generated by the compact sets of T.

Note that B0 � B;B0 � B00 and B � B0.By K we denote the family of compacts and by D, the family of open sets of T.It is well known that regularity is an important property of continuity, which connects measure theory and topology,

approximating general Borel sets by more tractable sets, such as compact and/or open sets. In what follows we recall severalimportant types of regularity, first for real valued set functions, and then for set multifunctions in Hausdorff topology:

Definition 3.4. If m : C ! Rþ is a set function, with m(;) = 0, then m is said to be:

(i) Inner regular if mðAÞ ¼ supK2K\C;K�AmðKÞ, for every A 2 C.(ii) Outer regular if mðAÞ ¼ infD2D\C;A�DmðDÞ, for every A 2 C.

(iii) Regular if it is inner regular and outer regular.(iv) R0l-regularif for every A 2 C and every e > 0, there is K 2 K \ C; K � A so that m(AnK) < e.(v) R0r-regular if for every A 2 C and every e > 0, there is D 2 D \ C; A � D so that m(DnA) < e.

(vi) R0-regular if for every A 2 C and every e > 0, there are K 2 K \ C; K � A and D 2 D \ C; A � D so that m(DnK) < e.

Definition 3.5.

(I) A set A 2 C is said to be:

(i) R-regular if for every e > 0, there are K 2 K \ C; K � A and D 2 D \ C; D � A so that h(l(A),l(B)) < e, for every B 2 C,

with K � B � D.(ii) Rl-regular if for every e > 0, there exists K 2 K \ C; K � A so that h(l (A), l(B)) < e, for every B 2 C, with K � B � A.(iii) Rr-regular if for every e > 0, there exists D 2 D \ C; D � A such that h(l(A),l(B)) < e, for every B 2 C, with A � B � D.

A.C. Gavrilut� / Information Sciences 259 (2014) 225–230 229

(iv) R0-regular if for every e > 0, there are K 2 K \ C; K � A and D 2 D \ C; A � D so that jl(B)j < e, for every B 2 C, withB � DnK.

(v) R0l-regular if for every e > 0, there is K 2 K \ C; K � A so that jl(B)j < e, for every B 2 C, with B � AnK.(vi) R0r-regular if for every e > 0, there is D 2 D \ C; A � D such that jl(B)j < e, for every B 2 C, with B � DnA.

(II) l is said to be:

(i) R-regular (respectively, Rl-regular, Rr-regular) if every set A 2 C is R-regular (respectively, Rl-regular, Rr-regular).(ii) R0-regular (respectively, R0l-regular, R0r-regular) if every set A 2 C is R0-regular (respectively, R0l-regular, R0r-regular).

Note these above definitions are consistent if, for instance, C is the ring (d-ring, r-ring, respectively) generated bythe compact/compact, Gd subsets of T.

Proposition 3.6.

(i) l is R0l-regular (R0r-regular, R0-regular, respectively) if and only if the same is m2;(ii) l is Rl-regular (Rr-regular, R-regular, respectively) if and only if both m1 and m2 are inner regular (outer regular, regular,

respectively).

Proof.

(i) The statements easily follow by the definitions and Proposition 3.1 (i). Indeed, l is R0l-regular if and only if for everyA 2 C and every e > 0, there is K 2 K \ C; K � A so that jl(B)j < e, for every B 2 C, with B � AnK.According to Proposition 3.1 (i) and the monotonicity of m2, this is equivalent to the existence of a set K 2 K \ C, sothat K � A and m2(AnK) < e, i.e., m2 is R0l-regular.The other statements follow analogously.

(ii) We shall use the equality

ð�Þhð½a; b�; ½c; d�Þ ¼maxfja� cj; jb� djg;

where a; b; c; d 2 Rþ; a 6 b and c 6 d.For instance, l is Rl-regular if and only if for every A 2 C and every e > 0, there is K 2 K \ C; K � A so that h(l(A),l(B)) < e, forevery B 2 C, with K � B � A.According to the above mentioned equality, this is equivalent to the existence of a set K 2 K \ C; K � A so that jm1(A) �m1(B)j < e and jm2(A) �m2(B)j < e, for every B 2 C, with K � B � A.Because both m1 and m2 are monotone, this is equivalent to m1(A) < m1(K) + e and m1(A) < m1(K) + e, whencem1ðAÞ ¼ supK2K\C;K�Am1ðKÞ and m2ðAÞ ¼ supK2K\C;K�Am2ðKÞ, i.e., m1 and m2 are inner regular.Same argues for Rr-regularity – outer regularity.Alos, by the definition, l is R-regular if and only if for every A 2 C and every e > 0, there are K 2 K \ C; K � A andD 2 D \ C; D � A so that h(l(A),l(B)) < e, for every B 2 C, with K � B � D.By (�), this is equivalent to the existence of K 2 K \ C; K � A and D 2 D \ C; D � A so that (��)jm1(A) �m1(B)j < e andjm2(A) �m2(B)j < e, for every B 2 C, with K � B � D.Because both m1 and m2 are monotone, one can easily see that (⁄⁄) is equivalent to the regularity of m1 and m2. h

4. Concluding remarks

This paper is dedicated to the study of the properties of interval-valued set multifunctions which are monotone in Guoand Zhang [7] order relation. Related results concerning variation and continuity properties in Hausdorff topology are alsoestablished.

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