Fuzzy measure based on decomposition theory

Fuzzy Sets and Systems 112 (2000) 187–205www.elsevier.com/locate/fss

Fuzzy measure based on decomposition theoryJing-Shing Yao∗, San-Chyi Chang

Department of Applied Mathematics, Chinese Culture University, Hwakang, Yangminshan, Taipei, Taiwan, ROC

Received June 1997; received in revised form March 1998

Abstract

On the real line R(= (−∞;+∞)), we considered the fuzzy sets with which the �-cut exists for every �∈ [0; 1]. Wecan de�ne a fuzzy measure M∗ for these fuzzy sets. By using the decomposition theory and some properties of measureM on R in crisp case, we �nd M∗ is an extension of M. If we restrict the fuzzy sets on R in a �xed �nite open interval,then the results of fuzzy measure de�ned by Zimmerman (1991) in fuzzy sense hold. c© 2000 Elsevier Science B.V. Allrights reserved.

Keywords: Measure; Fuzzy measure; Extended fuzzy measure; Fuzzy point; Fuzzy interval; Fuzzy number

1. Introduction

Suppose FS contains all fuzzy numbers (FN ), level 1 fuzzy intervals (FI (1)) and their countable unions.Let D be a fuzzy set in FS with membership function �D(x). For every �∈ [0; 1], the �-cut of D is an ordinarycrisp interval

D(�)= [Dl(�); Dr(�)]= {x∈R | �D(x)¿�};where Dl(�) and Dr(�) denote the left and right bounds of D(�). By using the decomposition theory, we have

D=⋃

�∈[0;1][Dl(�)�; Dr(�)�];

where [Dl(�)�; Dr(�)�] is a level � fuzzy interval which is in one–one correspondence to the crisp interval[Dl(�); Dr(�)] (�-cut of D) with measure M[Dl(�); Dr(�)]=Dr(�)−Dl(�). In Section 2, we de�ne the fuzzymeasure of D∈FN ∪FI (1) as follows:

M∗(D)=∫ 1

0[Dr(�)− Dl(�)] d�:

∗ Corresponding author.

0165-0114/00/$ - see front matter c© 2000 Elsevier Science B.V. All rights reserved.PII: S 0165 -0114(98)00074 -8

188 J.-S. Yao, S.-C. Chang / Fuzzy Sets and Systems 112 (2000) 187–205

Furthermore, on the real line, crisp set is a special case of fuzzy set. In fact, for every j=1; 2; : : : ; n, a level1 fuzzy interval [a( j)1 ; b

( j)1 ] is in one–one correspondence with the crisp interval [a

( j); b( j)] such that

M∗

n⋃j=1

[a( j)1 ; b( j)1 ]

=

n∑j=1

(b( j) − a( j))=M

n⋃j=1

[a( j); b( j)]

provided [a( j); b( j)], [a(k); b(k)] are nonoverlapping, ∀j 6= k. Thus the fuzzy measure and the crisp measure areequal in intervals. Therefore fuzzy measure M∗ is an extension of the crisp measure M and M∗ is called anextended fuzzy measure. Moreover we shall show that M∗ satis�es the condition of a fuzzy measure exceptM∗(R)=∞. (In Zimmermann [4], the fuzzy measure of R should be one.)Let X =(p; q), −∞¡p¡q¡∞, be a �nite open interval of R. In Section 3, we consider a family of fuzzy

sets in X , say FX . For each D∈FX , we de�ne

M′(D)=1

q− pM∗(D):

We shall show that M′ is a fuzzy measure which satis�es the de�nition of Zimmermann [4] in fuzzy sensebut not an extension of the crisp measure M. Consider the uniform distribution over X =(p; q). For fuzzynumber A=(a; b; c); p¡a¡b¡c¡q, we �nd that the fuzzy measure M′(A) equal to the fuzzy probabilityP(A).

2. Extension of real measure to fuzzy measure depend on decomposition theory

This section brie y introduces a fuzzy measure on FS . Some de�nitions relative to this study are stated asfollows:

De�nition 2.1. For each �∈ [0; 1], a fuzzy set a� is called a level � fuzzy point if the membership functionof a� is given by

�a�(x)={�; x= a;0; elsewhere:

(1)

The family of level � fuzzy points is denoted by Fp(�) and we denote Fp=⋃�∈[0;1] Fp(�).

De�nition 2.2. For each �∈ [0; 1], and a; b∈R with a¡b, a fuzzy set [a�; b�] is called a level � fuzzy intervalif the membership function of [a�; b�] is given by

�[a�; b�](x)={�; a6x6b;0; elsewhere:

(2)

Other level � fuzzy intervals are (a�; b�), [a�; b�), (a�; b�], (a�;∞), [a�;∞), (−∞; b�), (−∞; b�]. Theirmembership functions are de�ned accordingly as in (2). For example

�(a�; b�)(x)={�; a¡x¡b;0; elsewhere:

For convenience, for every �∈ [0; 1] we de�ne FI (�)= {[a�; b�] | ∀a; b∈R; a¡b}; F∗I (�)= {(a�; b�) |

∀a; b∈R; a¡b} and FI =⋃�∈[0;1] FI (�)∪F∗

I (�). Moreover, the characteristic function for (a; b) is C(a; b)(x)= 1, if a¡x¡b; and C(a; b)(x)= 0, if x6a or x¿b. On the other hand, the membership function for (a1; b1) is

J.-S. Yao, S.-C. Chang / Fuzzy Sets and Systems 112 (2000) 187–205 189

�(a1 ; b1)(x)= 1, if a¡x¡b; and �(a1 ; b1)(x)= 0, if x6a or x¿b. Thus �(a1 ; b1)(x)=C(a; b)(x), ∀x∈R. Therefore,for �=1, the level 1 fuzzy interval (a1; b1) and the real number interval (a; b) are the same thing but havedi�erent expressions. That is,

(a1; b1)↔ (a; b); ∀(a1; b1)∈F∗I (1); ∀(a; b)⊂R:

This is an one–one onto mapping. Also, [a1; b1]↔ [a; b]; ∀[a1; b1]∈FI (1); ∀[a; b]⊂R: This is also an one–oneonto mapping. Similarly, for every �xed �∈ [0; 1], there is one–one onto mapping

(a�; b�)↔ (a; b); ∀(a�; b�)∈F∗I (�); ∀(a; b)⊂R;

[a�; b�]↔ [a; b]; ∀[a�; b�]∈FI (�); ∀[a; b]⊂R:

Therefore, for every �xed �∈ [0; 1], the relation of intervals of R and F∗I (�) (or FI (�)) are similar. For

example, if a; b; c; d∈R then

(b; c)⊂(a; d)⇔ (b�; c�)⊂ (a�; d�); ∀�∈ [0; 1]:

2.1. A family of fuzzy sets in R

Some de�nitions for fuzzy sets of R and their properties are given by Kaufmann and Gupta [3]. We willlist them as follows which will be used henceforth.

De�nition 2.3. (1) A fuzzy subset A of R is convex i� every ordinary subset (�-cut of A)

A(�)= {x | �A (x)¿�}; �∈ [0; 1]

is convex; that is, if A(�) is a closed interval of R.(2) A fuzzy subset A of R is normal i� ∃ x∈R s:t: �A(x)= 1.(3) A convex and normal fuzzy subset of R is called a fuzzy number in R.The family of fuzzy numbers of R is denoted by FN and we let FS be the family of fuzzy sets in FN ∪FI (1)

and their countable unions, i.e.

