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On fuzzy number valued Choquet integral.

[section] 1. Introduction

After the formulation of fuzzy integral by Sugeno, various generalizations of fuzzy integral were introduced and investigated. Fuzzy number fuzzy integral (FNFI) were defined by various authors in [3], [5] and [6].

Zhang Guang-Quan [5] used the concept of Sugeno's fuzzy integral as [lambda]-cuts to define the fuzzy number valued fuzzy integral. He defined it as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Leechay Jang et al [3], defined fuzzy number valued fuzzy Choquet integral as the Choquet integral of fuzzy number valued function. But the concepts in [3] are all based on the interval-valued Choquet integrals.

We in this paper define the fuzzy number valued Choquet integral that is neither based on interval-valued Choquet integrals nor fuzzy valued functions. The properties are then investigated. Fuzzy number valued Choquet integral has many applications as indicated in [3]. For the basic definitions that are relevant to fuzzy number, the reader may refer [2].

[section] 2. Definition and properties

Definition 2.1. Let (X, [OMEGA] )be a measurable space where [OMEGA] is a non-empty class of subsets of X. A fuzzy number valued fuzzy measure (FNFM) [mu] on X is a set function [mu]: [OMEGA] [right arrow] [F'.sub.+] where [F'.sub.+] is the class of all fuzzy numbers in [R.sup.+.sub.0] with the following properties.

(1) [mu][phi] = [bar.0];

(2) A, B [member of] [OMEGA], A [subset or equal to] B [??] [mu]A [less than or equal to] [mu]B.

Definition 2.2. A FNFM [mu] is said to be continuous above if for [E.sub.i] (i = 1, 2, ...) [member of] [OMEGA] with [E.sub.1] [subset] [E.sub.2] ... and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Similarly A FNFM [mu] is said to be continuous below if for [E.sub.i] (i = 1, 2, ...) [member of] [OMEGA] with [E.sub.1] [contains] [E.sub.2] ... and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with [mu]([E.sub.1]) is finite, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

A FNFM which is both continuous above and continuous below is called continuous.

Definition 2.3. (Let (X, [OMEGA], [mu]) be a fuzzy number valued fuzzy measure space where [mu] is continuous. The fuzzy number valued Choquet integral of a measurable function f with respect to [mu] on a crisp subset A of X is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [F.sub.[alpha]] = {x : f(x) [greater than or equal to] [alpha]} and the integrals on the right side represent Lebesgue integrals.

Proposition 2.1.

(i) If [mu]A = [bar.0] then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for any f.

(ii) If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] then [mu]([F.sub.0+] [intersection] A) = [bar.0].

(iii) If [f.sub.1] [less than or equal to] [f.sub.2] then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(iv) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [[chi].sub.A] is the characteristic function of A.

(v) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for any constant a [member of] [0, [infinity]).

(vi) If A [subset or equal to] B then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(vii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(viii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(ix) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(x) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof.

(i) If [mu]A = [bar.0] then [mu]([F.sub.[alpha]], [intersection] A) = [bar.0] because of monotonicity of [mu]. Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(ii) Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for any [lambda] [member of] [0,1]

Suppose [mu]([F.sub.0+] [intersection] A) = [bar.c] > [bar.0]. As [F.sub.1/n], [intersection] A [??] [F.sub.0] [intersection] A, by continuity from below of [mu], ([bar.[rho]])lim [mu],([F.sub.1/n] [intersection] A) = [mu]([F.sub.0] [intersection] A) = [bar.c]. Hence by theorem 2.2 of [5] there exists a [[lambda].sub.0] [member of] [0,1] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence there exists a [n.sub.0] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], a contradiction. Hence the result.

(iii) Let [F.sub.[alpha]] = {x : [f.sub.1] (x) [greater than or equal to] [alpha]} and [F'.sub.[alpha]] = {x : [f.sub.2](x) [greater than or equal to] [alpha]} As [f.sub.1] [less than or equal to] [f.sub.2], [F.sub.[alpha]] [subset or equal to] [F'.sub.[alpha]] for all [alpha] [member of] [0, [infinity]].

Hence [mu]([F.sub.[alpha]] [intersection] A) [less than or equal to] [mu]([F'.sub.[alpha]] [intersection] A) and hence [mu][([F.sub.[alpha]] [intersection] A).sup.-.sub.[lambda]] [less than or equal to] [mu][([F'.sub.[alpha]] [intersection] A).sup.-.sub.[lambda]] and [mu][([F.sub.[alpha]] [intersection] A).sup.+.sub.[lambda]] [less than or equal to] [mu][([F'.sub.[alpha]] [intersection] A).sup.+.sub.[lambda]]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] from which the result follows.

