The Structures of Fuzzifying Measure

Transcription

1 Sesors & Trasduers Vol 7 Issue 5 May 04 pp 56-6 Sesors & Trasduers 04 by IFSA Publishig S L hp://wwwsesorsporalom The Sruures of Fuzzifyig Measure Shi Hua Luo Peg Che Qia Sheg Zhag Shool of Saisis Jiagxi Uiversiy of Fiae & Eoomis ahag 00 Chia Applied Saisis Researh Ceer Jiagxi Uiversiy of Fiae & Eoomis ahag 00 Chia Shool of Iformais Guagdog Uiversiy of Foreig Sudies Guagzhou 5040 Chia luoshihua@aliyuom Reeived: May 04 /Aeped: 0 May 04 /Published: May 04 Absra: A ew so-alled fuzzifyig measurable heory ha geeralizes he lassial measurable heory is esablished ad he sruures of suh ew heory are disussed very deailed I he las we sudy he produ of wo fuzzifyig measures ad osider a problem whih is like he hird of he ope problems i fuzzy measure preseed by Z Wag We have solved his problem saisfaorily i he ew heory Copyrigh 04 IFSA Publishig S L Keywords: Fuzzifyig measure Fuzzifyig ull-addiive Produ of fuzzifyig measure Fuzzifyig weakly absoluely oiuous Iroduio Sie M Sugeo [0] irodued he oep of fuzzy measure i 974 he sudy of fuzzy measure heory has gaied rih olusios [5-9] Bu mos of hem are oeraed o sigle valued fuios Moivaed by Auma s iegral of se - valued fuios Caimei Guo Deli Zhag e irodued fuzzy iegral of se-valued fuios ad he esablished o se-valued fuzzy measure [4] I heir paper he fuzzy measure perais o he se oed by PI( FL( X ( FL ( X ={ A A : X L} L is a omplee residual laie or L= P 0 ( R + I=(0 I 99 Yig Mig-Sheg used a semai mehod of oiuous-valued logi LX o propose he oep of fuzzifyig opology [-] I 99 She Ji-Zhog uses he same mehod o esablish he heory of fuzzifyig groups ad gai a lo of good algebrai properies [] They all suessful exeded he appliaio of suh heory Usig he same mehod fuzzifyig measure FI ( P( X ad fuzzifyig measure spae have bee esablished [ 4] ad beig proved useful i may applyig domai suh as fuzzy orol fiae model e Bu beig a ew heory he sruure of suh heory is o larified I his arile suh diffiul problem has bee sudied ad some saisfaio resuls have bee gaied Firs we display he fuzzy logial ad orrespodig se-heoreial oaios used i his paper [ ϕ]: = [ ϕ] [ ϕ ψ]: = mi([ ϕ][ ψ] [ ϕ ψ] = [ ϕ] α[ ψ]: = mi( [ ϕ] + [ ψ] [ ϕ ψ] = [ ϕ] β[ ψ] [( x ϕ( x] = if[ ϕ( x] x X [( x ϕ( x] = sup[ ϕ( x] x X where X is he uiverse of disourse [ x A]: = A ( x ϕ ψ : = ( ϕ ψ ϕ ψ : = ( ϕ ψ ( ψ ϕ A B: = ( x( x A x B 56 hp://wwwsesorsporalom/html/digest/p_rp_04hm

