Managing Editors

Shigeo Kusuoka

Toru Maruyama

The University of Tokyo

Tokyo, JAPAN

Keio University

Tokyo, JAPAN

Editors

Jean-Michel Grandmont

Robert Anderson

CREST-CNRS

University of California,

Malakoff, FRANCE

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Berkeley, U.S.A.

Kunio Kawamata

Keio University

Tokyo, JAPAN

Hiroshi Matano

Norimichi Hirano

Charles Castaing

The University of Tokyo

Yokohama National

Université Montpellier II

Tokyo, JAPAN

University

Montpellier, FRANCE

Yokohama, JAPAN

Kazuo Nishimura

Francis H. Clarke

Kyoto University

Université de Lyon I

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The Ohio State University

Ohio, U.S.A.

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Yoichiro Takahashi

University of Vienna

The University of Tokyo

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Israel

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Technology

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Kyoto, JAPAN

Berkeley, U.S.A.

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Fukuoka University

Fukuoka, JAPAN

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The University of Tokyo

Tokyo, JAPAN

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Shigeo Kusuoka • Toru Maruyama

Editors

Advances in

Mathematical Economics

Volume 19

123

Editors

Shigeo Kusuoka

Professor Emeritus

The University of Tokyo

Tokyo, Japan

Toru Maruyama

Professor Emeritus

Keio University

Tokyo, Japan

ISSN 1866-2226

ISSN 1866-2234 (electronic)

Advances in Mathematical Economics

ISBN 978-4-431-55488-2

ISBN 978-4-431-55489-9 (eBook)

DOI 10.1007/978-4-431-55489-9

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Contents

On the Integration of Fuzzy Level Sets . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

Charles Castaing, Christiane Godet-Thobie, Thi Duyen Hoang,

and P. Raynaud de Fitte

1

A Theory for Estimating Consumer’s Preference from Demand . . . . . . . . . . .

Yuhki Hosoya

33

Least Square Regression Methods for Bermudan Derivatives

and Systems of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

Shigeo Kusuoka and Yusuke Morimoto

Discrete Time Optimal Control Problems on Large Intervals . . . . . . . . . . . . . .

Alexander J. Zaslavski

57

91

Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 137

v

Adv. Math. Econ. 19, 1–32 (2015)

On the Integration of Fuzzy Level Sets

Charles Castaing, Christiane Godet-Thobie, Thi Duyen Hoang,

and P. Raynaud de Fitte

Abstract We study the integration of fuzzy level sets associated with a fuzzy

random variable when the underlying space is a separable Banach space or a

weak star dual of a separable Banach space. In particular, the expectation and the

conditional expectation of fuzzy level sets in this setting are presented. We prove

the SLLN for pairwise independent identically distributed fuzzy convex compact

valued level sets through the SLLN for pairwise independent identically distributed

convex compact valued random set in separable Banach space. Some convergence

results for a class of integrand martingale are also presented.

JEL Classification: C01, C02.

Mathematics Subject Classification (2010): 28B20, 60G42, 46A17, 54A20.

C. Castaing

Département de Mathématiques, Case courrier 051, Université Montpellier II, 34095 Montpellier

Cedex 5, France

e-mail: charles.castaing@gmail.com

C. Godet-Thobie

Laboratoire de Mathématiques de Bretagne Atlantique, Université de Brest, UMR CNRS 6295, 6,

avenue Victor Le Gorgeu, CS 9387, F-29238 Brest Cedex3, France

e-mail: christiane.godet-thobie@univ-brest.fr

T.D. Hoang

Quang Binh University, Quang Binh, Viet Nam

e-mail: hoangduyen267@gmail.com

P. Raynaud de Fitte ( )

Laboratoire Raphaël Salem, UFR Sciences, Université de Rouen, UMR CNRS 6085, Avenue de

l’Université, BP 12, 76801 Saint Etienne du Rouvray, France

e-mail: prf@univ-rouen.fr

1

2

C. Castaing et al.

Keywords Conditional expectation • Fuzzy convex • Fuzzy martingale •

Integrand martingale • Level set • Upper semicontinuous

Article type: Research Article

Received: November 5, 2014

Revised: December 1, 2014

1 Introduction

The study of fuzzy set-valued variables was initiated by Feron [10], Kruse [18],

Kwakernaak [19, 20], Puri and Ralescu [25], Zadeh [30]. In particular, Puri

and Ralescu [25] introduced the notion of fuzzy set valued random variables

whose underlying space is the d -dimensional Euclidean space Rd . Concerning the

convergence theory of fuzzy set-valued random variables and its applications we

refer to [15, 21–23, 25–27].

In this paper we present a study of a class of random fuzzy variables whose

underlying space is a separable Banach space E or a weak star dual Es of a

separable Banach space.

The paper is organized as follows. In Sect. 2 we summarize and state the needed

measurable results in the weak star dual of a separable Banach space. In particular,

we present the expectation and the conditional expectation of convex weak star

compact valued Gelfand-integrable mappings. In Sect. 3 we present the properties

of random fuzzy convex upper semi continuous integrands (variables) in Es . In

Sect. 4, the fuzzy expectation and the fuzzy conditional expectation for random

fuzzy convex upper semi continuous variables are provided in this setting. Section 5

is devoted to the SLLN for fuzzy convex compact (compact) valued random level

sets through the SLLN for convex compact (compact) valued random sets. The

above results lead to a new class of integrand martingales that we develop in Sect. 6.

Some convergence results for integrand martingales are provided.

Our paper provides several issues in Fuzzy set theory, but captures different tools

from Probability and Set-Valued Analysis and shows the relations among them with

a comprehensive concept.

