On the Fuzzy Measures and the Measures of Fuzziness for L-Fuzzy Sets

Copyright © IFAC Fuzzy Infonnation MalYille, France, 1983

ON THE FUZZY MEASURES AND THE MEASURES OF FUZZINESS FOR L-FUZZY SETS Zi-Xiao Wang Department of Mathematics, Northeast Normal University, Changchun, jilin, People's Republic of China

Abstract.

This paper is based

o~

the results of De Luca and Termini(1.2)

and Wang (6,7.8) . It includes the following three parts. (1) We extend the

fuzz~'

integrals taking value in the unit inteval (0, 1) to a fuzzy

integrals taking value in a lattice. When the lattice is total order. it is given that one group equivalent conditions such that the L-fuzzy measure can be represented as a L-fuzzy integral. (2) We extend the measure of

fu~ziness

for general fuzzy sets to the measure of fuzziness

for L-fuzzy sets , where L is a complement lattice. (3)

We investigate

the L-ruzzy inteeral representation for the measure of fuzziness on the L-fuzzy sets. Keywords

1

L-Fuzzy Measure

L-Fuzzy Integrals

L-Measure of Fuzziness.

TNTHOlJUCT10N

In 1968, Zadeh (11) first suggested the fuzzy

probability. In 1?72. De Luca :>.nd Termini

probability on fuzzy events and peoneared G.c\v e three princirles as a measure of fuzzi ·

the consideration of fuzzy measu re. Klement

ne~s.

et a1. (4) Harked important 1'e3u1 t ~; at this

This paper has itself generated a

direction. 'f heir work's characteristic is

number of papers in this arAa

that their tool i s the classical Lebesgqe

sugg ested an intuitive definition of the

integral. In 1974, Sugeno (5) suggested a

fuzzille"'" anci Yagor (10) and Dc Luc:>. a1,d

complete theory of fuzzy integral which is

Terlliini (2) investigated the measure o~

indepenJent of Lebesgue meas ure and integral.

These authors cOLsidor

L-fuzzy sets on the universal space is a

dation of Lebesgue i ntegral. Therefore, in

lattice. Thus their rather conSider the

(6. 7 ). we supplemented the fuzzy addi tivi ty r.

on lattice

the fuzziness of a la ttice because of the

Generally, Sugeno's fuzzy integral is exten-

for the fuzzy

~uzzine~~

YBEAr (g)

fuzziness of a lattice with the betweenness

easure such tha t the measure

of elements of a lattice than investigate

].s really a ex tenda tion of a " similarity" At the samel tillJe, vie g8-ve a new definition

the fuzziness on L-fuzzy sets.

which is eCiuivahnn wl ti; Sugeno' s fuzzy

In this paper, first of all. we conSider

integral ano. is !JlOre c:Ol.vellie,1t. This is

the fuzzy integral taking value in a lattice

anathel' deriction on the investigation of

~nd

fuzz; measure, it is more intuitive than

we investigate immed1atly the fuzziness of

the first deriction on fuzzy measure.

L-fuzzy sets. Here, we require the lattice

By a measure of

L is a metric lattice. At the Same time. an

fUt;.;ll1e~$,

we ,Lean a measur·

.e call it as L-fuzzy integral, Next,

axiomatic definition of the measure (If fuzzi-

men t of the fuzz:i grade for a fuzzy set. J t

ness for L-fuzzy sets is given, Finally, 341

342

Zi-Xiao Wang

we prove an integral representation of the htx)

measure of fuzziness. 2

=V'~l ~-

Coi.~

j\

.p

Ei (\ Ej =

'x. E i (x»)

( i /:. j)

L-FUZZY INTEGRAL then

In (5) . Sugeno first suggested complete theory of fuzzy integral. In (6,7) , we gave a new definition of the fuzzy integral

i E hlx)

dm

V

=

n

1.= 1

(0(1"

m(E.(I ~

El)

is called L-!u.zzy in tegral of h( x) on E ~ Il.

which is equivalent with Sugeno's integral.

(it) hex) is a general function. we take a

Now. we shall consider the fuzzy integral

monotone incroasing sequence of simple

taking value in a lattice.

functions such that hn(x)~h(X) (uniformly)

Throughout this paper . we suppose that L

then

denotes a complete distributive metric lattice with the maximum element I and the minimum element () . The universll l space is a measurable space ( X.$). where $ is ordinary Eorel field on X.

~ --..L.