2 Sesors & Trasduers Vol 7 Issue 5 May 04 pp 56-6 A B: = ( A B ( B A [a]= a if a [0] The some defiiios i lassi measure heory [5] ad i fuzzifyig measurable heory are all display Defiiio Le X is uiverse R P( X saisfy: X R A R A R { A } R A R = The R is alled σ-algebra ad (X R a measurable spae Defiiio Le X is a o-empy se ad R is a σ-algebra of X A se fuio m: R [0 ] is alled a ormal srog semi-measure if m( = 0 AB R ad A B m( A m( B { A } R ad A A m( A = lim m( A + = ' { A } R ad A A m( A = lim m( A + = Defiiio Le X is he uiverse R is a mappig from P(X o I=[0] ( R F I ( P( X saisfy he followig odiios: X R ( A( A R A R For ay { A } ( ( A R A R The R is alled a fuzzifyig σ-algebra ad (X R a fuzzifyig measurable spae If ( isead of (' he R is alled a fuzzifyig algebra [] (' For ay { A } ( ( A R A R k k k={ k} Defiiio 4 Le (X R is a fuzzifyig measurable spae is a mappig from P(X o I ( FI ( P( X he sig lim( A is deoed as: lim( A := ( ( m(( ( m Am ad saisfy he followig odiios: ( ( ( X ( A( B(( A R ( B R ( A B ( A B For ay { A } k ( A (( A R ( A A + (( A lim( A = ' For ay { A } ( A(( A R ( A+ A (( A lim( A = The is alled a fuzzifyig measure o fuzzifyig measurable spae (X R ad (X R is alled a fuzzifyig measure spae If oly saisfy ( ( ( he is alled a lower-oiuiy fuzzifyig measure If oly saisfy ( ( (' he is alled a upper-oiuiy fuzzifyig measure The Sruure of Fuzzifyig Measure Defiiio Le is a fuzzifyig measure defied o fuzzifyig measure spae (X R he is osidered havig fuzzifyig ull-addiive propery (deoed briefly as 0-add /or fuzzifyig ull-subraive propery [9] (deoed briefly as 0-sub if for ay A B PX ( : ( A( B( A R B R ( ( B m (( A B A /or ( A( B( A R B R ( ( B m (( A B A Theorem Le is a fuzzifyig measure defied o (X R he he followig olusios are equivalee is fuzzifyig ull-addiive ( A( B( A R B R ( A B= ( ( B (( A B A ( A( B( A R B R ( B A ( ( B (( A B A Proof ( ( is obvious ( (: If B AA = ( A B B ad ( A B B= he: (( A B B β( A B R( A B R( B ( ( B α0 ma ( βma ( B R ( A R ( B ( B A ( m ( B α0 ( (: ma ( B B βma ( B R ( AB R ( B ( B A B ( ( B α0 So ma ( βma ( B R( A B R( B 57

3 Sesors & Trasduers Vol 7 Issue 5 May 04 pp 56-6 ( mb ( α0 R( A R( B ( mb ( α0 Theorem Le is a upper-oiuiy fuzzifyig measure o (X R ad is 0-add R = { A R ( A A P( X} for ay ( A R { B } R B+ B We have ( A( B (( (lim( B (lim(( A B A Proof is defied o (X R so ( B = lim ( B Le B= B ad = = for { A B} A B + A B so ( ( AB = lim ( A B = The lim ma ( B β ma ( = ( ( A B β ( A = ma ( B β ma ( = mbα ( 0= lim mb ( α0 Theorem Le is a lower-oiuiy fuzzifyig measure o (X R ad is 0-sub for ay A R { B } R B+ B We have ( A( B (( (lim( B so (lim(( A B A B Proof Le B= is defied o (X = = ( B = lim ( B For { A B } we have AB A B + ( ( AB = lim ( A B = The R lim ma ( B β ma ( = ( ( A B β ( A = ma ( B β ma ( = mbα ( 0= lim mb ( α0 ow we wa o poi ou ha here has ay fuzzifyig measure ( whih has o fuzzifyig ull-addiive propery Example Le is a fuzzifyig measure defied o (X R R =X X={a b} A = X ma ( = If A={a} B={b} 0 A X ma ( B β ma ( = β 0=0 