Fuzzy Measure: A Fuzzy Set Theoretic Approach

(1)

Vol. 14, No. 1, 2017, 21-31

ISSN: 2279-087X (P), 2279-0888(online) Published on 16 June 2017

www.researchmathsci.org

DOI: http://dx.doi.org/10.22457/apam.v14n1a3

21

Annals of

Fuzzy Measure: A Fuzzy Set Theoretic Approach

P.S.Sivakumar1 and R.Kavitha2

1

Department of Mathematics, Government Arts College, Udumalpet, Tamil Nadu, India E-mail: sivaommuruga@rediffmail.com

2

Department of Mathematics, Sri Shakthi Institute of Engineering and Technology Coimbatore, Tamil Nadu, India.

2

Corresponding author. E-mail: brahulkavitha@gmail.com

Received 26 May 2017; accepted 12 June2017

Abstract. In this paper, the fuzzy measure which is generalized with underlying continuity concept and some examples are discussed.

Keywords: Fuzzy relation, fuzzy measure, membership function

AMS Mathematics Subject Classification(2010) : 28E10, 03E72, 26E50, 94D05

1. Introduction

Classical measure theory [3] is studied based on additive property. But, in many real time applications like fuzzy logic [2], artificial intelligence, decision making, data mining and re-identification problem etc. require the non-additive measure. In this situation, fuzzy measure was introduced by Sugeno [4] which is a non additive measure. Fuzzy measure was discussed by several authors [4, 6, 7, 8, 9,10,12] in different manner.

The fuzziness in the concept of fuzzy measure introduced by Sugeno is abandoning the additive property. However no membership was assigned for the measure as in the fuzzy set [5, 9,11]. Fuzzy measure, defined in [1], was studied with the membership value.

In this paper, we generalize the definition of fuzzy measure which is defined in [1] with underlying continuity conditions and we give some examples.

2. Fuzzy relation and fuzzy measure

It is clear that if R:A→B is a relation then R is a subset of A × B where A and B are any two sets. Hence R can be fuzzified like fuzzification of sets.

Definition 2.1. (Fuzzy relation)

If A and B are any two sets, then a relation R:A→B is said to be a fuzzy relation if

(i) D(R) = A, where D(R) is the domain of R _and

(2)

P.S.Sivakumar and R.Kavitha

22

Example 2.2. Let A=

{

1, 2,3

}

, B=

{

b, c

}

and the relation R be given by

{

(1, a), (2,a),(2,b),(3,c)

}

R =

The membership function µ_R :R→[0,1] is defined as 7 . 0 )] , 3 [( , 3 . 0 )] , 2 [( , 1 )] , 2 [( )] , 1

[( a = _R a = _R b = _R c =

R µ µ µ

µ

Then R is a fuzzy relation .

Remarks 2.3.

1. For our convenience

µ

R

[(

x

,

y

)]

is simply denotes by

µ

R

( y

x

,

)

.

2. If (x,y)∈ R, we write it as R(x)=y.

3. By

µ

R

(

x

,

y

)

=

a

,

we mean that a is the membership value in [ 0, 1 ] of the

case R(x)= y.That is µ(R(x)=y) = a is simply represented by

.

)

,

(

x

y

a

R

=

µ

Definition 2.4. (Fuzzy measure) Let X be a non empty set, Ω be a non empty class of subsets of X and ( X, Ω) be a measurable space. A fuzzy relation m:Ω →[0, ∞ ] is said to be a fuzzy measure, if the following conditions are satisfied

(i)

µ

_m

(

φ

,0

)

=1

(ii) For any two sets A and B in Ω, A⊆ B and A is a nonempty set implies

y B m x A m

y x

= = ≤ ( ) )

(

sup

sup and _

  

 

     

≤

   

 

     

=

= m B y

m x

A m

m A x B y

) ( )

(

sup , sup

,

µ

,

(iii)For a sequence of non empty sets

{ }

A

n

⊂

Ω

,

A

1

⊆

A

2

⊆

A

3

⊆

A

4

⊆

...

and ∞ ∈Ω

=

∪

1

n n

A

⇒

y

n n

A m x A m

n x y

n

=         _∞

= =  =

 

 

∪

1 )

(

sup sup

lim

and

   

 

   

 

=

   

 

   

 

=         _∞

= ∞

= =

y

n n

A m

n n

m x

A m n m

n A x A y

n

∪

1 1

) (

sup , sup

,

lim

µ

(iv)For a sequence of non empty sets

{ }

A

n

⊂

Ω

,

∞

<

⊇

₂ ₃ ₄

...