1E h n( x)

dm

is called L-fuzzy integral of h( x) on E" ~. PropoSition 2.1

The L-fuzzy integral

defined in the Def. 2.2 e quivalent with the ~o is the

Borel field of the unit inteval (0, ))

By a L-fuzzy measure m(.):

Definition 2.1

h( x) dm = lim

fuzzy integral defined by Sugeno,

L-fuzzy measure

I

'1 E

Proof.

Sugeno's fuzz y integral is defined

we mean that m(· ) satisfies the (s)

following conditions

•

~ E h( x) dm = suplo<. /\ m(l>~"E»)

wh.e re Fo< = {x : h( x ) ~ 0( } , 0 '- 01. ~ 1 . If hex) is a simple function, say, (2 ) I f

A.bEt>. then m(AU

.lJ)~.I1(A)V

m(.tJ)

is a monotone sequence. (3) If {A n I.J n9 1c:.f> ;:>

h( x) =Vi:1 (oI. i "

oil n

RE'ma.rk. (i)

=

lirr m(.'" )

~I.).

i.e. t:f' Ac.B.

A () P

1>

~

jB

wnere Fi =

0(2

~

A.FE~

Uk~l

and

(i

1>

f j)

~ o(n .

V i~1

Coli A m( F / ' E)j

Ek · It folluws that dm

=V~:1

(oIi":;:(E () El)

f E h( x ~

rr: 1 A U B) = :Ti( A ) V m( E)

t h,.,n

•..

h( x) dm =

\::;d E hex)

The condit i on (2) is equivalent to

following condition that H

i

1m

If h( x) is a g eneral fur:ctior:, the pro pOSi-

L - fuzz:' integral

11

( s)

th l'"

m( A) ~ m( A) V m( B) = m( B) • ( i ')

Ei () E j =

So that

n

Th,., COTId i t i on (2) i t:! )'lUE'S tl1,.,

::0noi;Ol"icitv of

~

No lest generality, we assume that

then m( lim A )

1 E~x)J

tion is imrr.cdiate re suI t of the monctonici ty In this section . we suppose that D(· ). L ~

(0, MJ is a distance on L

where M is a

firit p pOSitive real number t'1f'olo£,." ami ~

D

"

is the metric is the Borel field on 1-_

of "lueeno' s fuzzy in tesral ·qr.d L-fuzz/ intebra1s. Now l der.ote the set of all measurable fuz zy sets, and define that

~_

function h( x) is called measurable iff

h-1(c)E~ for all CE~ .

for all A E ~

o

PropOSition 2.1

Tt is c] par' t hqt m(. ) is a fuzzy mE'asurc on

If hex) is measurable. then

there exists a sequence {hn(x)}n~ 1

of

simple functions SUCh that lim h (x) = hex) ( unif
Defini tion 2.2

Let (X,1> .m) b" a fuzzy

n.

If L is also total ord'lr lattice, then we have

Theorem?l measure on

Let iii(.) ,b-+L he a I,-fuzzy

E'

then the following statements

are equivalent.

measure space, h(x) a measurable function ( i)

h( x) is a simple function

\ 1)

m(.) is represented as a L-fuzzy

343

On the Fuzzy Measures and the Measures of Fuzziness integral, i.e. there exists a L-fuzzy

intuitive definition of fuzziness, he asso-

measure m(· ): ~-L such that

ciates fuzziness with the lack of distintion

( i i)

A( x»

m( ( 0<. t\

V

between a proposition and its negation. In

for all A E ~

meA) = -) A( x) dm

(f3 /\ B( x) »)

(lOJ , Yager devoloped his concept to L-

=

fuzzy sets.

= (o<.i\m(A(x») V (j3Am(B(x»)

In ('7, 8 ) , we have pointed out th'l.t

for all o(.'{3 ELand A.B E ~ , (iii) and A e( i v)

ilrto<.AA(x»

=oI.i\m(A\x)

forallotEL

t>, For all 0<. , a ELand A E; D( m(

0<.

m(f3 /\

A (a /\ A) ),

SO:r:8

faults of De Luca and Termini's conditions of fuzziness measurement and have retieved it.

f, . (a /\ A) ) )

lim

In this section, first of al]. we sh'1.Jl give an axiomatic definition for measure

-° a ,. ° me a m(

K( ( a /\ A ), o() { :

/I A ) ~ 0(

11 A)

> 0(

Proof. The procedure of the proof is the

of fuzzlness on ordinary fuzzy sets that t he grades of menbership lie the unit inteval lO, 1) , then we shall extend this aefinition to L-fuzzy sets.

following ti1.at (i)~( ii)~( iii)~(

iv)==>( iii)~ i)