2 Integration of Convex Weak Star Compact Sets

in a Dual Space

Throughout this paper, . ; F ; P / is a complete probability space, E is a Banach

space which we generally assume to be separable, unless otherwise stated, D1 D

.ek /k2N is a dense sequence in the closed unit ball of E, E is the topological

On the Integration of Fuzzy Level Sets

3

dual of E, and B E (resp. B E ) is the closed unit ball of E (resp. E ). We denote

by cc.E/ (resp. cwk.E/) (resp. ck.E/) the set of nonempty closed convex (resp.

weakly compact convex) (resp. compact convex) subsets of E. Given C 2 cc.E/,

the support function associated with C is defined by

ı .x ; C / D supf< x ; y >; y 2 C g .x 2 E /:

We denote by dH the Hausdorff distance on cwk.E/. A cc.E/-valued mapping

C W ! cc.E/ is F -measurable if its graph belongs to F ˝ B.E/, where B.E/

is the Borel tribe of E. For any C 2 cc.E/, we set

jC j D supfkxk W x 2 C g:

1

We denote by Lcwk.E/

.F / the space of all F -measurable cwk.E/-valued

multifunctions X W ! cwk.E/ such that ! ! jX.!/j is integrable. A sequence

1

.Xn /n2N in Lcwk.E/

.F / is bounded if the sequence .jXn j/n2N is bounded in L1R .F /.

A F -measurable closed convex valued multifunction X W ! cc.E/ is integrable

if it admits an integrable selection, equivalently if d.0; X / is integrable.

We denote by Es , (resp. Eb ), (resp. Ec ) the vector space E endowed with

the topology .E ; E/ of pointwise convergence, alias w -topology (resp. the

topology s associated with the dual norm jj:jjEb ), (resp. the topology c of compact

convergence) and by Em the vector space E endowed with the topology m D

.E ; H /, where H is the linear space of E generated by D1 , that is the Hausdorff

locally convex topology defined by the sequence of semi-norms

Pn .x / D maxfjhek ; x ij W k Ä ng;

x 2 E ; n 2 N:

Recall that the topology m is metrizable, for instance, by the metric

dE .x ; y / WD

m

1

X

1

.jhek ; x i

2k

hek ; y ij ^ 1/; x ; y 2 E :

kD1

w

We assume from now on that dE is held fixed. Further, we have m

m

s : On the other hand, the restrictions of m , w , c to any bounded subset

c

of E coincide and the Borel tribes B.Es /, B.Em / and B.Ec / associated with

Es , Em , Ec , are equal, but the consideration of the Borel tribe B.Eb / associated

with the topology of Eb is irrelevant here. Noting that E is the countable union of

closed balls, we deduce that the space Es is a Lusin space, as well as the metrizable

topological space Em . Let K D cwk.Es / be the set of all nonempty convex

weak compact subsets in E . A K -valued multifunction (alias mapping for

short) X W

Ã Es is scalarly F -measurable if, 8x 2 E, the support function

ı .x; X.:// is F -measurable, hence its graph belongs to F ˝ B.Es /. Indeed,

let .fk /k2N be a sequence in E which separates the points of E , then we have

4

C. Castaing et al.

x 2 X.!/ iff hfk ; x i Ä ı .fk ; X.!// for all k 2 N. Consequently, for any Borel

set G 2 B.Es /, the set

X G D f! 2

W X.!/ \ G ¤ ;g

is F -measurable, that is, X G 2 F , this is a consequence of the Projection

Theorem (see e.g. [8, Theorem III.23]) and of the equality

X G D proj fGr.X / \ .

G/g:

In particular if u W

! Es is a scalarly F -measurable mapping, that is, if for

every x 2 E, the scalar function ! 7! hx; u.!/i is F -measurable, then the function

f W .!; x / 7! jjx

u.!/jjEb is F ˝ B.Es /-measurable, and for every fixed

! 2 ; f .!; :/ is lower semicontinuous on Es , i.e. f is a normal integrand. Indeed,

we have

jjx

u.!/jjEb D sup jhek ; x

u.!/ij:

k2N

As each function .!; x / 7! hek ; x

u.!/i is F ˝ B.Es /-measurable and

continuous on Es for each ! 2

, it follows that f is a normal integrand.

Consequently, the graph of u belongs to F ˝ B.Es /. Let B be a sub- -algebra

of F . It is easy and classical to see that a mapping u W

! Es is .B; B.Es //

measurable iff it is scalarly B-measurable. A mapping u W

! Es is said to be

scalarly integrable (alias Gelfand integrable), if, for every x 2 E, the scalar function

! 7! hx; u.!/i is F -measurable and integrable. We denote by GE1 ŒE.F / the

space of all Gelfand integrable mappings and by L1E ŒE.F / the subspace of all

Gelfand integrable mappings u such that the function juj W ! 7! jju.!/jjEb is

integrable. The measurability of juj follows easily from the above considerations.

1

1

More generally, by Gcwk.E

. ; F ; P / (or Gcwk.E

.F / for short) we denote the

s /

s /

space of all scalarly F -measurable and integrable cwk.Es /-valued mappings and

1

1

by Lcwk.E

. ; F ; P / (or Lcwk.E

.F / for short) we denote the subspace of

s /

s /

all cwk.Es /-valued scalarly integrable and integrably bounded mappings X , that

is, such that the function jX j W ! ! jX.!/j is integrable, here jX.!/j WD

supy 2X.!/ jjy jjEb , by the above consideration, it is easy to see that jX j is F measurable.

