L-fuzzy valued measure and integral

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1 USFLAT-LFA 2011 July 2011 Aix-les-Bains, France L-fuzzy valued measure and inegral Vecislavs Ruza, 1 Svelana Asmuss 1,2 1 Universiy of Lavia, Deparmen of ahemaics 2 Insiue of ahemaics and Compuer Science of Universiy of Lavia Absrac We coninue o develop a consrucion of an L-fuzzy valued measure by exending a measure defined on a σ- algebra of crisp ses o an L-fuzzy valued measure defined on a T -ribe in he case when operaions wih L-ses and L-fuzzy numbers are defined by using he minimum riangular norm T. We inroduce an L-fuzzy valued inegral over an L-se wih respec o an L-fuzzy valued measure, consider is properies and describe a mehod of L-fuzzy valued inegraion. Keywords: L-se, L-fuzzy real number, L-fuzzy valued measure, L-fuzzy valued inegral. 1. Inroducion One can find a lo of works regarding a fuzzy approach o measure and inegral. The mos imporan conceps and resuls concerning his opic are considered in [1], [2], [3]. Our ineres is in developing a heory where no only ses are fuzzy, bu also measure and inegral ake fuzzy real values. In he previous papers [4], [5] we suggesed he consrucion ha allows us o obain an L- fuzzy valued measure defined on a T -clan of L-ses by exension of a measure defined on a σ-algebra of crisp ses. We coninue o develop he resuls obained before and describe how an L-fuzzy valued measure defined on a T -ribe can be obained for a given σ-algebra Φ 2 X and a finie measure ν : Φ R +. On he nex sage we inroduce he concep of an L-fuzzy valued inegral over a measurable L-se. Some properies of L-fuzzy valued inegral are considered. We suppose ha L is a complee, compleely disribuive laice (see e.g. [6]) and operaions wih L-ses and L-fuzzy numbers are defined by using he minimum riangular norm T. We give our preference o he fuzzy real numbers as hey were firs defined by B. Huon [7] and hen sudied horoughly in a series of papers (see e.g. [8], [9], [10]). The preference of using his approach for defining fuzzy real numbers is moivaed by our inenion o develop resuls on approximaion from [11], [12]. For problems ha can be solved only approximaely he noion of he error of a mehod of approximaion plays he fundamenal role. In order o esimae he qualiy of approximaion on an L-fuzzy se, we need an appropriae L-fuzzy analogue of a norm. Our inension is o use he L-fuzzy valued inegral o define an L-fuzzy norm for invesigaion of he error of approximaion on an L-se. 2. Preliminaries 2.1. L-ses Given a (crisp) universe X and a complee, compleely disribuive laice L(,,0 L,1 L ), an L-subse A of X (or, briefly, an L-se A) is a funcion A : X L. The class of all L-subses of X is denoed L X. The operaions wih L-ses A,B are defined by using he minimum riangular norm T, is corresponding conorm S and decreasing involuion N: (A B)(x) T (A(x),B(x)), (A B)(x) S (A(x),B(x)), A c (x) N(A(x)). A finie family of L-ses A 1,A 2,,A n is said o be T -disjoin (see e.g.[1]) iff for each k {1,,n} we n have ( A i ) A k /0. A counable family of L-ses,i k is said o be T -disjoin iff every finie subfamily of his family is T -disjoin. In order o consider an L-fuzzy valued T -measure we consider classes of L-ses called T -clans and T -ribes (see e.g. [1]). Definiion 2.1. A subclass A L X is called a T -clan on X if he following properies are saisfied: /0 A; for all A A we have A c A; for all A,B A we have A B A. Definiion 2.2. A subclass Σ L X is called a T -ribe on X if he following properies are saisfied: /0 Σ; for all A Σ we have A c Σ; for all sequences (A n ) Σ we have 2.2. L-fuzzy real numbers A n Σ. For our purposes we use he L-fuzzy real numbers as hey were firs defined by B. Huon [7]. Definiion 2.3. An L-fuzzy real number is a funcion z : R L such ha z is non-increasing; z() 0 L, z() 1 L ; z is lef semi-coninuous, i.e. for all 0 R we have z() z( 0 ). < The auhors - Published by Alanis Press 127