The conditions for measure of fuzziness which De Luca and Terrr.tni suggested in (lJ

i")==}(iii). T,ot 0(. !l.€L and AE$. 1) 01. ;>m(A). By the condition (iv) "nil

I

the connecti vi tv of the inteval (m( a 1\ A( x) )

. 1) it follows that

m(o(/\ (a/\ A( x») is a

constant function on the inteval(m( a /\ A( x» , nand m( '" 1\ (a /\ A( x» = m( a " A( x». thUS. ITi(m(a/\Alx))/\(aAA(x))) =m(aAA(x». (?)o«m(.A). I f a
Indeed. if m(ai\A(x»

= m(b/\A(x». then

m( a J\ A( x » = m( 0( /\ A( x» = m( h J\ A( x ) ) . :"la that KtA, 0<. ) = ° for'1.1J o(.~ (a b) m(o
m( 0< /I A( x» =

< 0(

me 0(, A Al x »

'"

0(

~

!ll(' ):

~-L

(D ) 3

is a fuzzy mea>lure

AE

33

AXIOMATIC DEFINITION OF

In (1) . De Luca and Termini first suggested the measure of fuzziness and they gave three conditions what the measure of fUZ2iness ~atisfy

0

for all A E:

P( X) ,

~

d( A) 11 d(B) if

AeX)

when

A( x) ~ ~

iJ(x) '" A( x)

when

A( xl", X

x)

In (2J ,

iUZZINESS

should

=

~

De

Luca and Termini suggested

further the following definition,

m( A)

for all 3

:j( A)

( D ) d( A) is maximum of d(·) if A(x) 2 for all x E: X ,

at least, His intial paper

introduce in the inteval I = (0, 1)

~s

Let

the partial order relation '", defineo. for all x ,Y E I, as

such that

m( A)

11 , then d(·)

that

x) )

i\

p( X) the pOvler

satisfies at least tne following conditions

B(

( iii) ==* ( i). For any A c ~. define m(A) = on

ma pping dC,): r(x) __ (o,

then

this is impossible. Thus, we have

X,

set of X. A

mlm(o(AA(x»td",I\A(x»)

< m( 0( /\ A(

m( A), then

Let X be an universa l set, :rlX) the set of all ord inary fuzzy sets on

(D ) l

< rn' b i\ At x) ) .

Therefore, if

are the following that

X I~

.Y

So tllat A .. ' B ~ .A. ( x) ",' B( x )

for all XEX

for all A,BE f(X) and a real value mapping d( .) is called a meaSUJ'e of fuzziness, if

, lD ) l

d( A) = 0

(D~)

d(A)

has ge nerated a lot of papers un this area.

A = A',

In particular, in (9) • Yager suggested the

(D )

, 3

ift A i.s a crisp set,

reaches its maximum value iff

d( A) ~ d( B)

if

A '~ B •

Zi-Xiao Wang

344

It is obverous that (D ) l includes (Di), (D2 ) includes (D arid (D ) 3 includes lD we think that the condition

sequence, tnen

( Dl ), (D2 ) and (D ) are more reasonable than 3 the conditions (Di), (D;2.l and (D are.

cept to L-fuzzy sets, let us to

Remark.

(1)

2)

3).

3)

d(·) satisfying the condition (D ), l (D 2 ) and (D ) may be not a voluation on 3 (X).

Defini tiOD 3.2

13n (~,

A mapping

cC·) : L ---I' L is

L --+ L satisfies the following conditions

~n(o, X) =

= { An [0, ~l : AE~}' $2 =

r3 J

called a complementory operation, if cl· ):

Counterexample. Let X = (0, IJ , ~ is the

=

that the lattice 1 is complementary, Here, Dubois in

r

j31

su~pose

the complementary operatinn is defined by

( 2)

Borel field of X.

d( lim A ) = lim d( A ). n n

In order to extend the abovo mentioned con-

1J

AfM,

( 1)

c( 0) = I,

(?)

C fR

( "")

c is involutive ( co c = identify)

strictly decreasing,

tnen$lana 13 2 are Borel fields on [0, ~) and l~, 1) respE'cti.-.

Proposition 3.1

vely.

mentory metric lattice. Any element x and

1et m and m2 are fuzzy measures on ~l and l $2 respectively, so that a mapping

its complementory element x' is comparable.

=

{An

0;,

1) :

Ill(') : mlA) = m_,(A)V m lA) for all A.:!3

2 The following mapping

$ .

is a measure on

d( A) = 2-}1 A( x) where A.

{x:

=

A(

x)

and (x, e', x') ( see

f; .