1

For any X 2 Lcwk.E

.F /, we denote by SX1 .F / the set of all Gelfands /

integrable selections of X . The Aumann-Gelfand integral of X over a set A 2 F is

defined by

Z

Z

EŒ1A X D

X dP WD f

A

A

f dP W f 2 SX1 .F /g:

On the Integration of Fuzzy Level Sets

5

Let B be a sub- -algebra of F and let X be a K -valued integrably bounded

random set, let us define

SX1 .B/ WD ff 2 L1E ŒE. ; B; P / W f .!/ 2 X.!/ a.s.g

and the multivalued Aumann-Gelfand integral (shortly expectation) EŒX; B of X

Z

EŒX; B WD f

f dP W f 2 SX1 .B/g:

As SX1 .B/ is .L1E ŒE.B/; L1

E .B//-compact [7, Corollary 6.5.10], the expectation EŒX; B is convex .E ; E/-compact. We summarize some needed results

on measurability for convex weak -compact valued Gelfand-integrable mappings in

the dual space. A K -valued mapping X W Ã E is a K -valued random set if

X.!/ 2 K for all ! 2 and if X is scalarly F -measurable. We will show that

K -valued random sets enjoy good measurability properties.

Proposition 1. Let X W

! cwk.Es / be a convex weak -compact valued

mapping. The following are equivalent:

(a)

(b)

(c)

(d)

X V 2 F for all m -open subset V of E .

Graph.X / 2 F ˝ B.Es / D F ˝ B.Em /.

X admits a countable dense set of .F ; B.Es //-measurable selections.

X is scalarly F -measurable.

Proof. .a/ ) .b/. Recall that any K 2 K is m -compact and m

w and

the Borel tribes B.Es / and B.Em / are equal. Recall also that Em is a Lusin

metrizable space. By (a), X is an m -compact valued measurable mapping from

into the Lusin metrizable space Em . Hence Graph.X / 2 F ˝ B.Em / because

Graph.X / D f.!; x / 2

Em W dE .x ; X.!// D 0g

m

and the mapping .!; x / 7! dE .x ; X.!// is F ˝ B.Em /-measurable.

m

.b/ ) .a/ is obtained by applying the measurable Projection Theorem (see

e.g. [8, Theorem III.23]) and the equality

X V D proj fGraph.X / \ .

V /g:

Hence (a) and (b) are equivalent.

.b/ ) .c/. Since Es is a Lusin space, by [8, Theorem III-22], X admits a

countable dense set of .F ; B.Es //-measurable selections .fn /, that is, X.!/ D

w clffn .!/g for all ! 2 .

.c/ ) .d /. Indeed one has ı .x; X.!// D supn hx; fn .!/i for all x 2 E and for

all ! 2 , thus proving the required implication.

.d / ) .b/. We have already seen that .d / implies that Graph.X / 2 F ˝B.Es /.

As B.Es / D B.Em /, the proof is finished.

6

C. Castaing et al.

Corollary 1. Let X W ! cwk.Es / be a convex weak -compact valued mapping.

The following are equivalent:

(a)

(b)

(c)

(d)

X V 2 F for all w -open subset V of E .

Graph.X / 2 F ˝ B.Es /.

X admits a countable dense set of .F ; B.Es //-measurable selections.

X is scalarly F -measurable.

Proof. .a/ ) .d / is easy. The implications .d / ) .b/, .b/ ) .c/, .c/ )

.d /, .b/ ) .a/ are already known. For further details on these facts, consult

Proposition 5.2 and Corollary 5.3 in [5].

Let .Xn /n2N be a sequence of w -closed convex sets, the sequential weak upper

limit w -ls Xn of .Xn /n2N is defined by

w -ls Xn D fx 2 E W x D .E ; E/- lim xj I xj 2 Xnj g:

j !1

Similarly the sequential weak lower limit w -li Xn of .Xn /n2N is defined by

w -li Xn D fx 2 E W x D .E ; E/- lim xn I xn 2 Xn g:

n!1

The sequence .Xn /n2N weak star (w K for short) converges to a w -closed convex

set X1 if the following holds

w -ls Xn

X1

w -li Xn

a.s.

Briefly

w K- lim Xn D X1

n!1

a.s.

We need the following definition.

Definition 1. The Banach space E is weakly compactly generated (WCG) if there

exists a weakly compact subset of E whose linear span is dense in E.

Every separable Banach space is WCG, and every dual of a separable Banach space

(endowed with the dual norm) is WCG.

For the sake of completeness we recall the following [11]

Theorem 1. Suppose E is WCG (not necessarily separable) and let C and Cn .n D

1; 2; : : :/ be weak -closed, bounded, convex non empty sets of E .

Then ı .:; Cn / ! ı .:; C / pointwise on E if and only if the sequence .Cn / is

uniformly bounded with w K limit C .

Now we provide some applications.

On the Integration of Fuzzy Level Sets

7

1

Theorem 2. Let X; Xn.n 2 N/ be a sequence in Lcwk.E

.F / with the following

s /

property: jX j C jXn j Ä g for all n 2 N where g is positive integrable. Then the

following hold:

R

R

(a)

XdP; Xn dP , .n 2 N/, are convex weak -compact,

(b) If Xn w K converges to X , equivalently, ı .:; Xn / ! ı .:; X / pointwise on E,

then

Z

Z

w K- lim

Xn dP D

XdP:

n!1

1

1

Proof. (a) As jX j Ä g and jXn j Ä g for all n 2 N, SGe

.X /.F / and SGe

.Xn /.F /

1

1

are convex sequentially .LE ; LE /-compact [7, Corollary 6.5.10], so that

s

R

R

XdP , Xn dP , .n D 1; 2; : : :/, are convex weak -compact.

(b) Further by the Strassen theorem [8], we have that

Z

ı .x;

Z

XdP / D

Z

ı .x;

8x 2 E:

ı .x; X /dP;

(1)

Z

Xn dP / D

ı .x; Xn /dP;

8x 2 E:

(2)

Applying Lebesgue’s theorem and (1)–(2) gives

Z

n!1

Z

Xn dP / D lim

lim ı .x;

n!1

Z

D

ı .x; Xn /dP

ı .x; X /dP D ı .x;

Invoking Theorem 1, we conclude that

R

Z

XdP /:

Xn dP w K converges to

R

XdP .