2 In he original papers of his subjec (see [7], [8], [9]) L-fuzzy real numbers were defined no as order reversing funcions, bu as equivalence classes of such funcions. However each class of equivalence has a unique lef semi-coninuous represenaive and herefore an L-fuzzy real number can be idenified wih his represenaive. A deep heoreical jusificaion of viewing fuzzy numbers as disribuion funcion was given by U. Höhle [10], who showed ha such fuzzy real numbers can be obained from he se of raional numbers Q by means of Dedekind compleion in he same way as real numbers R are obained from Q if one applies he muliple-valued logic, insead of he binary logic which sands behind he Dedekind compleion in he classic case. The se of all L-fuzzy real numbers is called he L-fuzzy real line and i is denoed by R(L). An L-fuzzy number z is called non-negaive if z(0) 1 L. We denoe by R + (L) he se of all non-negaive L-fuzzy real numbers. Operaions wih L-fuzzy real numbers such as addiion and muliplicaion by a real posiive number r are defined as following: (z 1 z 2 )() {z 1 (τ) z 2 ( τ)}, (zr)() z( r ). τ The supremum and he infimum of a se of non-negaive L-fuzzy numbers F R + (L) are defined by he formulas (see e.g. [11], [12]): (In f F)() {z() z F}, R, Sup F In f {z z R(L), z z for all z F}. Taking ino accoun ha F is bounded from below i is easy o see ha In f F is an L-fuzzy real number. In case F is bounded from above (i.e. here exiss z 0 R(L) such ha z z 0 for all z F), Sup F is an L-fuzzy real number, oherwise he condiion does no necessarily hold. SupF() 0 L Going forward we will need also he counable addiion of non-negaive fuzzy real numbers. Given a sequence of non-negaive fuzzy real numbers (z n ) R + (L) we consider he counable sum z n Sup{z 1 z 2 z n n N}. For a R + and α L by z(a,α) we denoe a special ype of non-negaive L-fuzzy real numbers 1, 0, (z(a,α))() α, 0 < a, 0, > a, ha will play an imporan role in our work L-fuzzy valued measure We consider a measure ha is defined on a T -ribe and akes values in R + (L). Definiion 2.4. Le Σ be a T -ribe. A funcion µ : Σ R + (L) is called an L-fuzzy valued measure if i saisfies he following condiions: µ(/0) z(0,1 L ); µ is T -valuaion, i.e. for all A,B Σ i holds µ(a) µ(b) µ(a B) µ(a B); µ is lef T -coninuous, i.e. Sup{µ(A n ) n N} µ(a), for all (A n ) Σ and A A n. 3. Consrucion of L-fuzzy valued measure 3.1. easurable L-ses For a given σ-algebra Φ 2 X and a finie measure ν : Φ R + an L-fuzzy valued measure can be obained by he following schema (see [4], [5]): For Φ, α L we define an L-fuzzy se { α,x, ()(x) 0,x /. All hese L-ses form a class of L-ses ha we denoe by : { Φ,α L}. Noe ha he following properies hold for all L- ses A 1,A 2,,A n : n (i) A i ; (ii) here exis such T -disjoin L-fuzzy ses B 1,B 2,,B k ha n A i k B i. Nex we define an L-fuzzy valued funcion by he formula Obviously, m : R + (L) m() z(ν(),α). (i) for all ses A i A( i,α), i 1,2 : m(a 1 ) m(a 2 ) m(a 1 A 2 ) m(a 1 A 2 ); (ii) for all ( n ) Φ and n we have Sup{m(A( n,α)) n N} m(); 128