FS =

n⋃j=1

D( j) | D( j) ∈FN ∪FI (1); j=1; 2; : : : ; n; n is any integer or ∞ :

Obviously, if D(k; q)∈FN ∪FI (1), q=1; 2; : : : ; nk , k =1; 2; : : : ; m; then E(k) =⋃nkq=1 D

(k; q)∈FS , k =1; 2; : : : ; mand

⋃mk=1 E

(k) =⋃mk=1

⋃nkq=1 D

(k; q)∈FS .For every �∈ [0; 1], let CD(�)(z)= 1 if z ∈D(�) and CD(�)(z)= 0 if z =∈D(�). The decomposition theory can

be described as follows:

Theorem (The decomposition theory). Let D be a fuzzy set in FN ∪FI (1) with �-cut D(�)= [Dl(�); Dr(�)].Then

�D(z)=∨

�∈[0;1](� ∧ CD(�)(z))=

∨�∈[0;1]

�[Dl(�)�;Dr (�)�](z)= �B(z)

where B=⋃�∈[0;1] [Dl(�)�; Dr(�)�]:


Proof. By the de�nition of �-cut, we have

�D(z)=∨

�∈[0;1](� ∧ CD(�)(z)):

Also, by the de�nition of level � fuzzy interval [Dl(�)�; Dr(�)�], we have

�[Dl(�)�;Dr (�)�](z)= � and � ∧ CD(�)(z)= � ∧ 1= �; ∀z ∈D(�):

Furthermore, if z =∈D(�) then

� ∧ CD(�)(z)= 0= �[Dl(�)�;Dr (�)�](z):

Therefore, for every �∈ [0; 1] and z we have � ∧ CD(�)(z)= �[Dl(�)�;Dr (�)�](z) and hence

�D(z)=∨

�∈[0;1](� ∧ CD(�)(z))=

∨�∈[0;1]

�[Dl(�)�;Dr (�)�](z):

Remark 2.1. (a) For every �∈ [0; 1], there is an one–one onto mapping between level � fuzzy interval[Dl(�)�; Dr(�)�] and real interval [Dl(�); Dr(�)].(b) Let D(k) be fuzzy set in FN ∪FI (1) and [D(k)l (�); D(k)r (�)] denote the �-cut of D(k), k =1; 2; : : : ; n such

that D=⋃nk=1 D

(k). Then

D=n⋃k=1

D(k) =n⋃k=1

⋃�∈[0;1]

[D(k)l (�)�; D(k)r (�)�] =

⋃�∈[0;1]

n⋃k=1

[D(k)l (�)�; D(k)r (�)�]

and⋃nk=1[D

(k)l (�); D

(k)r (�)] is the �-cut of D.

2.2. Fuzzy measure on FS

Let [a; b] denote a �nite interval, then M([a; b])= b − a is the crisp measure of [a; b]. Also, M(R)=∞.Let D be a fuzzy set in FN ∪FI (1) and [Dl(�); Dr(�)] denote the �-cut of D, then D=

⋃�∈[0;1][Dl(�)�; Dr(�)�].

For every �∈ [0; 1], [Dl(�)�; Dr(�)�] is a level � fuzzy interval which is in one–one correspondence with acrisp interval [Dl(�); Dr(�)]. Since M([Dl(�); Dr(�)])=Dr(�)−Dl(�), ∀�∈ [0; 1]; then we can �nd its averagevalue by integration to extend the crisp measure to FN ∪FI (1) as the following de�nition.

De�nition 2.4. Let D be a fuzzy set in FN ∪FI (1) and [Dl(�); Dr(�)] denote the �-cut of D, then

M∗(D)=∫ 1

0[Dr(�)− Dl(�)] d�

is called the fuzzy measure of D.

Remark 2.2. (a) In De�nition 2.4, M∗(D) equals the area of the region bounded by �D(x) and the x-axis.(b) Let (a1; b1) be a level 1 fuzzy interval which is in one–one onto mapping to a crisp interval (a; b).

Then

M∗((a1; b1))=M∗([a1; b1])=∫ 1

0(b− a) d�= b− a=M((a; b))=M([a; b]):


Since b − a=(b − a) · 1, then M∗([a1; b1]) equals the area of a rectangle having width b − a and height 1.Also, M∗([a1; b1]) equals the area of the region bounded by �[a1 ; b1](x) and the x-axis and M([a; b]) equalsthe length of the interval [a; b]. Furthermore, let I(a�; b�) denote any level � fuzzy intervals such as (a�; b�),[a�; b�), (a�; b�]. Then for �=1, M∗(I(a1; b1))= b − a=M(I(a; b)). If a=−∞ or b=∞ or (a=−∞ andb=∞), then M∗(I(a1; b1))=∞.

Example 2.1. Let A be a triangular fuzzy number with membership function as follows:

�A(x)=

x − ab− a ; a6x6b;

c − xc − b ; b6x6c;

0; elsewhere;

where a¡b¡c. For every �∈ [0; 1], let Al(�)= a+ (b− a)�, Ar(�)= c− (c− b)�, then A(�)= [Al(�); Ar(�)]is the �-cut of A. By De�nition 2.4,

M∗(A)=∫ 1

0[c − (c − b)�− a− (b− a)�] d�= 1

2(c − a):

It is obvious that M∗(A) equals the area of the region bounded by �A(x) and the x-axis.

Proposition 2.1. Let D; E be fuzzy sets in FN ∪FI (1) and D⊂ E; then M∗(D)6M∗(E).

Proof. Let D(�)= [Dl(�); Dr(�)], E(�)= [El(�); Er(�)] denote the �-cuts of D, E respectively. Since D⊂ E,then El(�)6Dl(�)6Dr(�)6Er(�), ∀�∈ [0; 1]. Therefore

M∗(D)=∫ 1

0[Dr(�)− Dl(�)] d�6

∫ 1

0[Er(�)− El(�)] d�=M∗(E):

Let D(k) be fuzzy set in FN ∪FI (1) and let [D(k)l (�); D(k)r (�)] denote the �-cut of D(k), k =1; 2; : : : ; n. ThenD=

⋃nk=1 D

(k) and

n⋃k=1

D(k) =n⋃k=1

⋃�∈[0;1]

[D(k)l (�)�; D(k)r (�)�] =

⋃�∈[0;1]

n⋃k=1

[D(k)l (�)�; D(k)r (�)�];

where⋃nk=1[D

(k)l (�)�; D

(k)r (�)�] is a fuzzy set which is corresponding to the crisp set

⋃nk=1[D

(k)l (�); D

(k)r (�)].

Furthermore, if [D(k)l (�); D(k)r (�)], k =1; 2; : : : ; n are nonoverlapping intervals, then by De�nition 2.5 (below),

we have

M

(n⋃k=1

[D(k)l (�); D(k)r (�)]

)=

n∑k=1

∫ 1

0[D(k)r (�)− D(k)l (�) d�]:

Remark 2.3. (a) For every �∈ [0; 1], if a¡c¡b¡d then [a�; b�], [c�; d�] are overlapping intervals. Then wehave [a�; b� ]∪ [c�; d�] = [a�; c�]∪ [c�; d�] such that [a�; c�]; [c�; d�] are nonoverlapping intervals or [a�; b�]∪[c�; d�]= [a�; b�]∪ [b�; d�] such that [a�; b�]; [b�; d�] are nonoverlapping intervals. Similarly, for crisp case ifa¡c¡b¡d then [a; b] ∪ [c; d] = [a; c] ∪ [c; d] or [a; b] ∪ [c; d] = [a; b] ∪ [b; d].