But d(' )

~

A( x) Bl

B

x)

B( x)

f." '

X,

when

xf (

when

x r: (0, ~)

when

x E ( ~, 1) ,

11

of I"

and

b~

as tne following that

of all

$ -

'$

rn),

ji

°

i f A~

deAl

( F2 )

d( A) ~ d( B)

iff

(F J 3 lF 4 )

dlA) If

=

lA n \

n~

sothata~a'

~

( x' )' ~

X

I

=*

1

x'

E

12

L , then x' E: L , Thus, 2 l b and a' E 12~ b ~ a' ==> b

E

and b E: Ll =+ b"

b, and a

= b

a"'" a

a' b'

Since a< b is

prove that tne condi t~on is necessary,

we assume that any element x such that x< x' , if

if

~A')

~ x

2

impossi ble.

13 A(X)

>

B( x) )

B( x) ~ A( x)

Ll ' then

"'* a' b ~ a' "'* b' ~ a

'1'0

if

B( x) ~ A( x) V A '( x)

Eo

a ) b'

d(·) satisfios tne following conditions (}'r)

2

similarly, if x

A mapping d(·) :~~(o, I)

is called a measure of fuzziness on

l

x ---+ x'

denote the set

measurable fuzzy sets on X.

Definition 3.1

E 11 and b E: L , thus, a' a' 2 b'. It is sufficient to show a = b.

Indeed, if x

Assume tnat tne universal set is a fuzzy

X,D ,

1

By the definitions of 11 and 1 2 , we have x-x' : L ---+ L ; x---+ x' : 1 - 1

,we suggested the concept of

'lleasure space l

x E 1 } and

easily to prove a

$" .

fu~ziness

V {x :

x E L2 \ exist in L and they are denoted by a and b respectively. It is

and ~A) = d(B) = 1, this is to say d(·) is

mea sure 01'

it follows that

V {x:

d(AUB) = 1, d(AOB) = u,

In (8)

{x: x ~ x'}

Ll and 12 are non-empty. By the completeness

that

not a valuation on

)

{x:x~x'~

11 L2

x E (0, ~)

when

°0

(10)

Proof. Put

not a valuation on ~ • lndeed, let A : A( x)

So

di tions that if x < x', then there exist an

} ,

is a measure of fuzziness on 4S

To exist uniquely an element a sucn that a = a' it is necessary and sufficient conelement e f x and x' such that (x, e, x')

M x) I dm

~ ~

1et 1 be a complete comple-

~ ~

B( x) ~ ~~

for all H~ ,

l e i is a monotone

or

then the element a is an element satisfyinC the condition. Assume that 1 denote the lattice satisfying the condition of tiLe pro posi tion 3.1, then ;
By a meaSure of fuzziness

345

On the Fuzzy Measures and the Measures of Fuzziness d( -) :

're X)---+L

we mean that d( -) satisfies

dlA)

=0

d(A)

~

Proof. By h(O)

if A is a crisp set,

heAl xl>

d(B) i f

B( x) VB' ( x) "> A( x) V A•( x)

i f A( x) > a

orB(x»a

At

B( x) ~

x)

if

At x) <; a

and

for all A E:

j ( X) ,

If {An} n~ le 1(X) is a monotone

A )

n

= lim

By A and B satisfy the condition (1 2 ), therefore, h( A( x» "9 h( B( x» , so that deAl "9 d(B). Other conditions are clearly The measure of fuzziness defined in the proposition 4.i possesses the following If d(-) satisfies fuzzy adctitivity. i.e.

d(A ). n

d( ( 0( /\ A( x) ) V (~/\ B( x) ) )

To illustrate the above mentioned definition is reasonable, we have ~roposition

3.2

= for all

The element a is comparable

in the lattice 1 , in

~ ELand A and B Eo

For all E E $ and If

In order to com pare the fuzziness grades element ~

d( cl 1\ A( x) ) V d( ~ 1\ Et x) ) 01. ,

wi th any element x ~ 1. between two

therefore, it is

property.

sequence of 1-fuzzy sets, then d( iim

= h(I) = 0,

for all AEi·

= 0

satisfied.