3 (iii) for all pairwise disjoin ses ( n ) Φ: m(a( n,α)) m(a( n,α)). Now we exend m o he L-fuzzy valued funcion m : L X R + (L) as following: m () In f { m( n ) ( n ) : n } (m is an L-fuzzy valued analogue of an ouer measure). Le us noe ha (i) for all L X here always exiss such a sequence ( n ) ha n ; (ii) m is bounded from above in he following sense: m () z(ν(x),1 L ) for all L X ; (iii) for all we obain m () m(); (iv) for L-ses A,B L X we have m (A) m (B) m (A B) m (A B). Finally, we generalize o he fuzzy case he classical concep of m -measurabily (in he sense of Caraheodory) and consider Σ - he class of all so called m -measurable L-ses. Definiion 3.1. A se L X is called a m - measurable if i saisfies he following condiions for all L-ses B L X : m () m (B) m ( B) m ( B), m ( c ) m (B) m ( c B) m ( c B). Noe ha (i) c is m -measurable for all m -measurable L- ses ; (ii) all L-ses are m -measurable L-fuzzy valued measure of measurable L-ses We consider µ as he resricion of m o Σ: µ() m () for all Σ. Theorem 3.2. µ is an L-fuzzy valued T -measure such ha µ/ m. As i was shown in [5] all m -measurable L-ses form a T -clan. To obain ha he class Σ is a T -ribe, we consider a sequence ( n ) of m -measurable L-ses. Firs we noice ha ( n ) Sup{ ( n i ) n N}. n Now aking ino accoun ha i is m -measurable we obain ha for all L-ses B L X and for all n N: ( n i ) m (B) (( n i ) B) (( n i ) B). This means ha for all n N and hence ( n i ) m (B) (( i ) B) (( i ) B), Sup{ ( n i ) n N} m (B) (( i ) B) (( i ) B). Finally we obain ( n ) m (B) (( n ) B) (( n ) B). By analogy he resul can be proved for n. Thus by exension of a crisp measure ν we obain L- fuzzy valued measure such ha (i) µ/ m; (ii) µ/φ ν. µ : Σ R + (L) The las equaliy means ha for every Φ i holds 4. L-fuzzy valued inegral µ(a(,1 L )) z(ν(),1 L ) Definiion of L-fuzzy valued inegral Our aim is o define an L-fuzzy valued inegral f dµ, where Σ and f : X R is a non-negaive measurable funcion wih respec o σ-algebra Φ. By analogy wih he classical case (see e.g. [13]) we define an L-fuzzy valued inegral sepwise, firs considering he case of simple non-negaive measurable funcions (for shor SNF): whenever n ( c i χ Ci ) dµ (c i µ(c i )), c i R +,C i Φ for all i 1,,n, χ Ci is he characerisic funcion of C i, i 1,,n, 129

4 C 1,,C n are pairwise disjoin ses. Then considering he case of non-negaive measurable funcions f (for shor NF): f dµ Sup{ g dµ g f and g is SNF}. For I f f dµ due o properies of he supremum of a se of L-fuzzy numbers, we have I f is non-increasing, I f () 1 L, I f is lef semi-coninuous, i.e. < 0 I f () I f ( 0 ). Definiion 4.1. We say ha a non-negaive measurable funcion f is L-fuzzy inegrable iff I f () 0 L Properies of L-fuzzy valued inegral For L-fuzzy inegrable non-negaive funcions f, f 1, f 2,, f n, and measurable L-ses, 1, 2,, n, Σ he following properies of L-fuzzy valued inegral are rue. (I1) dµ µ() (I2) r R + r f dµ r f dµ (I3) f 1 f 2 f 1 dµ f 2 dµ (I4) 1 2 f dµ f dµ 1 2 (I5) ( f 1 + f 2 )dµ f 1 dµ f 2 dµ (I6) 1 2 /0 f dµ f dµ f dµ (I7) ( n ) : n n+1 and n f dµ Sup{ f dµ n N} n (I8) ( f n ) : f n f n+1 and lim f n f n f dµ Sup{ f n dµ n N} 5. Inegraion over a measurable fuzzy se In his secion we sugges a mehod of calculaion of he fuzzy valued inegral over a measurable fuzzy se in he case when L [0,1] and is NF (i.e. is measurable wih respec o σ-algebra Φ). The main idea of he mehod is based on he following reasoning. The fuzzy se we wan o inegrae over can be viewed as a non-negaive funcion. Le us assume ha his funcion is measurable wih respec o σ-algebra Φ. I is known ha every non-negaive measurable funcion can be presened as a limi of a non-decreasing sequence of SNF. Obviously, every fuzzy se ha is SNF can be presened as he union of T -disjoin fuzzy ses from he class. And he L-fuzzy valued inegral over an