(b) Let D(k)(�)= [D(k)l (�); D(k)r (�)] denote the �-cut of D(k)∈FN ∪FI (1), k =1; 2; : : : ; n and D=

⋃nk=1 D

(k).By Remark 2.1(b),

D=n⋃k=1

D(k) =n⋃k=1

⋃�∈[0;1]

[D(k)l (�)�; D(k)r (�)�] =

⋃�∈[0;1]

n⋃k=1

[D(k)l (�)�; D(k)r (�)�]

and⋃nk=1[D

(k)l (�); D

(k)r (�)] is the �-cut of D.

(b-1) If [D(k)l (�); D(k)r (�)], k =1; 2; : : : ; n are nonoverlapping intervals, ∀�∈ [0; 1]. Then D has �-cut⋃n

k=1[D(k)l (�); D

(k)r (�)] with length

∑nk=1[D

(k)r (�)− D(k)l (�)].

(b-2) If [D(k)l (�); D(k)r (�)], k =1; 2; : : : ; n are overlapping intervals, for some �∈ [0; 1]. By Remark 2.3(a),

for every �∈ [0; 1], we can rewriten⋃k=1

[D(k)l (�)�; D(k)r (�)�] into

n⋃k=1

[D(k)∗l (�)�; D(k)∗r (�)�]

such that [D(k)∗l (�)�; D(k)∗r (�)�], k =1; 2; : : : ; n are nonoverlapping intervals. Obviously,

D=n⋃k=1

D(k) =⋃

�∈[0;1]

n⋃k=1

[D(k)l (�)�; D(k)r (�)�] =

⋃�∈[0;1]

n⋃k=1

[D(k)∗l (�)�; D(k)∗r (�)�]:

Then D has �-cut⋃nk=1[D

(k)∗l (�); D

(k)∗r (�)] with length

∑nk=1[D

(k)∗r (�)− D(k)∗l (�)]. It is obvious that

n∑k=1

[D(k)∗r (�)− D(k)∗l (�)]6n∑k=1

[D(k)r (�)− D(k)l (�)]; ∀�∈ [0; 1]:

(b-3) (b-1) is a special case of (b-2) (set D(k)∗l (�)=D(k)l (�) and D

(k)∗r (�)=D

(k)r (�) in (b-2)). Therefore, if

D=⋃nk=1 D

(k) ∈FS , where D(k) ∈FN ∪FI (1), k =1; 2; : : : ; n. Then the �-cut of D is denoted by⋃nk=1 [D

(k)∗l (�);

D(k)∗r (�)] with length∑n

k=1[D(k)∗r (�)− D(k)∗l (�)]: Also,

D=n⋃k=1

D(k) =⋃

�∈[0;1]

n⋃k=1

[D(k)l (�)�; D(k)r (�)�] =

⋃�∈[0;1]

n⋃k=1

[D(k)∗l (�)�; D(k)∗r (�)�]

andn∑k=1

[D(k)∗r (�)− D(k)∗l (�)]6n∑k=1

[D(k)r (�)− D(k)l (�)]; ∀�∈ [0; 1]:

Therefore, by Remark 2.3(b) we can de�ne the measure of fuzzy set⋃nk=1 D

(k) ∈FS as follows:

De�nition 2.5. Let D=⋃nk=1 D

(k) ∈FS; where D(k)∈FN ∪FI (1); k =1; 2; : : : ; n and D has �-cut⋃nk=1 [D

(k)∗l (�);

D(k)∗r (�)] with length∑n

k=1[D(k)∗r (�) − D(k)∗l (�)]; where [D

(k)∗l (�)�; D

(k)∗r (�)�]; k =1; 2; : : : ; n are nonoverlapping

intervals. Then

M∗(D)=M∗(

n⋃k=1

D(k))=

n∑k=1

∫ 1

0[D(k)∗r (�)− D(k)∗l (�)] d�

is the fuzzy measure of fuzzy set D.


Remark 2.4. (1) De�nition 2.4 is a special case of De�nition 2.5 (when n=1 and D(1)∗l (�)=D(1)l (�),

D(1)∗r (�)=D(1)r (�)).

(2) By the de�nition of FS , if D(k)(�)∈FN ∪FI (1), k =1; 2; : : : ; then⋃∞k=1 D

(k) ∈FS , and the fuzzy measureof fuzzy set

⋃∞k=1 D

(k) may refer to Propositions 2.8 and 2.9 or Proposition 2.11.

Remark 2.5. If [a(k); b(k)], k =1; 2; : : : ; n are nonoverlapping intervals, then we have the following results:1. �∪nk=1 [a

(k)1 ; b

(k)1 ](x)=C∪nk=1 [a(k) ; b(k)](x); ∀x.

2. If −∞¡a(k)¡b(k)¡∞, then

M∗(

n⋃k=1

[a(k)1 ; b(k)1 ]

)=

n∑k=1

(b(k) − a(k))=M

(n⋃k=1

[a(k); b(k)]

)

3. M∗(R)=∞:

Proposition 2.2. If D=⋃nk=1 D

(k) ∈FS and D(k); k =1; 2; : : : ; n are nonoverlapping fuzzy sets in FN ∪FI (1);then

M∗(D)=M∗(

n⋃k=1

D(k))=

n∑k=1

M∗(D(k)):

Proof. D(k), k =1; 2; : : : ; n; are nonoverlapping intervals so that D(k)∗l (�)=D(k)l (�) and D

(k)∗r (�)=D

(k)r (�). By

De�nition 2.5 and Remark 2.3, we get

M∗(D)=M∗(

n⋃k=1

D(k))=

n∑k=1

∫ 1

0[D(k)r (�)− D(k)l (�)] d�=

n∑k=1

M∗(D(k)):

Proposition 2.3. For D=⋃nk=1 D

(k) and G=⋃mj=1 G

( j) ∈FS; where D(k); G( j) ∈FN ∪FI (1); k =1; 2; : : : ; n;j=1; 2; : : : ; m: If D⊂ G then M∗(D)6M∗(G):

Proof. Since D=⋃nk=1 D

(k) and D(k) ∈FN ∪FI (1), k =1; 2; : : : ; n; by (b-3) of Remark 2.3, we have

D=n⋃k=1

D(k) =⋃

�∈[0;1]

n⋃k=1

[D(k)l (�)�; D(k)r (�)�] =

⋃�∈[0;1]

n⋃k=1

[D(k)∗l (�)�; D(k)∗r (�)�];

where [D(k)∗l (�); D(k)∗r (�)], k =1; 2; : : : ; n are nonoverlapping intervals. Similarly,

G=m⋃j=1

G( j) =⋃

�∈[0;1]

m⋃j=1

[G( j)l (�)�; G( j)r (�)�] =

⋃�∈[0;1]

m⋃j=1

[G( j)∗l (�)�; Q( j)∗r (�)�]

and [G( j)∗l (�); Q( j)∗r (�)]; j=1; 2; : : : ; m; are nonoverlapping intervals. Since D⊂ G, then

n⋃k=1

[D(k)∗l (�)�; D(k)∗r (�)�]⊂

m⋃j=1

[G( j)∗l (�)�; Q( j)∗r (�)�]; ∀�∈ [0; 1]:

Basing on the one–one correspondence between level � fuzzy interval and real interval, for every �∈ [0; 1]we have

n⋃k=1

[D(k)∗l (�); D(k)∗r (�)]⊂

m⋃j=1

[G( j)∗l (�); G( j)∗r (�)]:


Thusn∑k=1

[D(k)∗r (�)− D(k)∗l (�)]6m∑j=1

[G( j)∗r (�)− G( j)∗l (�)]:

Therefore

M∗(D)=n∑k=1

∫ 1

0[D(k)∗r (�)− D(k)∗l (�)] d�6

m∑j=1

∫ 1

0[G( j)∗r (�)− G( j)∗l (�)] d�=M∗(G):

Remark 2.6. If n=m=1; then Proposition 2.3 coincides with Proposition 2.1.