,

(

₁

)

1

A

m

A

and ∞ ∈Ω

=

∩

1

n n

A

⇒

y

n n

A m x A m

n x y

n

=         _∞

= =  =

 

 

∩

1 )

(

sup sup

lim and

   

 

   

 

=

   

 

   

 

=         _∞

= ∞

= =

y

n n

A m

n n

m x

A m n m

n A x A y

n

∩

1 1

) (

sup , sup

,

lim

µ

(3)

23

In the following examples, P( X) denotes the power set of a non empty set X.

Example 2.5. Let X =

{

1,2,3,...n

}

, n is a finite value, and Ω = P( X). Let the fuzzy relation m:Ω →[0,∞ ] be defined as

m(S)= x iff x≤ S that is m(S)=

{

x / x∈

[

0, S

]

}

and the membership function

µ

_m :m→[0,1] be defined as

φ

µ

= x≤ S for A≠

S x x S

m( , ) , ,

and

1 ) 0 , (φ = µm

In particular case , when n = 3, X =

{

1,2,3

}

, and

{

1, 2, 3, 4, 5, 6, 7, 8

}

)

(

X

S

P

=

Ω

, where

{ }

1 ,

{ }

2 ,

{ }

3 ,

{

1,2

}

,

, ₂ ₃ ₄ ₅

1= S = S = S = S =

S

φ

{

}

S

{

}

S X S 6 = 1,3 , 7 = 2,3 , 8 =

Then the values of m and its corresponding membership values for each set in Ω are

,

0 )

(

S

₁

=

m

µ_m(S₁, 0)=1

[ ]

{

/

0 ,

1 }

)

(

)

(

)

(

S

₂

=

m

S

₃

=

m

S

₄

=

x

∈

m

1 if

)

,

(

)

,

(

)

,

(

S

₂

x

=

_m

S

₃

x

=

_m

S

₄

x

=

x

≤

m

µ

[ ]

{

/

0 ,

2 }

)

(

)

(

)

(

S

₅

=

m

S

₆

=

m

S

₇

=

x

∈

m

2 if

2 )

,

(

)

,

(

)

,

(

S

₅

x

=

_m

S

₆

x

=

_m

S

₇

x

=

x

≤

m

µ

[ ]

{

/

0 ,

3 }

,

)

(

S

₈

=

x

∈

m

3

3 )

,

(

8

=

if

x

≤

x

S

m

µ

The conditions in the definition can be verified below

,

0 )

(

S

₁

=

m

µ

_m

(

S

₁

,

0 )

=

1

hence condition ( i ) is true

5 2

S

⊆

⇒

sup

1 ,

sup

2

) ( )

( ₂ ₅

=

= =x mS y S

m

y

x

, and hence

y S m x S m

y x

= = < ( ) )

( 2 5

sup sup

also _

  

 

= =

   

 

   

 

=

= mS y

m x

S m

m S x S y

) ( 5 )

( 2

5 2

sup , 1

sup

,

µ

6 2

S

⊆

⇒

sup

1 ,

sup

2

)

( )

( 2 6

=

= =x mS y S

m

y

x

, and hence

y S m x S m

y x

= = < ( ) )

(2 6

sup sup

_

  

 

= =

   

 

   

 

=

= mS y

m x

S m

m S x S y

) ( 6 )

( 2

6 2

sup , 1

sup ,

also

µ

8 2

S

⊆

⇒

sup

1 ,

sup

3

)

( )

( 2 8

=

=x mS y

S m

y

x

, and hence

y S m x S m

y x

= = < ( ) )

( 2 8

(4)