bu mbα ( 0=0α 0=07 So has o fuzzifyig ull-addiive propery Defiiio Le is a fuzzifyig measure defied o (X R ad saisfies: For ay A B PX ( ( A( B ( A R B R ( (lim( B m (lim(( A B A ' ( A ( B ( A R B R ( (lim( B m (lim(( A B A The is osidered havig fuzzifyig auooiuiy propery (deoed briefly as Fauo If oly saisfies ( (or (' is osidered havig fuzzifyig upper-auo (or lower-auo oiuiy propery (deoed briefly as Fauo (or Fauo [ ] Theorem 4 Le is a fuzzifyig measure defied o (X R he If is Fauo is 0-add If is Fauo is 0-sub Proof ( For ay A B PX ( he AB = A B le B=B So lim ma ( B = supif ma ( B = ma ( B m lim ( B = sup if ( B = ( B Ad m ma ( B β ma ( = lim ma ( B β ma ( R( A ( R( B (lim mb ( 0 α = R ( A R ( B ( m ( B α0 Similarly we a prove ( Example is a fuzzifyig measure o (X R le R =X X={ } i ( / π arg( ara (/ A X ( A = i A A = X (ar A=he umber of he members i se A Le B={ + + } he is easy o hek: lim ma ( B β ma ( = lim mb ( α 0 = So is Fauo Bu lim ma ( B βma ( = 0β= 0 is o Fauo Proposiio If is a fuzzifyig measure defied o (X R ad X is fiie he is Fauo if ad oly if is Fauo Defiiio Le is a fuzzifyig measure defied o (X R is fuzzifyig uiformly upper-auo oiuous (deoed briefly as exis Fuauo if i saisfies: for ay ε > 0 58

4 Sesors & Trasduers Vol 7 Issue 5 May 04 pp 56-6 δ = δ( ε > 0 suh ha ( ( A( B( A R B R ( B δ + ( A B G ( A ε ( G ( A ε A B m + (Le [ G ( A ε] = mi( ( A + ε _ [ G ( A ε] = max(0 ( A ε is fuzzifyig uiformly lower-auo oiuous (deoed briefly as Fuauo if i saisfies: for ay ε > 0 exis δ = δ( ε > 0 suh ha (' ( A( B( A R B R ( B δ ( A B G ( A ε ( G ( A ε + ( A B If saisfies ( ad (' he is fuzzifyig uiformly auo oiuous (deoed briefly as Fuauo Theorem 5 Le is a fuzzifyig measure o (X R ad he is Fuauo if ad oly if is Fuauo (We mus poi ou ha if is defied o (X R suh proposiio is o value Proof : For ay ε > 0 exis δ= δ( ε > 0 A B PX ( By A= ( AB ( A B R So ad is defied o (X ma ( B mb ( ma ( B αδ mb ( αδ By is Fuauo we have: (( AB ( AB α + [ G (( A B ε] ma ( B αδ mb ( αδ So mi( (( AB ( A B + mi( ma ( B + ε mbαδ ( mi( ma ( + mi( ma ( B + ε mbαδ ( mi( max(0 ma ( ε + ma ( B mbαδ ( Therefore _ [ G ( A ε] α ( A B mbαδ ( Similarly [ (( G AB ε] α (( AB ( AB ma ( Bαδ mbαδ ( Therefore mi( max(0 ma ( B ε + ma ( mbαδ ( mi( ma ( B + ma ( + ε mbαδ ( mi( ma ( B + mi( ma ( + ε mbαδ ( + So ma ( B α[ G ( A ε] mbαδ ( _ ([ G ( A ε] α ( A B ( ma ( B α[ G ( A ε] mbαδ ( + is Fuauo : Usig he same mehod we a prove he par of resul easily Theorem 6 Le is a fuzzifyig measure defied o (X R he: If is Fuauo is Fauo If is Fuauo is Fauo Proof ( For ay A B PX ( { B } P( X is Fuauo so for ay ε > 0 exis δ = δ( ε > 0 ad ( ( AB α[ G + ( A ε] ([ G ( A ε] α( AB R ( A R ( B ( m ( B α0 For suh ε exis >0 if > we have ( ma ( B α[ G + ( A