Let F be the family of fuzzy sets in Fp ∪FS ∪FI and their arbitrary unions. In order to establish a fuzzytopology for R, we need to know the following de�nitions.

De�nition 2.6 (Chang [1], De�nition 2.2). A fuzzy topology is a family T of fuzzy set in X which satis�esthe following conditions:(a) �; X ∈T .(b) If A; B∈T , then A∩ B∈T .(c) If Ai ∈T , ∀i∈ I , where I is any index set, then

⋃i∈I Ai ∈T .

If T is a fuzzy topology for X then the pair (X; T ) is called a fuzzy topological space (FTS for short).

Let A= {(a�; b�); (a�;∞); (−∞; b�); (∞;∞) | ∀a¡b; a; b∈R; �∈ [0; 1]} (⊂F): We de�ne the openfuzzy set in F as follows:

De�nition 2.7. A fuzzy set O in F is an open fuzzy set i� for each x�⊂ O, there exists U ∈A such thatx�⊂ U ⊂ O.Let TS be the family of all open fuzzy sets in F . Obviously A⊂TS . Now we shall prove that TS is a

fuzzy topology as de�ned in De�nition 2.6.

Proposition 2.4. TS is a fuzzy topology for R and (R; TS) is a fuzzy topological space.

Proof. (a) It is obvious that �; R∈TS .(b) If A; B∈TS , then for each x�⊂ A∩ B, we have x�⊂ A and x�⊂ B. Since A; B are open fuzzy sets,

by De�nition 2.7, there exists U ; V ∈A such that x�⊂ U ⊂ A, and x�⊂ V ⊂ B. Furthermore x�⊂ U ∩ V im-plies U ∩ V 6= ∅. Moreover the intersection of U and V is an open fuzzy interval. Therefore U ∩ V ∈A andx�⊂ U ∩ V ⊂ A∩ B, i.e. A∩ B∈TS and De�nition 2.6(b) is satis�ed.(c) If Aj ∈TS , j∈ I (any index set), then for each x�⊂

⋃j∈I Aj, there exists some n∈ I such that x�⊂ An.

Since An is an open fuzzy set, there is an U ∈A such that x�⊂ U ⊂ An⊂⋃j∈I Aj. Therefore

⋃j∈I Aj ∈TS

and De�nition 2.6(c) is satis�ed.By De�nition 2.6, TS is a fuzzy topology and (R; TS) is a fuzzy topological space.

De�nition 2.8 (Chang [1], De�nition 2.3). A fuzzy set U in a FTS (X; T ) is a neighborhood (nbhd for short)of a fuzzy set A i� there exists an open fuzzy set O (∈T ) such that A⊂ O⊂ U .

De�nition 2.9 (Chang [1], De�nition 3.1). A sequence of fuzzy sets; say {An; n=1; 2; : : :}; is eventuallycontained in a fuzzy set A i� there exists a positive integer m such that whenever n¿m; then An⊂ A. If{An; n=1; 2; : : : ; } is a sequence of fuzzy sets in FTS (X; T ), then we say that this sequence converges to a


fuzzy set A i� it is eventually contained in each nbhd of A (i.e. if B is any nbhd of A; there is a positiveinteger m such that whenever n¿m; An⊂ B:)Now we let X =R, T = TS in De�nition 2.6, it will satisfy all conditions of De�nition 2.6. Also

De�nitions 2.8 and 2.9 hold for X =R and T = TS .

Proposition 2.5. Let A1 ⊂ A2 ⊂ · · · ⊂ An ⊂ · · · ⊂ A be a sequence of increasing fuzzy sets in FS . Iflimn→∞ �An(x)= �A (x); ∀x; then the sequence {An; n=1; 2; : : :} will converge to A and this is denotedby A= limn→∞ An=

⋃∞n=1 An.

Proof. It is obvious from De�nition 2.9.

De�nition 2.10. (a) Let D(k) =⋃�∈[0;1] [D

(k)l (�)�; D

(k)r (�)�]∈FN ∪FI (1), k =1; 2; : : : , then the sequence of

fuzzy sets (in FN ∪FI (1)) {D(k); k =1; 2; : : :} converges to

D=⋃

�∈[0;1][Dl(�)�; Dr(�)�]∈FN ∪FI (1)

i� ∀�∈ [0; 1],limk→∞

[D(k)l (�)�; D(k)r (�)�] = [Dl(�)�; Dr(�)�]: (3)

It will be denoted by limk→∞ D(k) = D:(b) Let G(k; q) ∈FN ∪FI (1), E(q) ∈FN ∪FI (1), q=1; 2; : : : ; mk , k =1; 2; : : : ; and

G(k) =mk⋃q=1

G(k; q) =⋃

�∈[0;1]

mk⋃q=1

[G(k; q)∗l (�)�; G(k; q)∗r (�)�]∈FS; k =1; 2; : : : :

Suppose E(q) ∈FN ∪FI (1), q=1; 2; : : : ; m and

E=m⋃q=1

E(q) =⋃

�∈[0;1]

m⋃q=1

[E(q)∗l (�)�; E(q)∗r (�)�]∈FS; k =1; 2; : : : ;

where m= limk→∞mk . We de�ne the convergence of fuzzy sets as following:(b-1) Sequence of fuzzy sets (in FS) {G(k); k =1; 2; : : :} converges to E i� ∀�∈ [0; 1],

limk→∞

mk⋃j=1

[G(k; j)∗l (�)�; G(k; j)∗r (�)�] =m⋃q=1

[E(q)∗l (�)�; E(q)∗r (�)�]: (3′)

It will be denoted by limk→∞ G(k) = E:(b-2) Sequence of fuzzy sets

n⋃k=1

G(k) =n⋃k=1

mk⋃q=1

⋃�∈[0;1]

[G(k; q)∗l (�)�; G(k; q)∗r (�)�] =⋃

�∈[0;1]

n⋃k=1

mk⋃q=1

[G(k; q)∗l (�)�; G(k; q)∗r (�)�]; n=1; 2; : : :

converges to E=⋃mq=1 E

(q) =⋃�∈[0;1]

⋃mq=1 [E

(q)∗l (�)�; E

(q)∗r (�)�] i� ∀�∈ [0; 1],

limk→∞

n⋃k=1

mk⋃j=1

[G(k; j)∗l (�)�; G(k; j)∗r (�)�] =m⋃q=1

[E(q)∗l (�)�; E(q)∗r (�)�]: (3′′)


Remark 2.7. (a) Under equality in (3), for each �∈ [0; 1], by the one–one correspondence between the level� fuzzy interval [D(k)l (�)�; D

(k)r (�)�] and the crisp interval [D

(k)l (�); D

(k)r (�)], we have some results as follows:

(i) 0¡�¡ 12 [Dr(�)− Dl(�)]⇒ (Dl(�) + �; Dr(�)− �)⊂ [D(k)l (�); D(k)r (�)]⊂ (Dl(�)− �; Dr(�) + �);

(ii) �¿ 12 [Dr(�)− Dl(�)]⇒ (Dr(�)− �; Dl(�) + �)⊂ [D(k)l (�); D(k)r (�)]⊂ (Dl(�)− �; Dr(�) + �);

(iii) ((Dr(�)− �)�; (Dl(�) + �)�); ((Dl(�) + �)�; (Dr(�)− �)�); ((Dl(�)− �)�; (Dr(�) + �)�)∈A.(b) For every k =1; 2; : : : ; set mk =1 in De�nition 2.10(b-1), then m=1 and D(k) = G(k;1) ∈FN ∪FI (1).