P.S.Sivakumar and R.Kavitha 24       = =             =

= m S y

m x

S m

m S x S y

) ( 8 ) ( 2 8 2 sup , 1 sup ,

also

µ

In general, it can be verified that

for i = 3, j = 5, 7 and for i = 4, j = 6, 7

j

i

S

⊆

⇒

sup

1 ,

sup

2

) ( ) (

=

= =x mS y S

m i j

y

x

, and hence

y S m x S

m i j

y x = = ≤ ( ) ) ( sup sup       = =             =

= m S y

j m x S m i m j i y S x S ) ( ) ( sup , 1 sup ,

also

µ

For i = 3, 4

8

S

_i

⊆

⇒

sup 1, sup 3

) ( ) ( ₈ = = = =x m S y S

m

y x

i

, and hence

y S m x S m y x

i = =

≤ ) ( ) ( 8 sup sup       = =             =

= mS y

m x

S m i

m S x S y

i ( )

8 ) ( ₈ sup , 1 sup ,

Also

µ

For i = 5, 6, 7

8

S

_i

⊆

⇒

sup

2 ,

sup

3

) ( ) ( 8

=

= =x mS y S

m

y

x

i

, and hence

y S m x S m y x

i = =

≤ ) ( ) ( 8 sup sup       = =             =

= m S y

m x

S m i

m S x S y

i ( )

8 ) ( ₈ sup , 1 sup ,

Also

µ

Hence condition (ii) is true.

Now, consider the sequence of non empty sets such that

...

5 4 3

2

1

A

⊆

and ₈ ,

1 Ω ∈ = ∞ = S A n n

∪

where

...

,

5 ,

4 ,

3 ,

,

₂ ₅ ₈

2

1

=

S

A

=

S

A

=

S

n

=

A

_n

When we consider

mAn x

x

=

) (

sup _{, for n = 1, 2, 3, 4, 5,… ,}

we get the sequence 1, 2,3,3,3,3,...and

3 sup sup lim 1 ) ( = =         =         _∞ = = y n n A m x n A m

n x y

∪

When we consider _

              =x n A m n

m A x

) (

sup ,

µ

, for n = 1, 2, 3, 4, 5,… ,

(5)

25

lim , sup , sup 1

1 1 ) ( =             =                 =         _∞ = ∞ = = y n n A m n n m x n A m n m

n A x A y

∪

µ

Now, consider another the sequence of non empty sets such that

...

5 4 3

2

1

A

⊆

and 8 ,

1

Ω ∈ =

∞

= A S

n n

∪ where

...

,

₈ 2,3, 4,5, 5

1

=

S

A

=

S

n

=

A

_n

When we consider

mAn x

x

=

) (

sup , for n = 1, 2, 3, 4, 5,… ,

we get the sequence 2,3, 3,3, 3,...and

3 sup

sup

lim

1 ) (

=













=         _∞ = = y n n A m x n A m

n

x

y

∪

              =x n A m n

m A x

) (

sup ,

µ

, for n = 1, 2, 3, 4, 5,… ,

we get the sequence 1,1,1,1,1,... and

1 sup , sup , lim 1 1 ) ( =             =                 =         _∞ = ∞ = = y n n A m n n m x n A m n m

n A x A y

∪

µ

Similarly for all other possible cases, the condition ( iii ) can be verified . Now , consider the sequence of non empty sets such that

...

5 4 3 2

1

⊇

A

⊇

A

⊇

A

⊇

A

⊇

A

and = ∈

∞ = 2 1 S A n n

∩

Ω, where

...

,

₂ ₅ ₂

3 ,

4 ,

5 ,

8

1

=

S

A

=

S

A

=

S

n

=

A

_n

When we consider

m An x

x

=

) (

sup

, for n = 1, 2, 3, 4, …. ,

We get the sequence 3, 2,1,1,1,1,...and

1 sup sup lim 1 ) ( = =         =         _∞ = = y n n A m x n A m

n x y

∩

              =x n A m n

m A x

) (

sup ,

(6)

26 we get the sequence 1,1,1,1,1,...and

1 sup

, sup

, lim

1 1

) (

=

   

 

   

 

=

    

  

    

  

=         _∞

= ∞

= =

y

n n

A m

n n

m x

n A m n m

n A x A y

∩

µ

Now, consider the sequence of non empty sets such that

...