ε] ([ G ( A ε] αma ( B R( A R( B ( ( B αδ R ( A R ( B ( m ( B α0 So lim ma ( B β ma ( R( A R( B (lim ( B α0 The seod olusio a be proved by he same mehod O he Produ of Two Fuzzifyig Measure Defiiio Le ad are all fuzzifyig measures defied o (X R we deoe he produ of ad as mm whih is defied as: for ay A R mm ( A = ( A ( A [6] ow we mus prove ha mm is fuzzifyig measures o (X R for ay A B PX ( : mm ( = ( ( = 0 mm ( X = ( X ( X = mm ( A αmm ( B = ( ( A ( A α( ( B ( B ( ( A α( B α R ( A R ( B ( A B ( m ( A m ( B Usig he same mehod we a prove mm saisfies he odiios ( ad (' of he Defiiio 4 Defiiio Le ad are all fuzzifyig 59

5 Sesors & Trasduers Vol 7 Issue 5 May 04 pp 56-6 measures defied o (X R is said o be fuzzifyig weakly absoluely oiuous wih respe o ad is deoed by ω if for ay A PX ( : ( A( A R ( ( A m ( A m is said o be srogly absoluely oiuous wih respe o ad is deoed by s if for ay ε > 0 exis δ = δ( ε > 0 suh ha for ay A PX ( : ( A( A R (( A m δ ( A m ε Defiiio is alled o have propery (s (or propery (s' if for ay {B} PX ( A PX ( : ( (( B R ( (lim( B m ( i( (( Bi s= i= s (Or ( (( B R ( A R ( (lim( B m ( (( A( B ( A i s= i= s i Theorem is a fuzzifyig measure o (X R ad is Fauo if ad oly if is 0-add ad has propery (s is a fuzzifyig measure o (X R ad is Fauo if ad oly if is 0-sub ad has propery (s' Proof We oly prove he olusio ( : We oly eed o prove ha has propery (s beause is 0-add has bee proved by Theorem 4 For ay {B} PX ( ε > 0 exis suh ha: (lim( B ( B ε / (* B by is Fauo we have For ( (( > ( (lim( B (lim(( B B B So exis > ad ( (( > ( (lim( B ((( B B mi( ( / 4 B + ε For equaio (* sele some we have ( (( > ( (lim( B ((( B B mi( / 4 ε Do like his agai ad agai we a fid ou suh olusio: (lim( B ( i(( B mi( i i ε = If ε = (lim( B ( i( (( B ( i i = If ε = / (lim( B ( i( (( B / ( i i = Geerally if ε = / (lim( B ( (( B / = i( ( i= i Therefore: (lim( B ( ( (( B i = i= i : For ay A PX ( {B} PX ( { B } { B } i suh ha ( ( ma ( B αma ( = m exis lim ma ( B i αma ( For { B i } by has propery (s we a fid { B } { B } ik = k= i he ( B α 0 lim ( B α 0 By is 0- ik ik add so ma ( ( B β ma ( = k= = k= ( B α0 lim mb ( α0 i k i k i k Ad (if sup ma ( B β ma ( m m = lim ma ( B αma ( = k= i ma ( ( B αma ( So i k (if sup ma ( B β ma ( lim mb ( 0 m i m k α The (supif ma ( B βma ( lim mb ( α0 m m lim ma ( B β ma ( lim mb ( α0 Theorem ad are fuzzifyig measures defied o (X R ad ω If ad are all 0-add he so is mm If ad are all Fauo he so is mm Proof ( For ay A B PX ( mm ( A B β mm ( A = ( ( AB ( AB β ( ( A ( A β β ( ( A B ( A ( m ( A B m ( A 60

