# On fuzzy number valued Lebesgue outer measure.

[section]1. Introduction

There are articles in the literature associated with fuzzy outer measure . 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

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

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

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

 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
Author: Printer friendly Cite/link Email Feedback Veluchamy, T.; Sivakkumar, P.S. Scientia Magna Report 9INDI Jan 1, 2009 2077 Generalized galilean transformations and dual Quaternions. A note to Lagrange mean value theorem. Fuzzy algorithms Fuzzy logic Fuzzy systems Measure theory Number theory