5 4

3 2

1

⊇

A

⊇

A

⊇

A

⊇

A

⊇

A

and = ∈Ω

∞

=

4 1

S A n

n

∩

, where

..

,

₄ 2 3, 4,5,... 6

1

=

S

A

=

S

n

=

A

_n

When we consider

m An x

x

=

) (

sup , for n = 1, 2, 3, 4, ……, we get the sequence

... , 1 , 1 , 1 , 1 ,

2 _and

1 sup

sup

lim

1 )

(

=













=         _∞

= =

y

n n

A m x n A m

n

x

y

∩

  

  

    

  

=x n A m n

m A x

) (

sup ,

µ

, for n = 1, 2, 3, 4, …..we get the sequence

... , 1 , 1 , 1 , 1 ,

1 and

lim , sup , sup 1

1 1

) (

=

   

 

   

 

=

    

  

    

  

=         _∞

= ∞

= =

y

n n

A m

n n

m x

n A m n m

n A x A y

∩

µ

Similarly, for all other possible cases, the condition ( iv ) can be verified.

Thus all the conditions which have been stated in the definition 2.4 are verified for the given fuzzy function m and its membership function

µ

_m and hence the given m is a fuzzy measure.

Hence for any finite value of n, the m is a fuzzy measure.

{

a,b ,c

}

, and

{

1

,

2

,

3

,

4

,

5

,

6

,

7

,

8

}

)

(

X

S

P

=

Ω

,

whereS1=

φ

,S 2 =

{ }

a , S 3 =

{ }

b , S4 =

{ }

c , S 5 =

{

a,b

}

,

{

a c

}

S

{

b c

}

S X S6 = , , 7 = , , 8 =

(7)

27

   

   

+

       

≤ ≤

= 0.2

3 0

/ ) (

2 S x x S

m _{, and}m(

φ

)=0

and the membership function µ_m : m→[0,1] be defined as

   

    

 

+

       

≤ ≤

       

− +

       

≤ ≤

       

=

2 . 0 3 3

2 . 0

2 . 0 3

3 0

3 )

, (

2 2

2

S x S if x

S

S x if

S x

x S m

µ

and

φ

≠ S

for

µ

m(

φ

,0) =1

, 0 ) ( S₁ =

m

µ

_m(S₁,0)=1

{

/

0

0 .

31 }

)

(

)

(

)

(

S

₂

=

m

S

₃

=

m

S

₄

=

x

≤

x

≤

m

     

≤ ≤ −

≤ ≤ =

= =

31 . 0 11

. 0 2

. 0

31 . 0

11 . 0 0

11 . 0 )

, ( ) , ( ) ,

( ₂ ₃ ₄

x if

x

x if x x

S x

S _m _m

m µ µ

µ

{

/

0

0 .

64 }

)

(

)

(

)

(

S

₅

=

m

S

₆

=

m

S

₇

=

x

≤

x

≤

m

     

≤ ≤ −

≤ ≤ =

= =

64 . 0 44

. 0 2

. 0 64 . 0

44 . 0 0

44 . 0 )

, )

, ( ) ,

( ₅ ₆ ( ₇

x if

x

x if x x

S x

S _m _m

m µ µ

µ

{

/

0

1 .

2 }

)

(

S

₈

=

x

≤

x

≤

m

,

   

≤ ≤

− ≤ ≤

= ₁ ₁_.₂

2 . 0

2 . 1

1 0 )

, ( ₈

x if x

x if x x

S m

µ

Conditions (i) - (iv) are satisfied for the fuzzy relation m and its membership function

m

µ

. Hence the fuzzy relation m is a fuzzy measure.