6 Sesors & Trasduers Vol 7 Issue 5 May 04 pp 56-6 ( ( B α0 ( ( B α0 By ω we have ( ( B α0 ( ( B α0 he ( B ( B So mm ( A B β mm ( A ( ( A α0 ( ( B α0 = ( B α0 ( ( B ( B α0 = mm ( Bα 0 We oly prove if ad are all Fauo he so is mm I is obvious ha mm is 0-add by Theorem we oly eed o prove ha has propery (s mm For {B} PX ( propery (s so exis { B } { B } = has i we have ( B 0 s i s i α lim ( B α0 For { B i } = = by has propery (s exis { B } { B } i i ( B i α 0 lim ( B α0 So = = m B α ( 0 = = i m ( B α 0 s= i= s lim ( B α0 The ( ( B α 0 ( ( B α0 i = = = = (lim ( B α0 m (lim ( B α0 ( ( B ( B α 0 = = i = = (lim ( B α0 mm ( B 0 i α = = (lim ( B α0 i i i (lim ( B α0 By ω we have B α m B So = = i B = lim mm ( B α0 The mm (lim m ( B α0 lim ( 0 lim ( B α0 ad lim ( lim ( B mm ( B α 0 lim ( B α0 = (lim ( lim m ( B α0 4 Colusios has propery (s Used a semai mehod of oiuous-valued logi sysem fuzzifyig measurable heory ha geeralizes he lassial measurable heory is esablished I his arile he sruure of he ew heory has bee sudied ad some saisfaio resuls have bee gaied We also sudy he produ of wo fuzzifyig measures ad osider a problem whih is like he hird of he ope problems i fuzzy measure preseed by Z Wag We have solved his problem saisfaorily i he ew heory Akowledgemes This researh is parially suppored by he SFC (6604 he Chia Posdooral SF (0M555 he SF of Jiagxi Provie (0BAB00 0BAB00 ad he Foudaio of he Offie of Eduaio Jiagxi Provie (KJLD0 Referees [] M S Yig A ew approah for fuzzy opology (I Fuzzy Ses ad Sysems 9 99 pp 0- [] M S Yig A ew approah for fuzzy opology (II Fuzzy Ses ad Sysems pp - [] J Z She Fuzzifyig Groups based o Compleed Laie valued Logi Iformaio Siees pp [4] C Guo D Zhag O se-valued fuzzy measure Iformaio Siees pp -5 [5] R Dusa S Aleksadr Z Mykhailo Fuzzy Prokhorov meri o he se of probabiliy measures Fuzzy Ses ad Sysems 75 0 pp [6] Q Jiag S Wag D Ziou A furher ivesigaio for fuzzy measures o meri spaes Fuzzy Ses ad Sysems pp 9 97 [7] J Li O Egoroff's heorems o fuzzy measure spae Fuzzy Ses ad Sysems 5 00 pp [8] Q Jiag O he uiform auooiuous fuzzy measures i Proeedigs of he 5h Chia aioal Fuzzy Mahemais ad Fuzzy Sysems 990 pp [9] Z Wag O he ull-addiiviy ad he auooiuiy of a fuzzy measure Fuzzy Ses ad Sysems pp 6 [0] M Sugeo Theory of fuzzy iegral ad is appliaios Tokyo Isiue of Tehology 974 [] G Zhag Theory of fuzzy measurable The Siee ad Tehology Publishig House Guizhou 994 [] M Ha C Wu Theory of fuzzy measure ad fuzzy iegral The Siee Publishig House Beiig 998 [] S H Luo J Z She Fuzzifyig sigma algebra ad is properies Fuzzy Sysems ad Mahemais 9 00 pp 0- [4] S H Luo J Z She Fuzzifyig measure spae ad fuzifyig measure Fuzzy Sysems ad Mahemais pp [5] C Zhu Basis of measurable heory The Siee Publishig House Beiig 998 [6] Z Wag G J Klir Fuzzy Measure Theory Pleum ew York Copyrigh Ieraioal Frequey Sesor Assoiaio (IFSA Publishig S L All righs reserved (hp://wwwsesorsporalom 6