A Statistical Integral of Bohner Type. on Banach Space

Transcription

1 Applied Mathematical cieces, Vol. 6, 202, o. 38, A tatistical Itegal of Bohe Type o Baach pace Aita Caushi aita_caushi@yahoo.com Ago Tato agtato@gmail.com Depatmet of Mathematics Polytechic Uivesity of Tiaa Tiaa, Albaia Abstact I this pape we popose oe type of Bohe itegatio i cotext of statistical covegece. This wats to costuct a ew covegece of fuctios i Baach space to defiite the measuable fuctios. We poved the Egoov theoem to get the elatios betwee st- measuability ad stog st- measuability. The mai esult is costuctio oe type of Bohe itegal as the statistical itegal. We give the example that thee ae fuctios that ae itegable by this type itegatio ad oitegable by classic defiitio ad pove some covegece theoems. Mathematics ubject Classificatio: 40A30, 40A99, 60B99 Keywods: statistical covegece, st-measuability, st-itegal

2 6858 A. Caushi ad A. Tato. Itoductio It is ow that the idea of statistical covegece was give by Zigmud [] sice 935 i his well ow wo Tigoometic seies. The cocept of was fomalized by teihaus[9] ad Fast [2]. ome yea latte the cocept was eitoduced by choebeg [7]. tatistically covegece has become a active aea of eseach i ecet yeas. This cocept appeas i vaious studies about umbe theoy, measue theoy, summability theoy, Baach space etc. I this pesetatio we follow cocepts itoduced by Fidy[4] about the covegece of sequeces ad the cocept of choebeg about itegatio the basic cocept is the statistically Cauchy covegece of Fidy[3]. O the Baach space we adopted this cocept fom the wo of Coo etc.[](989). We geealized to the Baach space some esults achieved by Goha [6] (2002) o eal lie ad study a statistically type of Boche itegal. 2. Pelimiaies Let be A a subset of odeed atual set Ν. It said to have desity δ(a) if A δ ( A) = lim, whee A = {< : A} ad with A deotes the cadiality of the set A. It is clea that the fiite sets have the desity zeo ad δ(a )= -δ(a) if A'=Ν-A. If a popety P()={ : A} holds fo all A with δ(a)=, we say that popety P holds fo almost all that is a.a.. The vetoial sequece x is statistically coveget to the vecto(elemet) L of a vectoial omed space if fo each ε >0 lim { : x L ε} = 0 i.e. x L <ε a.a.. We wite st-lim x =L. I same mae, the sequece x is a statistically Cauchy sequece if fo evey ε>0, thee exists a umbe N=N(ε) such that x x N <ε a.a.. Now, we deals with geealizatio of poitwise statistically covegece of fuctios o omed space. The sequece of fuctios {f } cotais the fuctios with value i vectoial space. Fo each x of the domai we coside the fuctioal sequece (f (x)). We

3 tatistical itegal of Bohe type 6859 deote with the set of x whee the sequece {f (x)} coveges. The fuctio f defied as f ( x) = lim f( x) ; x is called the limit fuctio of the sequece {f }, we say that sequece {f } coveges poit wise to f fo evey x of. This meas that fo evey poit x i ad fo evey ε>0,thee exists N(depedig o both x ad ε) such that >N implies f (x)-f(x) <ε. Defiitio. A sequece of fuctios {f (x)} is said to be poitwise statistically coveget to f if fo evey ε>0 lim { : f ( x) f( x) ε, x } = 0, i.e. fo evey x, f (x)-f(x) <ε. st a.a.. We wite st-lim f (x)=f(x) o f f o. This meas that fo evey δ>0, thee exists itege N such that () lim { : f ( x) f( x) ε, x } < δ Fo all >N = (N(ε,δ,x)) ad fo evey ε>0. If the iequality i () holds fo all but fiitely may, the usual limes, lim f (x)=f(x) o. It follows that this limes implies st- lim f( x) = f( x).but the covese of this is ot tue. Example 2. We use the well ow example that poves this oe. 2 x u =m x = 2 0 u m =,2,... is q diveget sequece i usual cocept of limes of fuctios but it coveges to zeo by the statistically covegece. Futhe, we deote (,,μ) the pobability measue space, whee is ay set ad Σ sigma algeba of Boel. Defiitio 3. A fuctio f : X, whee X is a vectoial omed space is called simple fuctio by μ, if thee is a fiite sequece measuable sets {E i }, such that E i, i=,..., E i E j = fo i j, = U Ei ad f(s) = x i fo s E i, i=