B(x)(,a d( A) = d( A ')

The mapping d(') is a

measure of fuzziness.

the following conditions (1 ) 1 (12 )

PrOpoSition 4.1

(ii)

0(

~

0(

~

f;,

then

L

a. then d( 0( 1\ E( x » = d(

Ifo«a, then there

exist~

0(' /\

Et x ) )

an

element A( E) such that

(10) , Yager suggested the following

concept. Definition 3.5

when h(e() <

Given two elements x and y

in a lattice, we shall say t hat element x

where A ( .) : $-+L and ,\IAUB)

i s at least as fuzzy a s y. denoted xfy, if

for all

the following two conditions hold. (1)

(y,

( 2 )

(

,B E1J , =

h(O
y, x~, y*)

,;here x" anO y" are the nega tions of x and y respectively. Proposition 3 .3

A

X(A)UA(B)

h(') satisfies that

y*)

Y.,

A(E)

Assume that A and Bare

h(~)

V h(~)

for allo(.~~ a. Conversely, we have Theorem 4.1

Let d(·) be a mea sure of

two 1-fuzzy sets on X. 'l'hen AfB iff A and B

fuzziness satisfying thp fuzzy additivity

satisfy the condition (L ). where A" is the

and the condition

complement of A.

fuzzy measure m(·) : f,-L such that

2

THE MEaSURE OF FUZZINESS FOR L-FUZZY SETS

for all A E Proof.

In th1s section, we shail consider the

dm :

First of all, we define the

following mapping m(·) :

(ii)

L--+L

(iii)

h( 0) = 0

h( u)

=

for all u E L

h(') is strictly monotone increa-

sing on (0, a).

.

tJ

dlA) = ~h(A(X» i)

A = rf.../\ E( x) lA =

h( u·)

that

Now, we shall show that

satisfies the following conditions ( i)

$ __ L

for all A E $ meaSure on

:t ---,>L.

.where the continuous function h(x)

")

It is clear that the mapping ~-) is a fuzzy

}'or all A Et, we define that d( A) = ~ h( A( x»

(

i.

relation between the fuzzy integ ral and the measure of fuzziness.

there exists a

~ h( A) dm

d( A)

4 THE INTEGRAL REPRESENTATION OF

(1~),then

dm

for all AE:j •

where

lh(A(x»

EE$, dm

S

h( 0(.,) /\ m( E)

t

h( o() t\ m( E)

Therefore, it is obverously that

'" ~ a 0(

<

~A)

a

= lA

Zi-xiao Wang

346

by the condition (L ) ii)

VoIi "

A( x) =

and E.Sanchez (Editors), North-

5

Holland Publishing Company, 1982.

Ei\ x) is a simple

71-78

function. 'I'he formula \ *) holds s ince d(' ) possesses the fuzzy additivity. iii)

A(x} is

a sequence taat

{An}

A) x) __ h(

An \

X») ---+

~ h( A( x)

measures of fuzziness, Presented at

of simple func"tions such.

" Procedings of the Second World

a( x; \ uniformly). By the

continuity of the function n(· ) , i t tha t

h(

X)

Conference on Mathematics at Severce th th of Man", June 28 to Jlily 3 ,

follow~

1982, Canary Islands, Spain.

and

lim ~ h( A (x)

dm

n

dm

( 9)

R.R.Yager, On the measure of fuzziness and negation part I : Membership

lim d( A } n

in the uni t in tevB1, In t. J. Gen eral

d( A) •

Remark.

, Fuzzy measures and

(8)

a general function. We take

Systems 5, 221-229.

The function h(' ) may be regarded

as a subjective principle of the measurement

. On the measure of fuzziness

(10)

of fuzzincss, thus, it is very intereflting

and negation

pr oblem how h(') i '; determined .

Information and Control, 44(1980)

~rt

JJ : T,l'tttice.

236-260

REl-'ERl!:NCES (1)

(ID

A. De Luca and 'l'ermini, It d efini tion of a n on -probabilistic entropy in tne setting of fuzzy se ts tneory, Inform. awl Contr ., 20 (1972) 301--312.

( 2)

, On some ale ebrai c as pects of the measure of fuzzine s s, " Fuzzy Information ani' Decision Pro cesses " M.tvl.Gupta and l!:. Sanchez \ Editors ) Nortn-·Hulland

(3)

Publishin~

Company , l':;ltlc .

D. DuboiS, Uncertainty in tran sportation network design: some new techniques, Presented at the IV Euro pean congress on

(4)

0

t il

pera tion re s each.

B.P.Klement et al., E'uzzy probability measures, J. Int.

Sets and

~ uzzy

Sys tems 5(1981) 21-30. (5)

M. Sugeno, '2heory of fnzzy integrals and its applications , Ph.D.Dissertati on, Tokyo Institute of Tech. , 1'::J74.

(6 )

Zi-Xiao Wang,

Note on the fuzzy

mea s ure, It Monthly J. of SCience, Vol. 26, No. 11 \Beijing, Cnina) 961- 964.

(7 )

The structure of fuzzy Lebesgue measures, " Fuzzy Information and Decision

Proc ess e~

", M.M.Gupta

L.A.Zadeh, Probability measure on fuzzy events. J. Math.Anal.Appl. (1968) .