Example 2.7. Let , ,

3 2 1, 

 

 

= a a a

X and

{

,

}

,

)

(

X

S

₁

S

₂

S

₃

S

₄

S

₅

S

₆

S

₇

S

₈

P

=

(8)

28

S

1

=

φ

,

S

2

=

{ }

a

1

,

S

3

=

{ }

a

2

,

S

4

=

{ }

a

3

,

S

5

=

{

a

1

,

a

2

}

,

{

a a

}

S

{

a a

}

S X S ₆ = ₁ , ₃ , ₇ = ₂ , ₃ , ₈ =

Let the fuzzy relation m:Ω →[0,∞ ] be defined as







∉

∈

=

,

0

1 )

(

0 0

i i

S

x

if

S

x

if

S

m

_∀S _i_∈_Ω and x₀ is a fixed point in X

and the membership function µm : m→[0,1] be defined as

φ

µ

≠

+ −

= for S

x S x

S

m ,

1 1 ) , (

and

1 ) 0 , (

φ

=

µ

m

As a particular case, we take x0=a1.

Then the values of m and its corresponding membership values for each set in Ω are ,

0 ) 1 (S =

m m(S2)=1,m(S3)=0, m(S 4)=0,

, 1 ) (S5 =

m m(S6)=1,m(S7)=0, m(S8)=1

(

S₁ ,0

)

=1,

m

µ

_m

(

S

₂

,

1 )

=

0 .

5 ,

µm

(

S ₃,0

)

=0 , µm

(

S4,0

)

= 0,

(

S₅,1

)

=0.67, m

µ µ_m

(

S ₆ ,0

)

=0.67, µ_m

(

S ₇ ,0

)

=0.5, µ_m

(

S₈ ,0

)

=0.75

Conditions (i) - (iv) in the definition are satisfied for the fuzzy relation m and its membership function

µ

_m.

In general, for any x₀ in X,the conditions are satisfied. Hence the fuzzy relation m is a fuzzy measure.

{

1,2,3,...n

}

, and Ω=P( X). Let the fuzzy relation

] , 0 [ :Ω → ∞

m be defined as

m

(

S

)

=

max

S

,

for

S

≠

φ

and

m

(

φ

)

=

0

and the membership function µm : m→[0,1] be defined as

φ

µ

= − for A≠

x x

S

m ,

1 1 ) , (

and

1 ) 0 , (

φ

=

µ

m

In particular case , when n = 3, X =

{

1,2,3

}

, and

{

,

}

,

)

(

X

S

₁

S

₂

S

₃

S

₄

S

₅

S

₆

S

₇

S

₈

P

=

Ω

where S₁=

φ

, S ₂ =

{ }

1 , S₃ =

{ }

2 , S ₄ =

{ }

3 , S₅ =

{

1,2

}

,

{

}

S

{

}

S X S 6 = 1,3 , 7 = 2,3 , 8 =

, 0 )

(S ₁ =

m m(S₂)=1, m(S₃)=2, m(S 4)=3, m(S5)=2, m(S6)=3,

, 3 ) (S7 =

(9)

29

,

1 )

0 ,

(

S

₁

=

m

µ

_m

(

S

₂

,

1 )

=

0 ,

µ

_m

(

S

₃

,

2 )

=

0 .

5 ,

,

7 .

0 )

3 ,

(

S

₄

=

m

µ

_m

(

S

₅

,

2 )

=

0 .

5 ,

,

7 .

0 )

3 ,

(

S

₆

=

m

µ

_m

(

S

₇

,

3 )

=

0 .

7 ,

µ

_m

(

S

₈

,

3 )

=

0 .

7

Conditions (i) - (iv) in the definition are satisfied for the fuzzy relation m and its membership function

µ

m.

In general, for any value finite of n, the conditions are satisfied. Hence the fuzzy relation m is a fuzzy measure.

Remark: In the first two examples, measure takes values from an interval and the next

two examples measure is a non negative finite real number. In general, fuzzy measure need not be an interval.

Secondly, we discuss some examples [4] which are not a fuzzy measure.

Example 2.9. Let X₀ =

{

1,2,3,...