Thus (3′) and (3) are the same. Therefore De�nition 2.10(a) is a special case of De�nition 2.10(b-1) form=1, mk =1, ∀k and G(k;1)∗l (�)=G(k;1)l (�), G(k;1)∗r (�)=G(k;1)r (�), E(k;1)∗l (�)=E(k;1)l (�), E(k;1)∗r (�)=E(k;1)r (�).

Proposition 2.6. (a) Let D(k) =⋃�∈[0;1] [D

(k)l (�)�; D

(k)r (�)�]∈FN ∪FI (1); k =1; 2; : : : : Then sequence of fuzzy

sets {D(k); k =1; 2; : : :} converges to D= ⋃�∈[0;1] [Dl(�)�; Dr(�)�]∈FN ∪FI (1); i� ∀�∈ [0; 1]; sequence of realintervals {D(k)(�)= [D(k)l (�); D(k)r (�)]; k =1; 2; : : :} converges to real interval D(�)= [Dl(�); Dr(�)] i� ∀�¿0;∀�∈ [0; 1]; there is a positive integer N such that whenever k¿N;

Dl(�)− �¡D(k)l (�)¡Dl(�) + � and Dr(�)− �¡D(k)r (�)¡Dr(�) + �:

It will be denoted by

limn→∞ D

n= D i� limk→∞

D(k)(�)=D(�); ∀�∈ [0; 1]

(b) Let G(k; q) ∈FN ∪FI (1); E(q) ∈FN ∪FI (1); q=1; 2; : : : ; mk ; mk¿1; k =1; 2; : : : ; and

G(k) =mk⋃q=1

G(k; q) =⋃

�∈[0;1]

mk⋃q=1

[G(k; q)∗l (�)�; G(k; q)∗r (�)�]∈FS; k =1; 2; : : : :

Suppose E(q) ∈FN ∪FI (1); q=1; 2; : : : ; m; and

E=m⋃q=1

E(q) =⋃

�∈[0;1]

m⋃q=1

[E(q)∗l (�)�; E(q)∗r (�)�]∈FS; k =1; 2; : : : ;

where m= limk→∞mk . We have the following results:(b-1) Sequence of fuzzy set {G(k) ∈FS ; k =1; 2; : : :} converge to E i� for each �∈ [0; 1]

limk→∞

mk∑q=1

(G(k; q)∗r (�)− G(k; q)∗l (�))=m∑q=1

(E(q)∗r (�)− E(q)∗l (�)):


limk→∞

G(k) = E i� limk→∞

mk∑q=1

(G(k; q)∗r (�)− G(k; q)∗l (�))=m∑q=1

(E(q)∗r (�)− E(q)∗l (�)):

(b-2) Sequence of fuzzy set

n⋃k=1

G(k) =⋃

�∈[0;1]

n⋃k=1

mk⋃q=1

[G(k; q)∗l (�)�; G(k; q)∗r (�)�]; n=1; 2; : : :


converges to E=⋃mq=1 E

(q) =⋃�∈[0;1]

⋃mq=1 [E

(q)∗l (�)�; E

(q)∗r (�)�] i� for each �∈ [0; 1]

limn→∞

n∑k=1

mk∑q=1

(G(k; q)∗r (�)− G(k; q)∗l (�))=m∑q=1

(E(q)∗r (�)− E(q)∗l (�)):


limk→∞

n⋃k=1

G(k) = E i� limn→∞

n∑k=1

mk∑q=1

(G(k; q)∗r (�)− G(k; q)∗l (�))=m∑q=1

(E(q)∗r (�)− E(q)∗l (�)):

Proof. (a) For every �∈ [0; 1],[D(k)l (�)�; D

(k)r (�)�] ↔ [D(k)l (�); D

(k)r (�)]; ∀k =1; 2; : : :

[Dl(�)�; Dr(�)�] ↔ [Dl(�); Dr(�)]

are one–one mapping, then we obtain (a).(b-1) For every �∈ [0; 1],[G(k; j)∗l (�)�; G(k; j)∗r (�)�] ↔ [G(k; j)∗l (�); G(k; j)∗r (�)]; ∀j=1; 2; : : : ; mk ; ∀k =1; 2; : : :

[E(q)∗l (�)�; E(q)∗r (�)�] ↔ [E(q)∗l (�); E

(q)∗r (�)]; ∀q=1; 2; : : : ; m

are one–one mapping. Where, by (b) of Remark 2.3,

[G(k; j)∗l (�)�; G(k; j)∗r (�)�]; ∀j=1; 2; : : : ; mk ; ∀k =1; 2; : : :

are nonoverlapping; also, [E(q)∗l (�)�; E(q)∗r (�)�]; ∀q=1; 2; : : : ; m are nonoverlapping. Therefore

limk→∞

mk∑q=1

(G(k; q)∗r (�)− G(k; q)∗l (�))=m∑q=1

(E(q)∗r (�)− E(q)∗l (�)) i�

limk→∞

mk⋃j=1

[G(k; j)∗l (�)�; G(k; j)∗r (�)�] =m⋃q=1

[E(q)∗l (�)�; E(q)∗r (�)�]; ∀�∈ [0; 1]:

By (b-1) of De�nition 2.10, we obtain (b-1).(b-2) Similar to (b-1).

Proposition 2.7. Suppose(1) D(k); E(q) ∈FN ∪FI (1); k =1; 2; : : : ; q=1; 2; : : : ; m and D=

⋃mq=1 E

(q) ∈FS .(2) D(1)⊂ D(2)⊂ · · · ⊂ D(n)⊂ · · ·⊂ D be a sequence of increasing fuzzy sets and limn→∞ �D(n) (x)=

�D(x); ∀x.Then M∗(limn→∞ D(n))= limn→∞ M∗(D(n)).