The following concerns the continuous dependence of the Aumann-Gelfand

1

multivalued integral of a cwk.Es /-valued mapping X˛ 2 Lcwk.E

.F / depending

s /

on a parameter ˛ 20; 1. It has some importance in applications.

Theorem 3. Let X W

0; 1 Ã Es be a convex weakly compact valued mapping

satisfying the properties:

(i) jX.!; ˛/j Ä g.!/; 8!; 2 ; 8˛ 20; 1, where g is a positive integrable

function,

1

(ii) For every ˛ 20; 1, X.:; ˛/ 2 Lcwk.E

.F /,

s /

(iii) For every ! 2 , ˛ 7! X.:; ˛/ from 0; 1 into cwk.Es / is scalarly left

continuous.

R

Then the mapping ˛ 7!

X.:; ˛/dP from 0; 1 into cwk.Es / is scalarly

left continuous.

8

C. Castaing et al.

Proof. Follows the lines of the proof of Theorem 2. By (i) jX˛ j Ä g for all ˛ 20; 1

with X˛ WD X.:; ˛/. Let ˛n ! ˛. Then X.!; ˛n / scalarly converges to X.!; ˛/,

that is, ı .x; X.!;

R ˛n // ! ı .x; X.!; ˛// for every x 2 E and for every ! 2 .

Remember that X˛ dP is convex weakly compact for every ˛ 20; 1. Further by

Strassen’s theorem, we have

Z

Z

ı .x; X˛ /dP; 8x 2 E:

(3)

ı .x; X˛ dP / D

Applying Lebesgue’s theorem and (3) gives

Z

n!1

Z

D

Z

X˛n dP / D lim

lim ı .x;

n!1

ı .x; X˛n /dP

Z

ı .x; X˛ /dP D ı .x;

X˛ dP /:

3 Random Fuzzy Convex Upper Semicontinuous Integrands

Thanks to measurable properties developed in Sect. 2 we present now some applications to a special class of random upper semicontinuous integrands (variables).

We recall some definitions that are borrowed from the study of normal integrands

(alias random lower semi-continuous integrands) on a general locally convex

Suslin space E. A random lower semicontinuous (resp. upper semicontinuous)

integrand is a F ˝ B.E/-measurable function X defined on

E such that

X.!; :/ is lower semicontinuous (resp. upper semicontinuous). The study of random

lower semicontinuous integrands occurs in some problems in Convex Analysis and

Variational convergence. See e.g. [28] and the references therein. In the following

we will focus on a special class of random upper semicontinuous integrands. Here

the terminologies are borrowed from the theory of fuzzy sets initiated by Zadeh [30]

and random fuzzy sets initiated by Feron [10] and Puri-Ralescu [25]. According to

[30] a fuzzy convex upper semicontinuous variable is a mapping X W E ! Œ0; 1

such that

(i) X is upper semicontinuous,

(ii) fx 2 E W X.x/ D 1g 6D ;

(iii) X is fuzzy convex, that is, X. x C .1

2 Œ0; 1 and for all x; y 2 E.

/y/

min.X.x/; X.y//, for all

A random fuzzy convex upper semicontinuous variable is an F ˝B.E/-measurable

mapping X W

E ! Œ0; 1 such that each ! 2 , a mapping X! W E ! Œ0; 1 is

On the Integration of Fuzzy Level Sets

9

a fuzzy convex upper semicontinuous variable. By upper semicontinuity and fuzzy

convexity, for each ! 2 and for each ˛ 20; 1, the level set

X˛ .!/ WD L˛ .X /.!/ WD fx 2 E W X! .x/

˛g

is closed convex. It is clear that the graph of this multifunction belongs to

F ˝ B.E/. In particular, this mapping is F -measurable by the measurable

projection theorem [8, Theorem III. 23]. Further, thanks to [8, Lemma III.39],

for any F ˝ B.E/-measurable mapping ' W

E ! Œ0; C1, the function

m.!/ WD supf'.!; x/ W x 2 L˛ .X /.!/g is F -measurable. Similarly the graph of

the multifunction fx 2 E W X.!; x/ > 0g WD ŒX! > 0 belongs to F ˝ B.E/.

In particular, this mapping is F -measurable by the measurable projection theorem

[8, Theorem III. 23]. Further, thanks to [8, Lemma III.39], for any F ˝ B.E/measurable mapping ' W

E ! Œ0; C1, the function ! 7! supf'.!; x/ W

x 2 ŒX! > 0g is F -measurable. In particular, if the underlying space E is a

separable Banach space, then ŒX! > 0 D fx 2 E W X.!; x/ > 0g is F measurable and so is the mapping ŒX! > 0 so that the mapping ! 7! supfjjxjj W

x 2 ŒX! > 0g is F -measurable, further assume that ŒX! > 0 is compact and

! 7! g.!/ WD supfjjxjj W x 2 ŒX! > 0g integrable, then L˛ .X / is convex compact

valued and integrably bounded: jL˛ .X /j Ä g for all ˛ 20; 1, here measurability

of g is ensured because of the above measurable properties. Similarly, for each

˛ 2 Œ0; 1Œ, L˛C .X /.!/ WD ŒX! > ˛ is compact valued and F -measurable. The

above considerations still hold when the underlying space is the weak star dual Es

of a separable Banach space E because Es is a Lusin space, by measurability results

developed in Sect. 2. Now we present some convergence properties of the level sets

associated with a random fuzzy convex upper semicontinuous integrand.

Proposition 2. Let X W

E ! Œ0; 1 random fuzzy convex upper semicontinuous

integrand with the following properties:

(1) fx 2 E W X.!; x/ > 0g is compact, for each ! 2 ,

(2) g WD jL0C .X /j 2 L1 ,

(3) The ck.E/-valued mapping ˛ 7! L˛ .X / is scalarly left continuous on 0; 1.