elemen from he class can be easily calculaed. This observaion gives a reason for he following heorem. Theorem 5.1. If : X [0,1] is a measurable funcion wih respec o σ-algebra Φ, hen fuzzy se is measurable wih respec o T -ribe Σ. We describe he mehod gradually depending on he ype of a fuzzy se : firs considering he case when is an elemen of he class, hen exend i o he case when is SNF or a finie union of elemens from he class and, finally, he case when is NF Inegraion over To show ha for all i holds f dµ z( f dν,α), we use some special properies of he addiion of fuzzy numbers z(a,α) described in subsecion 2.2. a 1,a 2 R + z(a 1,α) z(a 2,α) z(a 1 + a 2,α); c R + cz(a,α) z(ca,α); a i R +,i J Sup{z(a i,α) i J} z(sup{a i i J},α). For f n c i χ Ci we ge n c i χ Ci dµ (c i µ(c i )) (c i z(ν( C i ),α)) n z( c i ν( C i ),α) z( f dν,α). In he case when f is NF we have f dµ Sup{z( g dν,α) g f and g is SNF} z(sup{ g dν,α g f and g is SNF},α) z( f dν,α) Inegraion over SNF If is SNF hen (R) {α 1,,α n }. We assume ha and denoe α 1 > α 2 > > α n i 1 (α i ), i 1,,n. 130

5 Then Denoing I f dµ and I n n f dµ we ge i j i j /0; n i R; n A( i,α i ); α i i j j1 where α i as he α i -cu of fuzzy se. Taking ino accoun he propery of addiion of fuzzy numbers: ( n z(a i,α i ))() 1, 0, α 1, 0 < a 1, α i+1, a a i < a a i+1, 0, > a a n, we obain f dµ (α i, i ) f dµ 1, f dν, 1 i α i, f dν < i+1 f dν, j1 j j1 j 0, > n f dν, j1 j 1, f dν, α 1 α i, f dν < f dν, α i α i+1 α n, f dν < f dν, α n 1 αn 0, oherwise Inegraion over NF z( f dν,α i ) i As was already menioned every NF can be presened as he limi of a non-decreasing sequence of SNF. To describe his sequence we use he same logic as in he previous subsecion. Le us ake a sequence ( n ) such as: for all n N : n (R) {α n 1,,αn k n }; for all n N α n i > α n i+1, i 1,,k n 1; 1 n {x (x) αn 1 }, i n {x α n i (x) < α n i 1 }, i 2,,k n; n k n (α n i,n i ), n N; n. n I Sup{ n f dµ n N} Sup{I n n N}. From he las equaliy we can ge an approximae value of I by fixing n. Obviously, he inegral accuracy in his case will be dependen on n. Acknowledgemen The paper was parly suppored by he SF projecs 2009/0138/1DP/ /09/IPIA/VIAA/004 and 2009/0223/1DP/ /09/APIA/VIAA/008. References [1]. P. Klemen, D. Bunariu, Triangular norm based measures. In:. Pap, edior, Handbook of easure Theory, Chaper 23, pages , lsevier, Amserdam, [2] R. esiar, Fuzzy measures and inegrals, Fuzzy Ses and Sysems, 156: , [3] Z. Wang, G. J. Klir, Fuzzy measure heory, Plenum Press, New York, [4] V. Ruzha, S. Asmuss, A consrucion of a fuzzy valued measure based on minimum -norm. In: New Dimensions in Fuzzy Logic and Relaed Technologies, Proceedings of he 5 h USFLAT Conference (USFLAT 2007), pages , Osrava (Czech Republic), [5] V. Ruzha, S. Asmuss, A consrucion of an L- fuzzy valued measure of L-fuzzy ses. Proceedings of he 5 h IFSA-USFLAT 2009 Conference (IFSA- USFLAT 2009), pages , Lisbon (Porugal), [6] B. A. Davey, H. A. Priesley, Inroducion o Laices and Order, Cambridge Universiy Press, [7] B. Huon, Normaliy in fuzzy opological spaces, J.ah.Anal.Appl, 50:74 79, [8] R. Lowen, On (R(L), ), Fuzzy Ses and Sysems, 10: , [9] S.. Rodabaugh, Fuzzy addiion and he L-fuzzy real line, Fuzzy Ses and Sysems, 8:39 52, [10] U. Höhle, Fuzzy real numbers as Dedekind cus wih respec o muliple-valued logic, Fuzzy Ses and Sysems, 24: , [11] S. Asmuss, A. Šosak, xremal problems of approximaion heory in fuzzy conex, Fuzzy Ses and Sysems, 105: , [12] S. Asmuss, A. Šosak, On cenral algorihms of approximaion under fuzzy informaion, Fuzzy Ses and Sysems, 155: , [13] P. Halmos, easure heory, IL, oscow,