Proof. Since D(1)⊂ D(2)⊂ · · · ⊂ D(n)⊂ · · ·⊂ D and limn→∞ �D(n) (x)= �D(x), ∀x, by Proposition 2.5, it isobvious that sequence of fuzzy sets D(n) =

⋃�∈[0;1] [D

(n)l (�)�; D

(n)r (�)�]; n=1; 2; : : : ; converges to D. Thus

D= limn→∞ D(n) =⋃∞n=1 D

(n) ∈FS . Also, by Remark 2.3(b-3) we have D=⋃mq=1 E

(q) =⋃�∈[0;1]

⋃mq=1


[E(q)∗l (�)�; E(q)∗r (�)�]: By (b-1) of Proposition 2.6, for mk =1, ∀k and D(n)∗l (�)=D

(n)l (�), D

(n)∗r (�)=D

(n)r (�), ∀n,

∀�∈ [0; 1], we obtain

limn→∞[D

(n)r (�)− D(n)l (�)]=

m∑q=1

(E(q)∗r (�)− E(q)∗l (�)); ∀�∈ [0; 1]: (4)

Furthermore, for every n=2; 3; : : :

D(n−1)⊂ D(n)⇒D(n)l (�)6D(n−1)l (�)6D(n−1)r (�)6D(n)r (�)

⇒{D(n)r (�)− D(n)l (�) | n=1; 2; : : :} is an increasing sequence; ∀�∈ [0; 1]:Therefore, by Eq. (4), De�nitions 2.4, 2.5 and the integral theory we have

M∗(limn→∞ D

(n))=M∗(D)=

m∑q=1

∫ 1

0(E(q)∗r (�)− E(q)∗l (�)) d�

= limn→∞

∫ 1

0[D(n)r (�)− D(n)l (�)] d�= lim

n→∞M∗(D(n)):

Example 2.2. Let A(n); n=1; 2; : : : ; and let A be fuzzy numbers with membership functions

�A(n) (x)=

x − 6− 1=2n2− 1=2n ; 6 + 1=2n6 x68;

12− 1=3n − x4− 1=3n ; 86 x612− 1=3n;

0; elsewhere;

�A (x)=

x − 62; 66 x68;

12− x4

; 86 x612;

0; elsewhere;

respectively. Then A(1)⊂ A(2)⊂ · · · ⊂ A(n)⊂ · · · ⊂ A and limn→∞ �A(n) (x)= �A (x); ∀x. By Proposition 2.5 wehave A= limn→∞ A(n). Furthermore, by the decomposition theory we have

A (n) =⋃

�∈[0;1][(6 + 1=2n + (2− 1=2n)�)�; (12− 1=3n − (4− 1=3n)�)�];

A=⋃

�∈[0;1][(6 + 2�)�; (12− 4�)�]:

Thus,

M∗(A)=∫ 1

0(12− 4�− 6− 2�) d�=3;

M∗(A (n)) =∫ 1

0

[12− 1

3n−(4− 1

3n

)�−

(6 +

12n

)−(2− 1

2n

)�]d�

= 3−(12n+13n

)/2; n=1; 2; : : : :

Therefore

limn→∞M∗(A(n))=M∗(A)=M∗

(limn→∞ A

(n)):


Proposition 2.8. Let⋃∞k=1 D

(k) ∈FS; where E(q); D(k) ∈FN ∪FI (1); k =1; 2; : : : ; q=1; 2; : : : ; m. If D(k); k =1; 2; : : : ; are nonoverlapping fuzzy sets such that

limn→∞ �∪

nk=1D(k)(x)= �D(x); ∀x and

n⋃k=1

D(k)⊂ D=m⋃q=1

E(q); ∀n=1; 2; : : : ;

then M∗(⋃∞k=1 D

(k))=∑∞

k=1M∗(D(k)):

Proof. Since

D(1)⊂2⋃k=1

D(k)⊂ · · · ⊂m⋃k=1

D(k)⊂ · · · ⊂ D and limn→∞ �∪nk=1D(k) (x)= �D(x); ∀x

by Proposition 2.5 we have D= limn→∞⋃nk=1 D

(k) =⋃∞k=1 D

(k) ∈FS . Also, D=⋃mq=1 E

(q) =⋃�∈[0;1]

⋃mq=1

[E(q)∗l (�)�; E(q)∗r (�)�]: Since D(k); k =1; 2; : : : ; are nonoverlapping fuzzy sets, by (b-2) of Proposition 2.6 for

mk =1; ∀k; ∀�∈ [0; 1]; D(k)∗l (�)=D(k)l (�); D

(k)∗r (�)=D

(k)r (�); then we have

limn→∞

n∑k=1

[D(k)r (�)− D(k)l (�)]=m∑q=1

(E(q)∗r (�)− E(q)∗l (�)); ∀�∈ [0; 1]: (5)

Moreover,∑n

k=1[D(k)r (�) − D(k)l (�)] is an increasing sequence for n which satis�es Eq. (5). Therefore, by

Eq. (5), De�nition 2.4, 2.5 and the integral theory, we obtain

M∗( ∞⋃k=1

D(k))=M∗(D)=

m∑q=1

∫ 1

0[E(q)∗r (�)− E(q)∗l (�)] d�

= limn→∞

n∑k=1

∫ 1

0[D(k)r (�)− D(k)l (�)] d�= lim

n→∞

n∑k=1

M∗(D(k))=∞∑k=1

M∗(D(k)):

Remark 2.8. M∗(⋃∞k=1 D

(k))=∑∞

k=1M∗(D(k)) is positive �nite or in�nity.

Proposition 2.9. Let D=⋃mj=1 E

( j); D(k) =⋃mkq=1 D

(k; q) ∈FS; where D(k; q); E( j) ∈FN ∪FI (1); q=1; 2; : : : ; mk ;k =1; 2; : : : ; j=1; 2; : : : ; m. If D(k; q); ∀k; ∀q are nonoverlapping fuzzy sets such that

limn→∞ �∪nk=1D(k) (z)= �D(z); ∀z and

n⋃k=1

D(k)⊂ D; ∀n=1; 2; : : : ;


(k))=∑∞

k=1M∗(D(k)):

Proof. Since

D(1)⊂2⋃k=1

D(k)⊂ · · · ⊂n⋃k=1

D(k)⊂ · · · ⊂ D and limn→∞ �∪nk=1D(k) (z)= �D(z); ∀z;

by Proposition 2.5 we have

D= limn→∞

n⋃k=1

D(k) =∞⋃k=1

D(k) ∈FS: (6)


Thus, by (6) and the decomposition theory, ∀n=1; 2; : : : ;n⋃k=1

D(k) =n⋃k=1

mk⋃q=1

D(k; q) =n⋃k=1

mk⋃q=1

⋃�∈[0;1]

[D(k; q)∗l (�)�; D(k; q)∗r (�)�]

=⋃

�∈[0;1]

n⋃k=1

mk⋃q=1

[D(k; q)∗l (�)�; D(k; q)∗r (�)�]:

And the sequence⋃nk=1 D

(k); n=1; 2; : : : ; converges to D=⋃�∈[0;1]

⋃mq=1 [E

(k; q)∗l (�)�; E

(k; q)∗r (�)�]: Since D(k; q);

k =1; 2; : : : ; q=1; 2; : : : ; mk ; are nonoverlapping fuzzy sets, by (b-2) of Proposition 2.6 for D(k; q)∗l (�)=D(k; q)l (�);

D(k; q)∗r (�)=D(k; q)r (�); we have

limn→∞

n∑k=1

mk∑q=1

(D(k; q)r (�)− D(k; q)l (�))=m∑q=1

(E(k; q)∗r (�)− E(k; q)∗l (�)); ∀�∈ [0; 1]: (7)

Moreover,∑n

k=1

∑mkq=1[D

(k; q)r (�)−D(k; q)l (�)] is an increasing sequence for n which satis�es Eq. (7). Therefore,

by Eqs. (6) and (7), De�nitions 2.4 and 2.5, Proposition 2.2 and the integral theory, we obtain

M∗( ∞⋃k=1

D(k))=

m∑q=1

∫ 1

0[E(q)∗r (�)− E(q)∗l (�)] d�= lim

n→∞

n∑k=1

mk∑q=1

∫ 1

0[D(k; q)r (�)− D(k; q)l (�)] d�

= limn→∞

n∑k=1

mk∑q=1

M∗(D(k; q))=∞∑k=1

M∗

mk⋃q=1

D(k; q)

= ∞∑

k=1

M∗(D(k)):

Proposition 2.10. Let D(1)⊂ D(2)⊂ · · · ⊂ D(n)⊂ · · ·⊂ D be a sequence of increasing fuzzy sets in FS; andlimn→∞ �D(n) (x)= �D(x); ∀x; then M∗(limn→∞ D(n))= limn→∞ M∗(D(n)).

