# On fuzzy number valued Lebesgue outer measure.

[section]1. Introduction

There are articles in the literature associated with fuzzy outer measure [3]. We construct fuzzy number valued outer measures and measurable sets that is similar to that of Carathrodory construction. In section 3 we deal with fuzzy number valued Lebesgue outer measure in the real line R and in section 4, the results obtained in section 3 are carried to arbitrary fuzzy number valued measure space (X, [Omega], m).

In section 2 we give preliminary ideas relevant to fuzzy numbers.

[section]2. Basic Definitions

Definition 2.1. Let F = {n\n : R [right arrow] [0, 1]} . For every n [member of] F, n is called a fuzzy number if it satisfies the following properties:

a) n is normal i.e there exists an x [member of] R such that n(x) = 1

b) whenever [lambda] [member of] [0, 1] , the [lambda]-cut , [n.sub.[lambda]] = {x : n [greater than or equal to] [lambda] is a closed interval denoted by [[n.sup.-.sub.[lambda]], [n.sup.+.sub.[lambda]]] We denote the set of all fuzzy numbers on R by F*.

Remark 2.2. By decomposition theorem of fuzzy sets [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Remark 2.3. The fuzzy number [a, b] is defined by [bar.[a, b]](x) = 1, iff x [member of] [bar.[a, b]] and [bar.[a, b]] (x) 0, iff x [??] [bar.[a, b]]. Similarly we can define [bar.(a, b)].

If a = b then [a, b] is simply denoted by [??] i.e [??] is defined as a(x) = 1, iff x = a and [??](x) = 0 iff x [not equal to] a. Obviously then [bar.[a, b]], [??] [member of] F*.

Further [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Remark 2.4. A fuzzy number n which is increasing in the interval (a, b), n([b, c]) = 1 and is decreasing in the interval (b, c) is simply denoted by (a, b, c, d). Here n([b, c]) = 1 means n(x) = 1 for every x [member of] [b, c].

The fuzzy number (a, b, c, c+ [??]) where [??] > 0 is denoted by (a, [bar.[b, c]]) and the fuzzy number (a - [??], a, b, c) where [??] > 0 is denoted by ([bar.[a, b]], c).

Similarly the fuzzy number (a, b, b, b + [??]) where [??] > 0 is denoted by (a, [??]) and the fuzzy number (a - [??], a, a, b) where [??] > 0 is denoted by ([??], b).

Definition 2.5. For every a, b [member of] F* the sum a + b is defined as c where [c.sup.-.sub.[lambda]] = [a.sup.-.sub.[lambda]] + [b.sup.-.sub.[lambda]] and [c.sup.+.sub.[lambda]] = [a.sup.+.sub.[lambda]] + [b.sup.-.sub.[lambda]] for every [lambda] [member of] (0, 1].

Definition 2.6. For every a, b [member of] F* we write a [less than or equal to] b if [a.sup.-.sub.[lambda]] [less than or equal to] [b.sup.-.sub.[lambda]and [a.sup.+.sub.[lambda]] [less than or equal to] [b.sup.+.sub.[lambda] and for every [member of] (0,1].

[section]3. Fuzzy number valued Lebesgue outer measure

Definition 3.1. Let [mu] : R [right arrow] [0, 1] be a fuzzy subset of the real line. The fuzzy number valued Lebesgue outer measure for the fuzzy subset [mu] is defined as m*([mu]) = (0, [??]) where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the infimum is taken over all countable collection ([I.sub.n]) of open intervals covering R.

Proposition 3.2. If [[mu].sub.1] [less than or equal to] [[mu].sub.2], then m*([[mu].sub.l]) [less than or equal to] m*([[mu].sub.2]).

Proof. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Lemma 3.3. If [[mu].sub.1] and [[mu].sub.2] are any two fuzzy sets and [mu] = [[mu].sub.1] V pz, then

m*([mu]) [less than or equal to] m*([[mu].sub.1]) + m* ([[mu.sub.2])

Proof. If m*([[mu].sub.1]) (0, [??] or m*([[mu].sub.2]) = (0, [??]) then the lemma is trivial.