(iv) Let [F.sub.[alpha]] = {x : f(x) [greater than or equal to] [alpha]} and [J.sub.[alpha]] = {x : f(x)[[chi].sub.A](x) [greater than or equal to] [alpha]} When [alpha] [member of] [0, [infinity]), x [member of] [J.sub.[alpha]], f(x)[[chi].sub.A](x) [greater than or equal to] [alpha] [??] f(x) [greater than or equal to] [alpha] [??] x [member of] [F.sub.[alpha]]

Therefore [F.sub.[alpha]] [intersection] A = [J.sub.[alpha]] and hence [mu]([F.sub.[alpha]] [intersection] A) = [mu]([J.sub.[alpha]]) = [mu]([J.sub.[alpha]] [intersection] X)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(v) As [F.sub.[alpha]] = {x : f(x) [greater than or equal to] [alpha]}, [F.sub.[alpha]] is X when a [greater than or equal to] [alpha] and is [phi] when a [less than or equal to] [alpha],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(vi) As [F.sub.[alpha]] [intersection] A [subset or equal to] [F.sub.[alpha]] [intersection] B, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(vii) Let [M.sub.[alpha]] = {x : ([f.sub.1] [disjunction] [f.sub.2])(x) [greater than or equal to] [alpha]} [F'.sub.[alpha]] = {x : [f.sub.1](x) [greater than or equal to] [alpha]}

[F.sup.2.sub.[alpha]] = {x : [f.sub.2](x) [greater than or equal to] [alpha]}

Clearly [M.sub.[alpha]] = [F'.sub.[alpha]] [union] [F.sup.2.sub.[alpha]]

Therefore [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Similarly [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence the result.

(viii) By setting [m.sub.[alpha]] = {x : ([f.sub.1] [and] [f.sub.2])(x) [greater than or equal to] [alpha]}, the proof of the result is made analogously to that of (vii).

(ix) As [mu]([F.sub.[alpha]] [intersection] A), [mu]([F.sub.[alpha]] [intersection] B) [less than or equal to] [mu]([F.sub.[alpha]] [intersection] (A [union] B)), we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(x) The proof is analogous to that of (ix).

Definition 2.3. Let (X, [OMEGA], [mu]) be a fuzzy number valued fuzzy measure space. Suppose that f : X [right arrow] [0, [infinity]) and h : X [right arrow] [0, 1] is a fuzzy set on X. If h is measurable, then the fuzzy number valued Choquet integral of f on the fuzzy set h is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The following results are immediate.

Proposition 2.2.

(i) ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for a [greater than or equal to] 0.

(ii) If f : X [right arrow] [0,1] is measurable then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(iii) If [h.sub.1] [less than or equal to] [h.sub.2] then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(iv) If [f.sub.1] [less than or equal to] [f.sub.2] then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(v) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(vi) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(vii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(viii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By (ii) of above proposition it is clear that the definition 2.3 is well defined.

References

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

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

[3] Leechae Jang, Taekyun Kim, Jongduek Jeon , Wonju Kim, On Choquet integrals of Measurable Fuzzy number valued Functions, Bull. Korean Math. Soc., 41(2004), No.1, 95-107.

[4] Ladislav Misik and Janos T. Toth, On Asymptotic behaviour of Universal Fuzzy measures, Kybernetika, 42(2006), No.3, 379-388.

[5] Zhang Guangquan, Fuzzy number-valued fuzzy measure and fuzzy number-valued fuzzy integral on the fuzzy set, Fuzzy Sets and Systems, 49(1992), 357-376.

[6] Congxin Wu, Deli zhang, Caimei Guo, Cong Wu, Fuzzy number fuzzy measures and fuzzy integrals (I). Fuzzy integrals of functions with respect to fuzzy number fuzzy measures, Fuzzy Sets and Systems, 98(1998), 355-360, 4(2008), No.2, 80-95.

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

sivaommuruga@rediffmail.com

([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
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Author:Veluchamy, T.; Sivakkumar, P.S.
Publication:Scientia Magna
Article Type:Report
Geographic Code:9INDI
Date:Dec 1, 2008
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