Proof. Similar to the treatment in Propositions 2.7 and 2.9.

Remark 2.9. Proposition 2.7 is a special case of Proposition 2.10 if D(k) ∈FN ∪FI (1); ∀k:

Basing on De�nitions 2.4 and 2.5 and Propositions 2.3 and 2.9, we give a summary of the fuzzy measureM∗ as follows:(1) Let D=

⋃nk=1 D

(k) ∈FS; where D(k) ∈FN ∪FI (1); k =1; 2; : : : ; n; thenM∗(D)¿0: (8)

(2) Let D=⋃nk=1 D

(k); E=⋃mj=1 E

( j) ∈FS; and D⊂ E; where D(k); E( j) ∈FN ∪FI (1); k =1; 2; : : : ; n; j=1; 2; : : : ; m; then

M∗(D)6M∗(E): (9)

(3) Let D(k) =⋃mkq=1 E

(k; q) ∈FS; where E(k; q) ∈FN ∪FI (1); q=1; 2; : : : ; mk ; k =1; 2; : : : be nonoverlappingfuzzy sets. If

limn→∞ �∪nk=1D(k) (z)= �D(z); ∀z and

n⋃k=1

D(k)⊂ D; ∀n=1; 2; : : : ;


then

M∗( ∞⋃k=1

D(k))=

∞∑k=1

M∗(D(k)): (10)

(4) M∗(�)= 0 and M∗(R)=∞.

Thus we have shown that M∗ satis�es the condition of a measure de�ned by Halmos [2] in fuzzy sensebut does not satisfy ‘M∗(R)= 1’ which is a condition of fuzzy measure de�ned by Zimmermann [4] in fuzzysense (see De�nition 3.1 in this paper and B is replaced by FS). Furthermore,1. M∗(I(a1; b1))= b− a=M(I(a; b)):2. If −∞¡a(k)¡b(k)¡∞ and I(a(k)1 ; b

(k)1 ); k =1; 2; : : : ; n are nonoverlapping intervals, then

M∗(

n⋃k=1

I(a(k)1 ; b(k)1 )

)=

n∑k=1

(b(k) − a(k))=M

(n⋃k=1

I(a(k); b(k))

):

3. M∗(R)=M(R)=∞.Therefore M is a special case of M∗ and M∗ is called an extended fuzzy measure.

Proposition 2.11. Let D(k); E(q) ∈FN ∪FI (1); k =1; 2; : : : ; q=1; 2; : : : ; m; andn⋃k=1

D(k)⊂ D=m⋃q=1

E(q); ∀n=1; 2; : : :

such that

limn→∞ �∪nk=1D(k) (x)= �D(x); ∀x


(k))6∑∞

k=1M∗(D(k)):

Proof. Similar to the proof of Proposition 2.8, we have limn→∞⋃nk=1 D

(k) =⋃∞k=1 D

(k) = D: By (b-3) ofRemark 2.3 we have

n⋃k=1

D(k) =⋃

�∈[0;1]

n⋃k=1

[D(k)∗l (�)�; D(k)∗r (�)�] and D=

⋃�∈[0;1]

m⋃q=1

[E(q)∗l (�)�; E(q)∗r (�)�]:

By (b-2) of Proposition 2.6, for mk =1; ∀k; we have

limn→∞

n∑k=1

(D(k)∗r (�)− D(k)∗l (�))=m∑q=1

(E(q)∗r (�)− E(q)∗l (�)); ∀�∈ [0; 1]:

Therefore, similar to the treatment in Proposition 2.8, by De�nitions 2.4 and 2.5 and the integral theory, weobtain

M∗( ∞⋃k=1

D(k))=M∗(D)=

m∑q=1

∫ 1

0[E(q)∗r (�)− E(q)∗l (�)] d�

= limn→∞

n∑k=1

∫ 1

0[D(k)∗r (�)− D(k)∗l (�)] d�


6 limn→∞

n∑k=1

∫ 1

0[D(k)r (�)− D(k)l (�)] d�

= limn→∞

n∑k=1

M∗(D(k))=∞∑k=1

M∗(D(k)):

3. Fuzzy measure in Zimmermann’s sense

De�nition 3.1 (Zimmermann [4], De�nition 4.1). Let B be a Borel �eld of the arbitrary (universal) set X .A set function g de�ned on B which has the following properties is called a fuzzy measure(a) g(�)= 0; g(X )= 1:(b) If A; B∈B and A⊂B; then g(A)6 g(B).(c) If Ai ∈B; ∀i=1; 2; : : : ; and A1⊆A2⊆ · · · ; then limn→∞ g(An)= g(limn→∞ An):If p; q∈R; p¡q; let X =(p; q)= {x |p¡x¡q; x∈R} be a �nite open interval on R. De�ne

F1 =⋃

�∈[0;1]{a� | a∈X }; corresponding to Fp;

F2(�)= {(a�; b�) | a¡b; a; b∈X }; corresponding to F∗I (�); �∈ [0; 1];

F3(�)= {[a�; b�] | a¡b; a; b∈X }; corresponding to FI (�); �∈ [0; 1];F3 =

⋃�∈[0;1]

F2(�)∪F3(�); corresponding to FI ;

F4 = {D | D∈FN ∩X }; corresponding to FN :Corresponding to FS; let FX be a family of fuzzy sets contained in F4 ∪F3(1) and their countable unions,i.e. FX = {D=⋃n

j=1 D( j) | D( j) ∈F4 ∪F3(1); j=1; 2; : : : ; n; n is any integer or ∞}: It is clear that FX ⊂FS . In

order to coincide the condition g(X )= 1; we will normalize M∗ as follows:

De�nition 3.2. Let D be a fuzzy set in F4 ∪F3(1) and [Dl(�); Dr(�)] denote the �-cut of D; then

M′(D)=1

q− p∫ 1

0[Dr(�)− Dl(�)] d�

is called the fuzzy measure of D.

Remark 3.1. It is obvious that 06M′(D)61; M′(D)=M∗(D)=(q− p) and M′(X )= 1.Similar to De�nition 2.5, let D(k) ∈F4 ∪F3(1); k =1; 2; : : : ; n; then

M′(

n⋃k=1

D(k))=

1q− pM∗

(n⋃k=1

D(k))

Corresponding to F; let F∗ be a family of fuzzy sets contained in F1 ∪FX ∪F3 and their countable unions;obviously, F∗ ⊂F . Let

AX = {(a�; b�); (a�; q); (p; b�); (p; q) | ∀a¡b; a; b∈X; �∈ [0; 1]} (⊂F∗):

Similar to De�nition 2.7, we de�ne the open fuzzy set in F∗ as follows:


De�nition 3.3. A fuzzy set O in F∗ is an open fuzzy set i� for each x�⊂ O; there exists I ∈AX such thatx�⊂ I ⊂ O.