Suppose that m*([[mu].sub.1]) [not equal to] (0, [??]) and m*([mu].sub.2]) [not equal to] (0, [??]). Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Choosing [??] > 0 we can find countable covers {[I'.sub.1],[I'.sub.2],...} and {[I".sub.1],[I".sub.2],...} of open intervals for R such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Set another sequence {[J'.sub.1],[J'.sub.2],...} of pairwise disjoint open intervals such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (say) is countable and such that each [J'sub.n] is contained in some [I'.sub.i] and [I".sub.j]. Choose a sequence of open intervals [J".sub.1],[J".sub.2],...] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Setting {[I.sub.1],[I.sub.2],...} = {[J'.sub.1],[J".sub.1],[J'.sub.2],[J".sub.2]...} we can find {[I.sub.1],[I.sub.2],...} is a cover of R.

Also

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Similarly we find [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and hence K [less than or equal to] [K.sub.1] + [K.sub.2].

Therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

m*([mu]) = (0, [??] [less than or equal to] (0, [??] = (0, [??] + (0, [??] = m*([mu].sub.1] + m*([mu].sub.2]

Remark 3.4. By using the induction on n the result of above lemma can be extended as m*([mu]) [less than or equal to] m*([mu].sub.1] + m*([mu].sub.2] + m*([mu].sub.n] whenever [mu] = [[mu].sub.1] [disjunction] [[mu].sub.2] [disjunction] ... [disjunction] [[mu].sub.n].

Lemma 3.5. m*([mu]) = (0, [??]) where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the infimum is taken over all sequences ([A.sub.n]) of Lebesgue measurable subsets of R such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We have to prove that K = K' for which it is enough to show that K [less than or equal to] K'.

Suppose that K' < [infinity]. Choosing [??] > 0, we can find a sequence {[[??].sub.n]} with {[[??].sub.n]} > 0 for all n such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Accordingly we can have a sequence of Lebesgue measurable subsets of R such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

To each n, find a sequence of pairwise disjoint open intervals such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and such that [??] [I.sub.n.sub.k][Delta] [A.sub.n] = [M.sub.n] (say) is countable.

Finding a sequence {[J.sub.n]} of open intervals such that [??] [M.sub.n] [[subset].bar] [??] [J.sub.n] and such that [??] l([J.sub.n]) < [??]/4.

Clearly {[I.sub.n.sub.k], [J.sub.n] : k = 1, 2, ... , n = 1, 2, ...} is a cover for R and we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

as 0 [less than or equal to] [mu](x) [less than or equal to] 1, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and {[I.sub.n.sub.k]}, k = 1, 2 ... are pairwise disjoint for every n.

Therefore K = K'. and hence the result follows.

Theorem 3.6. If E is a Lebesgue measurable subset of the real line and [mu] is a fuzzy subset of R then m*([mu]) = m*([mu] [intersection] E) + m*([mu] [intersection] [E.sup.c]).

Proof.

m*([mu]) = m*([mu] [intersection] (E [union] [E.sup.c])) = m*([mu] [intersection] E) [union] ([mu] [intersection] [E.sup.c]))

[less than or equal to] m*([mu] [intersection] E) + m*([mu] [intersection] [E.sup.c]).

Suppose that m*([mu]) = (0, K), m*([mu] [intersection] E) = (0, [K.sub.1]) and m*([mu] [intersection] [E.sup.c]) = (0, [K.sub.2]).

If K = [infinity], the result is trivial.

Suppose that K [not equal to] [infinity]. Choosing [??] > 0 we can find a sequence {[A.sub.n]} of pairwise disjoint measurable subsets of R such that [??] [A.sub.n] = R and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore by previous lemma,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore

m*([mu] [intersection] E) + m*([mu] [intersection] [E.sup.c]) = (0,[K.sub.1]) + (0,[K.sub.2]) = (0,[??])[less than or equal to] (0,[??]) = m*([mu]).

Hence the result.

Remark 3.7. Using above theorem we can define Lebesgue measurable fuzzy subset [lambda] as follows:

A fuzzy subset [lambda] is Lebesgue measurable iff m*([mu]) = m*([mu] [intersection] [lambda]) + m*([mu] [intersection] [[lambda].sup.c]) for some complement [lambda].sup.c] of [lambda].