Then the following hold

R

R

L˛ .X /dP D f f dP W f 2 SL1 ˛ .X / g .˛ 20; 1/ is convex compact.

(a)

R

(b) The ck.E/-valued mapping ˛ 7!

L˛ .X /dP is scalarly left continuous on

0; 1.

Proof. With the properties (1)–(2), we see that the level sets L˛ .X / (˛ 20; 1)

belong to the space Lck.E/ . ; F / of all convex compact valued integrably bounded

multifunctions so that the Aumann integral of L˛ .X /

Z

Z

L˛ .X /dP D f

f dP W f 2 SL1 ˛ .X / g

10

C. Castaing et al.

is convex compact,1 where SL1 ˛ .X / denotes the set of all integrable selections of the

convex compact valued multifunction L˛ .X /. We only sketch the proof.

See [2, 8]

R

for details. Indeed, SL1 ˛ .X / is convex weakly compact in L1E so that L˛ .X /dP is

convex weakly compact in E. Making use of Strassen’s formula we have

Z

ı .x ;

Z

L˛ .X /dP / D

ı .x ; L˛ .X //dP;

8x 2 E :

Applying Lebesgue’s dominated convergence theorem shows that

Z

x 7! ı .x ;

L˛ .X /dP /

is continuous on the closed unit ball equipped with the topology of compact

convergence = weak star topology that is compact metrizable with respect to these

topologies. From the Banach-Dieudonné theorem we conclude that this mapping is

continuous on Ec . Thus .a/ is proved. Taking account of (3), .b/ follows easily.

The following is a dual variant of the preceding result.

Proposition 3. Let X W

Es ! Œ0; 1 be a random fuzzy convex upper

semicontinuous variable with the following properties:

(1) fx 2 Es W X.!; x / > 0g is weak -compact, for each ! 2 ,

1

(2) g WD jLC

0 .X /j 2 L ,

(3) The cwk.Es /-valued mapping ˛ 7! L˛ .X / is scalarly left continuous on 0; 1.

Then the following hold

R

R

L˛ .X /dP D f f dP W f 2 SL1 ˛ .X / g .˛ 20; 1/ is convex weak -compact.

(a)

R

(b) The cwk.Es /-valued mapping ˛ 7!

L˛ .X /dP is scalarly left continuous

on 0; 1.

Proof. With the properties (1)–(3), we see that the level sets L˛ .X / (˛ 20; 1)

belong to the space Lcwk.E / .F / of all convex weak -compact valued integrably

bounded multifunctions so that the Aumann integral of L˛ .X /

Z

Z

L˛ .X /dP D f

f dP W f 2 SL1 ˛ .X / g

is convex weak -compact because the set SL1 ˛ .X / is convex sequentially

.L1E ; L1

E /-compact [7, Corollary 6.5.10]. Thus .a/ is proved, .b/ follows from

s

Theorem 3.

R

The compactness of

L˛ .X/dP according to Debreu integral is not available here, see also the

remarks of Theorem 8 in Hiai-Umegaki [13].

1

On the Integration of Fuzzy Level Sets

11

4 Expectation and Conditional Expectation of Level Sets

Now we proceed to the study of the expectation and conditional expectation of the

level sets associated with random fuzzy convex upper semicontinuous integrands.

The following lemma is crucial for this purpose. Compare with related results by

Puri-Ralescu [25] dealing with fuzzy sets on Rd .

Lemma 1. Let X be a random fuzzy convex upper semicontinuous integrand X W

Es ! Œ0; 1 with the following properties:

(1) fx 2 Es W X.!; x / > 0g is weak -compact, for each ! 2

(2) g WD jL0C .X /j 2 L1 ,

(3) Assume that 0 < ˛1 < ˛2 < : : : < ˛k ! ˛ and that

,

lim ı .x; L˛k .X /.!// D ı .x; L˛ .X /.!//

k!1

for all ! 2

and for all x 2 E.

Then we have

\Z

Z

L˛k .X /dP D

L˛ .X /dP:

k 1

Proof. Using (3) and applying Strassen’s theorem [8] and Lebesgue’s dominated

convergence theorem gives

Z

k!1

Z

X˛k .!/dP / D lim

lim ı .x;

k!1

Z

D

ı .x; X˛k .!//dP

ı .x; X˛ .!//dP D ı .x;

(4)

Z

X˛ .!/dP /:

Since the w -topology coincides with the metrizable topology m , by Theorem 15

in [5] we have

Z

Z

\Z

X˛n dP D m -LS

X˛n dP D

X˛k dP

C WD w - ls

k 1

so that C D w K limk!1

R

X˛k .!/dP . Applying Theorem 1 and (4), we have

Z

ı .x; C / D lim ı .x;

k!1

Z

X˛k .!/dP / D ı .x;

X˛ .!/dP /

12

for all x 2 E, so that

C. Castaing et al.

R

X˛ dP D C by the separability of E, that is,

\Z

Z

X˛k dP D

X˛ dP:

k 1

The following theorem yields a crucial property of the expectation of the level

sets associated with a random fuzzy convex upper semicontinuous integrand.

Theorem 4. Let X be a random fuzzy convex upper semicontinuous integrand X W

Es ! Œ0; 1 with the following properties:

(1) fx 2 Es W X.!; x / > 0g is weak -compact, for each ! 2 ,

(2) g WD jL0C .X /j 2 L1 ,

(3) For every fixed ! 2 , the cwk.Es /-valued mapping ˛ 7! L˛ .X /.!/ D

X˛ .!/ is scalarly left continuous on 0; 1.

Then

R

R the following

T hold

X˛ dP D k 1 X˛k dP whenever 0 < ˛1 < ˛2 < : : : : < ˛k ! ˛.

Proof. Follows from Lemma 1 using the continuity property of the level sets.