Let TX be the family of all open sets in F∗. Obviously TX ⊂TS . Similar to the treatment in Proposition 2.4,it can be shown that TX is a fuzzy topology and (X; TX ) is a fuzzy topological space. Also, many propositionsin Section 2 hold if FN ∪FI (1); FS ; M∗ are replaced by F4 ∪F3(1); FX ; M′ respectively. In De�nition 3.1,B is replaced by FX . Corresponding to Eqs. (8)–(10) and Proposition 2.3, 2.10 and 2.11, we give a summaryas follows:(1) 06M′(D)61 and 06M′(

⋃nj=1 D

( j))61; ∀D; D( j) ∈FX ; j=1; 2; : : : ; n.(2) M′(�)= 0 and M′(X )= 1 (condition (a) in De�nition 3.1).(3) If D=

⋃nj=1 D

( j); G=⋃mk=1 G

(k) ∈FX ; and D⊆ G; where D( j); G(k) ∈F4 ∪F3(1); j=1; 2; : : : ; n; k =1; 2; : : : ; m. Then M′(D)6M′(G) (condition (b) in De�nition 3.1).(4) If D( j) ∈FX ; j=1; 2; : : : ; n and D(1)⊆ D(2)⊆ · · · ⊆ D(n)⊆ · · · ⊆ D; then limn→∞ M′(D(n))=

M′ (limn→∞ D(n)) (condition (c) in De�nition 3.1).(5) If D( j) ∈FX ; j=1; 2; : : : ; n; then M′(

⋃nj=1 D

( j))6∑n

j=1M∗(D( j)) and “=” holds if D( j); j=1; 2; : : : ; n;

are nonoverlapping.(6) If D( j) ∈FX ; j=1; 2; : : : ; and limn→∞ �∪nj=1D( j) (x)= �D(x); ∀x; where D=

⋃∞j=1 D

( j); then M′(⋃∞j=1 D

( j))

6∑∞

j=1M′(D( j)) and “=” holds if D( j); j=1; 2; : : : ; n, are nonoverlapping.

Therefore M′ satis�es all conditions of De�nition 3.1 (see Zimmermann [4] for fuzzy sense) where B isreplaced by FX . Since

M′([a; b])=b− aq− p 6= b− a=M([a; b]) if p¡a¡b¡q;

M′ is not an extension of the crisp measure M.

Example 3.1. Let X =(p; q); p¡q; p; q∈R. For every triangular fuzzy number A=(a; b; c) in X; p¡a¡b¡c¡q; then the membership function of A is given by

�A (x)=

x − ab− a ; a6 x6b;

c − xc − b ; b6 x6c;

0; p¡x¡a or c¡x¡q:

By Example 2.1, M∗(A)= 12 (c − a); thus

M′(A)=1

q− pM∗(A)=c − a

2(q− p) :

Consider the uniform distribution over X =(p; q). Then the probability density function of the uniform dis-tribution over X =(p; q) is given by

f(x)=

1q− p; p6 x6q;

0; elsewhere.

Therefore the fuzzy probability of A can be found as follows:

P(A)=∫ ∞

−∞�A (x)f(x) dx=

1q− p

∫ q

p�A (x) dx=

c − a2(q− p) =M′(A):


Let Ak =(ak ; bk ; ck); p¡ak¡bk¡ck¡q; k =1; 2; : : : ; n are nonoverlapping fuzzy numbers in X . Consider theuniform distribution over X =(p; q); by Proposition 2.2 we have M∗(

⋃nk=1Ak)=

∑nk=1M

∗(Ak). Thus

M′(

n⋃k=1

Ak

)=

n∑k=1

M′(Ak)=n∑k=1

ck − ak2(q− p) :

From the property of fuzzy probability, we have

P

(n⋃k=1

Ak

)=

n∑k=1

P(Ak)=n∑k=1

ck − ak2(q− p) =M′

(n⋃k=1

Ak

):

Similarly, let Ak =(ak ; bk ; ck); p¡ak¡bk¡ck¡q; k =1; 2; : : : ; are nonoverlapping fuzzy numbers in X . Con-sider the uniform distribution over X =(p; q); by Proposition 2.8 we have M∗(

⋃∞k=1 Ak)=

∑∞k=1M

∗(Ak).Thus

M′( ∞⋃k=1

Ak

)=

∞∑k=1

M′(Ak)=∞∑k=1

ck − ak2(q− p) :

Also,

P

( ∞⋃k=1

Ak

)=

∞∑k=1

P(Ak)=∞∑k=1

ck − ak2(q− p) =M′

( ∞⋃k=1

Ak

):

Specially, for level 1 fuzzy interval [a1; b1] in X; p¡a¡b¡q; it is obvious that M∗([a1; b1])= b − a;M′([a1; b1])= (b− a)=(q− p). Also,

P([a1; b1])=∫ ∞

−∞�[a1 ; b1](x)f(x) dx=

1q− p

∫ q

p�[a1 ; b1](x) dx=

b− aq− p =M′([a1; b1]):

Similarly, Let [a(k)1 ; b(k)1 ]; p¡a

(k)1 ¡b

(k)1 ¡q; k =1; 2; : : : ; are nonoverlapping level 1 fuzzy intervals in X .

Consider the uniform distribution over X =(p; q); we have

M′(

n⋃k=1

[a(k)1 ; b(k)1 ]

)=

n∑k=1

M′([a(k)1 ; b(k)1 ])=

n∑k=1

[b(k) − a(k)]q− p

=n∑k=1

P([a(k)1 ; b(k)1 ])= P

(n⋃k=1

[a(k)1 ; b(k)1 ]

)

and

M′( ∞⋃k=1

[a(k)1 ; b(k)1 ]

)=

∞∑k=1

M′([a(k)1 ; b(k)1 ])=

∞∑k=1

[b(k) − a(k)]q− p

=∞∑k=1

P([a(k)1 ; b(k)1 ])= P

( ∞⋃k=1

[a(k)1 ; b(k)1 ]

):

4. Conclusion

On the real line, for fuzzy sets in FS; we de�ne an extended fuzzy measure M∗ which is an extension ofthe crisp measure M. Since M(R)=∞; then M∗(R)=∞ and hence M∗ is not a fuzzy measure de�ned by


Zimmermann [4] (in [4], the measure of R should be one). In order to coincide the de�nition of Zimmermann[4], we normalize M∗ to obtain M′. M′ is a fuzzy measure on FX and satis�es the conditions of De�nition3.1 in Zimmermann [4] (in fuzzy sense) when B is replaced by FX . But M∗ is not an extension of the crispmeasure M.

References

[1] C.L. Chang, Fuzzy topological space, J. Math. Anal. Appl. 41 (1968) 182–190.[2] P.R. Halmos, Measure Theory, Van Nostrand, New York, 4th Printing.[3] A. Kaufmann, M.M. Gupta, Introduction to Fuzzy Arithmetic Theory and Applications, Van Nostrand, New York, 1991.[4] H.J. Zimmermann, Fuzzy Set Theory and Its Applications, 2nd Revised ed., Kluwer Academic Publishers, Boston=Dordrecht=London,

1991.

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Fuzzy measure based on decomposition theory