[section]4. General Fuzzy number valued Lebesgue outer measure

We shall assume X as a nonempty set, [Omega] denotes a [sigma]- algebra of subsets of X and m denotes a positive measure on [Omega].

Definition 4.1. Let [mu] : X [right arrow] [0, 1] be a fuzzy subset of X . The fuzzy number valued outer measure for the fuzzy subset [mu] is defined as m*([mu]) = (0, [??]) where K = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the infimum is taken over all countable collection ([A.sub.n]) in [Omega] which cover X.

The following result can be proved that is analogous to lemma 3.3.

If [[mu].sub.1], [[mu].sub.2], ... [[mu].sub.n] are fuzzy subsets of X and if [mu] = [[mu].sub.1] [disjunction] [[mu].sub.2] [disjunction] ... [disjunction] [[mu].sub.n] then m*([mu]) [less than or equal to] m*([[mu].sub.1]) + m*([[mu].sub.2]) + ... m*([[mu].sub.n]).

Let [[Omega].sub.1] = E [[subset].bar] X : m*(A) = m*(A [intersection] E) + m*(A [intersection] [E.sup.c]) for every A [[subset].bar] X. Then [[Omega].sub.1] is [sigma]- algebra, [[Omega].sub.1] [[subset].bar] [Omega]. If m(A) = m*(A) for every A [member of] [[Omega].sub.1] then m is a positive measure on [[Omega].sub.1]. If m* is finite([sigma]- finite) then m is finite([sigma]- finite). We use m(A) instead of m*(A) whenever A [member of] [[Omega].sub.1].

Lemma 4.2. If [mu] is fuzzy set on X then m*([mu]) = (0, K), where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the infimum is taken over all sequences ([A.sub.n]) of sets in [[Omega].sub.1] such that X = [??] [A.sub.n].

Proof. The proof is similar to that of lemma 3.5 with the following modifications.

To each n, such that [??] [mu](x) [not equal to] 0, choose a sequence such that [A.sub.n] [[.subset].bar] [??] [A.sub.n.sub.k], [A.sub.n.sub.k] [member of] [Omega], [A.sub.n.sub.k] are pairwise disjoint, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let B be the set obtained from X after removing all [A.sub.n.sub.k]. Then {[A.sub.n.sub.k] : k = 1, 2, ...} [union] {B} is a cover for X which leads to the conclusion of the as in lemma 3.5.

Theorem 4.3. If E [member of] [[Omega].sub.1] and [mu] is a fuzzy subset of X, then

m*([mu]) = m*([mu] [intersection] E) + m*([mu] [intersection] [E.sup.c])

Proof. Analogous to the proof in lemma 3.6

Remark 4.4. Using above theorem if we wish to define measurable fuzzy subset [lambda] of X as those [lambda] which satisfies m*([mu]) = m*([mu] [intersection] [lambda]) + m*([mu] [intersection] [[lambda].sup.c]) for some complement [[lambda].sup.c] of [lambda] then [lambda] must be [[Omega].sub.1]- measurable.

References

[1] Z. Wang, G.J. Klir, Fhzzy Measure Theory, Plenum Press, New York, 1992.

[2] Vimala, Fizzy probability via fuzzy measures, Ph.D Dissertation, Karaikudi Alagappa University, 2002.

[3] Minghu Ha and Ruisheng Wang, Outer measures and inner measures on fuzzy measure spaces, The journal of Fizzy Mathematics, 10(2002), 127-132.

[4] Arnold Kaufmann, Madan M.Gupta, Introduction to Fizzy Arithmetic Theory and Applications, International Thomson Computer Press, 1985.

T. Veluchamy ([dagger]) and P. S. Sivakkumar ([double dagger])

([dagger]) Dr. S. N. S Rajalakshmi College of Arts and Science Coimbatore, Tamil Nadu India

([double dagger]) Department of Math. Government Arts College Coimbatore, Tamil Nadu India

sivaommuruga@rediffmail.com