Now we proceed to the conditional expectation of the level sets associated with

a random fuzzy convex upper semicontinuous integrand defined on the dual space

Es . For this purpose we need to recall and summarize the existence and uniqueness

1

of the conditional expectation in Lcwk.E

.F / [4, 13, 29]. In particular, existence

s /

results for conditional expectation in Gelfand integration can be derived from the

multivalued Dunford-Pettis representation theorem, see [4]. A fairly general version

of conditional expectation for closed convex integrable random sets in the dual of

a separable Fréchet space is obtained by Valadier [29, Theorem 3]. Here we need

only a special version of this result in the dual space Es .

Theorem 5. Let be a closed convex valued integrable random set in Es . Let B

be a sub- -algebra of F . Then there exists a closed convex B-measurable mapping

† in Es such that:

(1) † is the smallest closed convex B-measurable mapping ‚ such that 8u 2 S1 ,

E B u.!/ 2 ‚.!/ a.s.

(2) † is the unique closed convex B-measurable mapping ‚ such that 8v 2

L1

R .B/,

Z

Z

ı .v; /dP D

ı .v; ‚/dP:

(3) † is the unique closed convex B-measurable mapping such that S†1 D

cl .E B .S1 // where cl denotes the closure with respect to .L1E .B/; L1

E .B//.

On the Integration of Fuzzy Level Sets

13

Theorem 5 allows to obtain the weak compactness of the conditional expectation

of convex weakly compact valued integrably bounded mappings in E with strong

separable dual. Indeed if F WD Eb is separable and if is a convex weakly

compact valued measurable mapping in E with .!/

˛.!/B E where ˛ 2 L1R ,

B

then applying Theorem 4 to F gives †.!/ D E .!/

E with †.!/

E B ˛.!/B E where B E is the closed unit ball in E . As S1 is .L1E ; L1

E /compact, S†1 D E B .S1 / L1E . Whence †.!/ E a.s. See [29, Remark 4, page

10] for details.

The following existence theorem of conditional expectation for convex weak compact valued Gelfand-integrable mappings follows from a multivalued version

of the Dunford-Pettis theorem in the dual space [4, Theorem 7.3]. In particular, it

provides the weak -compactness of conditional expectation for integrably bounded

weak -compact valued scalarly measurable mappings with some specific properties.

1

.F / and a sub- -algebra B of F , there

Theorem 6. Given 2 Lcwk.E

s /

1

.B/, that is

exists a unique (for equality a.s.) mapping † WD E B 2 Lcwk.E

s /

the conditional expectation of with respect to B, which enjoys the following

properties:

R

R

(a)

ı .v; †/dP D

ı .v; /dP for all v 2 L1

E .B/.

B

(b) † E jjB E a.s.

1

(c) S†1 .B/ is .L1E ŒE.B/; L1

E .B//- compact (here S† .B/ denotes the set of

1

all LE ŒE.B/ selections of †) and satisfies

ı .v; E B S1 .F // D ı .v; S†1 .B//

for all v 2 L1

E .B/.

(d) E B is increasing: 1

2 a.s. implies E B 1

E B 2 a.s.

Now we need at first a conditional expectation version for Lemma 1.

Lemma 2. Let B be a sub- -algebra of F and let X be a random fuzzy convex

upper semicontinuous integrand X W

Es ! Œ0; 1 with the following

properties:

(1) fx 2 Es W X.!; x / > 0g is weak -compact, for each ! 2

(2) g WD jL0C .X /j 2 L1 ,

(3) Assume that 0 < ˛1 < ˛2 < : : : : < ˛k ! ˛ and that

,

lim ı .x; L˛k .X /.!// D ı .x; L˛ .X /.!//

k!1

for all ! 2

and for all x 2 E.

Let E B X˛ be the conditional expectation of the level sets L˛ .X / WD X˛ , ˛ 20; 1.

Then we have

\

E B X˛k D E B X˛ :

k 1

14

C. Castaing et al.

Proof. We follow some lines of the proof of Lemma 1. But here we need a careful

look using some convergence results in conditional expectation. By (1)–(3) we have

L˛ .X /.!/ WD X˛ .!/

Y .!/ for all ˛ 20; 1 and for all ! 2

where Y is a

weak -compact valued integrably bounded multifunction. By Theorem 6 and by our

assumption (1)–(3) the conditional expectation E B X˛ is convex weak -compact

valued, B-measurable and integrably bounded with E B X˛ .!/ .E B g/.!/ B E

for all ˛ 20; 1 and for all ! 2 . Since the weak -topology coincides with the

metrizable topology m , by Theorem 5.4 in [5] the multifunction

V .!/ D w - ls E B X˛n .!/ D m -LS E B X˛n .!/ D

\

E B X˛k .!/

k 1

is w -compact and B-measurable so that

V .!/ D w K lim E B X˛k .!/:

k!1

By Theorem 1, we have

lim ı .x; E B X˛k .!// D ı .x; V .!//

k!1

for all ! 2

and for all x 2 E. By the dominated convergence theorem for

conditional expectations, we have

ı .x; V .!// D lim ı .x; E B X˛k .!// D lim E B ı .x; X˛k .!//

k!1

k!1

D E B ı .x; X˛ .!// D ı .x; E B X˛ .!//

so that E B X˛ D V by the separability of E, noting that E B X˛ .!/ is convex

weak -compact, and that

E B X˛ .!/ D

\

E B X˛k .!/

k 1

for all ! 2

.

Now we proceed to the conditional expectation of the level sets associated with

such a random fuzzy convex upper semicontinuous integrand X W

Es ! Œ0; 1.

Theorem 7. Let B be a sub- -algebra of F and let X be a random fuzzy

convex upper semicontinuous integrand X W

Es ! Œ0; 1 with the following

properties:

(1) fx 2 Es W X.!; x / > 0g is weak*-compact, for each ! 2 ,

(2) g WD jL0C .X /j 2 L1 ,

(3) For every fixed ! 2 , the cwk.Es /-valued mapping ˛ 7! L˛ .X /.!/ D

X˛ .!/ is scalarly left continuous on 0; 1.