}

, X = X₀ ×X₀ =

{

(

x, y

)

/ x,y∈X₀

}

and, Ω = P( X).

m

(

S

)

=

Pr

oj

S

,

for

S

≠

φ

and

∀

S

∈

P

(

X

)

(

)

{

/ ,

}

and ( ) 0 Pr

where ojS = x x y ∈S m

φ

=

and the membership function µ_m:m→[0,1] be defined as

φ

µ

≠

+ −

= for S

x x

S

m ,

1 1 1 ) ,

( ₂

and

1 ) 0 , (

φ

=

µ

m

Then m satisfies the conditions (i), (ii) and (iii) but it does not satisfy the condition (iv).

For if we take S _n=

{ } {

1 × n,n+1,n+2,....

}

, then

n

S

m

S

₁

⊇

₂

⊇

₃

⊇

₄

⊇

....

and

(

_n

)

=

1 for

any

Therefore lim sup₍ ₎ _ = 1  

  

=x n S m

n x

but =φ

∞

=

∩

1

n n S

and sup 0

1

= =         _∞

= y

n n

S m

y

∩

Thus

lim

sup

.

1 )

(

y

n n

S m x n S m

n

x

y

=         _∞

= =



≠













∩

Hence the fuzzy relation m is not a fuzzy

measure.

{

a₁, a₂, a₃

}

, and

{

,

}

,

)

(

X

S

₁

S

₂

S

₃

S

₄

S

₅

S

₆

S

₇

S

₈

P

=

(10)

30

{ }

{

}

{

}

{

a

}

S

X

S

a

S

a

S

a

S

a

S

a

S

=

8 3 2 7

3 1 6 2 1 5

3 4

2 3

1 2 1

,

φ







∉

∈

=

,

0

1 )

(

0 0

i i

S

x

if

S

x

if

S

m

_∀S _i_∈_Ω and x 0 is a fixed point in X

and the membership function µ_m :m→[0,1] be defined as

(

)

2

1 1 )

, (

S x x

S m

− + = µ

Ω ∈ ∀Si

As a particular case, we take x ₀ = a ₁

Then the values of m and its corresponding membership values for each set in Ω are ,

0 ) (S ₁ =

m m(S₂)=1, m(S₃)=0, m(S ₄)=0, m(S₅)=1, m(S₆)=1,

, 0 ) (S7 =

m m(S₈)=1

,

1 )

0 ,

(

S

₁

=

m

µ

_m

(

S

₂

,

1 )

=

1 ,

µ

_m

(

S

₃

,

0 )

=

0 .

5 ,

,

5 .

0 )

0 ,

(

S

₄

=

m

µ

_m

(

S

₅

,

1 )

=

1 ,

µ

_m

(

S

₆

,

1 )

=

1

,

µ

_m

(

S

₇

,

0 )

=

0 .

2 ,

5 .

0 )

1 ,

(

S

₈

=

m

µ

This is not a fuzzy measure. For if

2 . 0 ) ( 5

. 0 ) ( but 0 ) ( )

( ₃ ₇ ₃ ₇

7

3 ⊂ S ⇒ m S = m S = S = > S =

S µ_m µ_m .

3. Conclusion

In this paper, we have attempted to give the definition of fuzzy measure, using the

fuzzy set approach and we cited some examples in which m is fuzzy measure and some examples in which m is not a fuzzy measure . A detailed investigation on the structure of fuzzy measures and fuzzy integrals can be then made using the present definition.

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2. Michael Gr. Voskoglou, An application of fuzzy logic to computational thinking, Annals of Pure and Applied Mathematics, 2(1) (2012) 18-32.

3. P.R. Halmos, Measure theory, Van Nostrand Renshold, New York, 1950. 4. Z.Wang and G.J.Klir, Fuzzy Measure Theory, Plenum Press, New York, 1992. 5. D.Ghosh and A.Pal, Use of fuzzy relational maps and intuitionistic fuzzy sets to

analyze health problem of agricultural labourers, Annals of Pure and Applied Mathematics, 5(1) (2013) 1-10

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7. T.Murofushi and M.Sugeno, An interpretation of fuzzy measures and the Choquet integral as an integral with respect to a fuzzy measure, Fuzzy Sets and Systems 29 (1989) 201-227.

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Edition.

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