On the Integration of Fuzzy Level Sets

15

Then the following

Thold

E B X˛ .!/ D k 1 E B X˛k .!/ for every ! 2

˛1 < ˛2 : : : < ˛k ! ˛.

and every ˛ 20; 1, whenever

Proof. Here we will use again the monotonicity of the conditional expectation and

X˛ and E B X ˇ

the monotonicity of the level sets, namely for ˛ Ä ˇ; X ˇ

B

E X˛ . We have to check that

\

E B X˛k .!/

( )

E B X˛ .!/ D

k 1

for every ! 2

whenever 0 < ˛1 < ˛2 < : : : < ˛k ! ˛. By the continuity

of the level sets (3) and the dominated convergence theorem for the conditional

expectation, we have

lim ı .x; E B X˛k .!// D lim E B ı .x; X˛k .!//

k!1

k!1

D E B ı .x; X˛ .!// D ı .x; E B X˛ .!//

for all x 2 E, so that the desired inclusion follows from the arguments developed

in first part of the proof of Lemma 2.

With the above considerations, we produce a general result on the conditional

1

.F / depending on

expectation of a convex weak -compact valued X˛ 2 Lcwk.E

s /

the parameter ˛ 20; 1.

Theorem 8. Let B be a sub- -algebra of F , and let X W

0; 1 Ã Es be a

convex weak -compact valued mapping with the following properties:

(1) jX.!; ˛/j Ä g 2 L1 for all .!; ˛/ 2

0; 1,

(2) For each ! 2 , X.!; :/ is scalarly left continuous on 0; 1,

(3) For every fixed ˛ 20; 1, X:.; ˛/ is scalarly F -measurable.

Then the convex weak -compact valued conditional expectation E B X˛ of the

mapping X˛ enjoys the properties

(a) For each ! 2 , ˛ 7! E B X˛ .!/ is scalarly left continuous on 0; 1,

(b) For each ˛ 20; 1, ! 7! E B X˛ .!/ is scalarly B-measurable on ,

(c) Assume further that ˛ 7! X.!; ˛/ is decreasing, for every fixed !, i.e. ˛ < ˇ 2

0; 1 implies X.!;

T ˇ/ X.!; ˛/, then 0 < ˛1 < ˛2 < : : : < ˛k ! ˛ implies

E B X˛ .!/ D k 1 E B X˛k .!/:

Similarly we have

Theorem 9. Let X W

0; 1 Ã Es be a convex weak -compact valued mapping

with the following properties:

(1) jX.!; ˛/j Ä g 2 L1 for all .!; ˛/ 2

0; 1,

(2) For each ! 2 , X.!; :/ is scalarly left continuous on 0; 1,

(3) For every fixed ˛ 20; 1, X.:; ˛/ is scalarly F -measurable on

.

16

C. Castaing et al.

Then the convex weak -compact valued mapping EX˛ WD

the properties

R

X.!; ˛/dP enjoys

(a) ˛ 7! EX˛ is scalarly left continuous on 0; 1,

(b) Assume further that ˛ 7! X.!; ˛/ is decreasing,

for every fixed !, then 0 <

T

˛1 < ˛2 < : : : < ˛k ! ˛ implies EX˛ D k 1 EX˛k :

Corollary 2. Assume that E D Rd and that . ; F ; P / has no atoms. Let X W

0; 1 Ã Rd be a compact valued mapping with the following properties:

(1) jX.!; ˛/j Ä g 2 L1 for all .!; ˛/ 2

0; 1,

(2) For each ! 2 , X.!; :/ is scalarly left continuous on 0; 1,

(3) For every fixed ˛ 20; 1, X.:; ˛/ is scalarly F -measurable on .

R

Then the convex compact valued mapping EX˛ WD

X.!; ˛/dP enjoys the

properties

(a) ˛ 7! EX˛ is scalarly left continuous on 0; 1,

(b) Assume further that ˛ 7! X.!; ˛/ is decreasing,

for every fixed !, then 0 <

T

˛1 < ˛2 < : : : < ˛k ! ˛ implies EX˛ D k 1 EX˛k :

Proof. Since E D Rd and . ; F ; P / has no atoms, by [8, Theorem IV-17], EX˛

is convex compact in Rd .

Using the notations of the preceding results and the Negoita-Ralescu representation theorem [25, Lemma 1] or Höhle and Šostak [14, Lemma 6.5.1], we mention a

useful result.

Lemma 3. Let .C˛ /˛2Œ0;1 be a family of convex weak -compact subsets in E with

the properties:

(1)

(2)

(3)

(4)

C0 WD Es ,

Cˇ C˛ ; 8˛ < ˇ 2 Œ0; 1,

C˛ rB E for all ˛ 20; 1,

˛ 7! ı .x; C˛ / is left continuous on 0; 1 for all x 2 E.

There is a unique fuzzy convex upper semicontinuous variable ' W Es ! Œ0; 1 such

that fx 2 Es W '.x / ˛g D C˛ , for every ˛ 20; 1, where ' is given by

'.x / D supf˛ 2 Œ0; 1 W x 2 C˛ g:

Proof. Step 1. We use at first some arguments and results developed in Lemma 1.

Let 0 < ˛1 < ˛2 ; : : : < ˛k ! ˛. By (4) we have

lim ı .x; C˛k / D ı .x; C˛ /

k!1

(5)

On the Integration of Fuzzy Level Sets

17

for all x 2 E. Note that by (2) and (3), .C˛n / is uniformly bounded and

decreasing. Since the w -topology coincides with the metrizable topology m

on bounded subsets in the weak dual, by Theorem 5.4 in [5]

C WD w - ls C˛n D m -LS C˛n D

\

C˛k

k 1

so that C D w K limk!1 C˛k . Applying Theorem 1, we have

ı .x; C / D lim ı .x; C˛k /

(6)

k!1

for all x 2 E, so that using (5) and (6) we have that

ı .x; C˛ / D ı .x; C /

for all x 2 E, whence C˛ D C by the separability of E, that is,

\

C˛k D C˛ :

(7)

k 1

Step 2. fx 2 Es W '.x / ˛g D C˛ , for every ˛ 20; 1.

Here we may apply the arguments given in the Negoita-Ralescu representation

theorem and the results in Step 1 to get this equality. Let x 2 C˛0 . Then

˛0 2 f˛ 2 Œ0; 1 W x 2 C˛ g

which implies

'.x / D supf˛ 2 Œ0; 1 W x 2 C˛ g

thus x 2 Œ'

˛0

˛0 . To show the converse inclusion, let x 2 Œ'

'.x / D supf˛ 2 Œ0; 1 W x 2 C˛ g

˛0 . Then

˛0 :

If '.x / > ˛0 , there exists ˛1 ˛0 with x 2 C˛1 . But then we have C˛1 C˛0

by (2), thus x 2 C˛0 . Assume that '.x / D ˛0 . Then there exists .˛k / such that

x 2 C˛k for each k and that ˛k " ˛0 . But then

x 2

1

\

nD1

by Step 1.

C˛k D C˛0

18

C. Castaing et al.

Applying Theorem 4 and Lemma 3, we get the expectation of random fuzzy

convex integrands.

Theorem 10. Let B be a sub- -algebra of F and let X be a random fuzzy

convex upper semicontinuous integrand X W

Es ! Œ0; 1 with the following

properties:

(1) fx 2 Es W X.!; x / > 0g is weak -compact, for each ! 2 ,

(2) g WD jL0C .X /j 2 L1 ,

(3) For every fixed ! 2 , the K -valued mapping ˛ 7! L˛ .X /.!/ D X˛ .!/ is

scalarly left continuous on 0; 1.

Q / on Es

Then there exist a unique fuzzy convex upper semicontinuous function E.X

satisfying

Q /˛ D

ŒE.X

Z

X˛ dP; 8˛ 20; 1I

Q / is the fuzzy expectation of X .

E.X

R

R

X˛ dP /˛2Œ0;1 with C0 WD

X0 dP

Proof. Apply Lemma 3 to the family .C˛ D

D E by taking account of Proposition 3 and Theorem 4.

Now it is possible to provide the fuzzy conditional expectation of random fuzzy

convex upper semicontinuous integrand.

Theorem 11. Let B be a sub- -algebra of F and let X be a random fuzzy

convex upper semicontinuous integrand X W

Es ! Œ0; 1 with the following

properties:

(1) fx 2 Es W X.!; x / > 0g is weak -compact, for each ! 2 ,

(2) g WD jL0C .X /j 2 L1 ,

(3) For every fixed ! 2 , the K -valued mapping ˛ 7! L˛ .X /.!/ D X˛ .!/ is

scalarly left continuous on 0; 1.

Let E B X˛ D E.X˛ jB/ be the conditional expectation of X˛ for every ˛ 2 Œ0; 1

and let

'! .x / WD supf˛ 2 Œ0; 1 W x 2 E B X˛ .!/g

for every ! 2

and for every ˛ 2 Œ0; 1. Then the following hold

fx 2 Es W '! .x /

˛g D E B X˛ .!/

Q jB/ WD ' is the fuzzy conditional

for every ! 2

and for every ˛ 20; 1; E.X

expectation of X .

On the Integration of Fuzzy Level Sets

Proof. Step 1.

Theorem 7

19

1

By Theorem 6, recall that E B X˛ 2 Lcwk.E

/ .B/ and by virtue of

s

E B X˛ .!/ D

\

E B X˛k .!/

k 1

for every ! 2 and every ˛ 20; 1, whenever ˛1 < ˛2 ; : : : < ˛k ! ˛.

Step 2. Let ˛0 20; 1 and let ! 2 . Let x 2 E B X ˛0 .!/. Then

˛0 2 f˛ 2 Œ0; 1 W x 2 E B X˛ .!/g

which implies

'! .x / D supf˛ 2 Œ0; 1 W x 2 E B X ˛ .!/g

thus x 2 Œ'!

˛0

˛0 . To show the converse inclusion, let x 2 Œ'!

'! .x / D supf˛ 2 Œ0; 1 W x 2 E B X˛ .!/g

˛0 . Then

˛0 :

˛0 with x 2 E B X˛1 .!/. But then we have

If '! .x / > ˛0 , there exists ˛1

B

B ˛0

E X˛1 .!/

E X .!/ by the monotonicity of the conditional expectation,

thus x 2 E B X˛0 .!/. Assume that '! .x / D ˛0 . Then there exists .˛k / such

that x 2 E B X˛k .!/ for each k and that ˛k " ˛0 . But then

1

\

x 2

E B X˛k .!/ D E B X˛0 .!/

kD1

by using Step 1.

It is worth to state the relationships between the fuzzy expectation and the

conditional expectation of a random upper semicontinuous fuzzy convex integrand

X . When E is a reflexive separable Banach space and X 2 LE1 . ; F ; P / and B

is a sub- -algebra of F , EX is the expectationRof X and E B XR the conditional

expectation of X , then, for any B 2 B, we have B E B XdP D B XdP , now we

have a similar equality if we deal with a random upper semicontinuous fuzzy convex

Q / and fuzzy conditional expectation E.X

Q jB/.

integrand X , fuzzy expectation E.X

Namely the following equality holds

Z

Q jB/˛ dP D

ŒE.X

B

Z

B

Z

Q

X˛ dP D ŒE.X1

B /˛

E X˛ dP D

B

for every B 2 B and for every ˛ 20; 1.

B

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