4 6860 A. Caushi ad A. Tato It epeseted i a fom f = x iχ i= Ei, whee χe i is a chaacteistic fuctio of E i. We deote T(μ,X) the set of simple fuctios with domai. T(μ,X) is a vectoial space with the additio of simple fuctio ad multiple with the eal umbe. The simple fuctios as it is ow ae the measuable fuctios. Defiitio 4. The fuctio f: X is called statistically measuable by μ o set ( i shot fom st- measuable) if thee exists a sequece of simple fuctios (f ) T(μ,X) that fo evey s ad evey ε>0 holds: lim { : f ( s) f( s) ε, x } = 0. fo almost all s. Pepositio 5. A liea combiatio of st-measuable fuctios is st-measuable fuctio.. Poof. Let (f ) ad (g ) be two sequeces of fuctios of the set T(μ,X) with domai such that st-lim f (x) = f(x) ad st-lim g (x) = g(x) with α,β Ρ. Let we obseve the otivial case whe α 0 ad β 0. Let ε>0 be give ad ote that, { : αf (x) + βg (x) (αf(x) + βg(x) ε x } { : f (x) - f(x) ε 2 α x } { : g ε (x) g(x) 2 β x } ice αf (x) + βg (x) (αf(x) + βg(x) α f (x) - f(x) + β. g (x) g(x) Hece, we get st lim (αf (x) + βg (x)) = αf(x) + βg(x). Defiitio 6. The fuctio f : X is called statistically stog measuable by μ o if evey δ >0 ad evey ε>0 thee exists a itege N(ε, δ) such { : f ( ) ( ) } s f s ε < δ fo > N(ε, δ) almost fo evey s. I this case is said the sequece statistically stog coveges almost eveywhee uifomly by μ to the fuctio f o. Now we modify some techiques developed fom [8] to pove the followig pepositio. Theoem 7.(Theoem Egoov). If a fuctio f : X is st-measuable by μ, the it is st- stog measuable uifomly almost eveywhee o.

5 tatistical itegal of Bohe type 686 Poof: ice the fuctio f : X is st-measuable by μ, the thee exists the measuable set Z with μ(z)=0 ad the sequece of simple fuctios (f ) such that fo evey s /Z it coveges statistically i poitwise way to f(s), lim { : f ( s) f( s) } = 0 We costuct the sets E, = {s \ Z : f (s) f(s) <, A ' } whee ' A =Ν \ A It is clea that se E, E, + ice the sequece (f (s)) coveges to f(s) fo evey s \Z ad evey, we wite U E =, \Z o I ( \ E ) =, Z Fom this iequality follows that fo evey ε>0 ad evey Ν we ca fid a itege such that ε μ ( \ E, ) < 2. We set Aε = I E, the = =, =, μ ( \ A ) = μ ( \ E ) = μ ( ( \ E ) ε ε I U This yields that f (s) f(s) <, fo evey A '.This meas that the sequece (f ) statistically uifomly coveges to f(s) o \A ε. 3. The statistically itegal of Boche type Defiitio 8. The itegal of the simple fuctio f : X, is called the elemet of vectoial omed space x μ( E ), symbolically i= i f () sdμ = = xiμ( Ei) i i I case whe E is a measuable set ad E, the itegal of simple fuctio f o E is the itegal of fuctio f χ E, we wite f () sdμ= ( f. χ )() sdμ E E

6 6862 A. Caushi ad A. Tato We defie the map * T : T(μ,E) Ρ ; f = f( s) dμ, T It is easy to pove that f T is a semiom. Followig the defiitio of Cauchy sequeces itoduced by Fidy [3] ad thei extesio to the fuctioal sequeces (see fo example to [6]). The sequece (f ) is called the statistically Cauchy sequece if fo evey ε>0 thee exits a itege N(=N(ε,x)) with lim { : f ( x) fn( x) ε x } =0 O the set of st-cauchy simple sequece we defie the equivalece elatio: (f ) (g ) st lim f -g = 0. We the followig theoem we exted i case of Baach space the esult peseted i [6]. Theoem 9.[3],[6] Let (f ) be a sequece of fuctios o a set with value to Baach space X. The followig statemets ae equivalet: a) the sequece (f ) is poitwise statistically coveget o ; b) the sequece (f ) is statistically Cauchy sequece o. Poof. It easy to pove that a) implies b) because we ca do as i classic case whee evey coveget sequece is a Cauchy sequece. Assume that st lim f ( s) = f ( s) o ad let be ε>0. The lim { : f ( s) f( s) ε} = 0 o f N (s)-f(s) < 2 ε a.a. ad we choose a idex N such that fn (s)-f(s) < 2 ε. The pove follows fom the iequality ε ε f (s)-f N (s) f (s)-f(s) + f N (s)-f(s) < a.a. ad evey s. Hece (f ) is statistically Cauchy. Let suppose that b) holds ad choose a idex N such that closed ball B(f N (s), ) cotais f (s) a.a.. ad evey s. We use agai b) ad choose a idex M such that the boule B (f M (s), 2 ) cotais f (s) a.a.. ad evey s. Deote B =B B it seems that B cotais f (s) a.a.. ad evey s. We have

7 tatistical itegal of Bohe type 6863 { : f (s)} B } { : f (s)} B} { : f (s)} B } so lim { : f ( s) B fo evey s } lim { : f ( s) B fo evey s } + lim { : f ( s) B' fo evey s } =0 We tae hee the close ball B with adius less tha o equal to that cotais f (s) a.a.. ad evey s. Latte we poceed by choosig a idex N(2) such that the closed B (f N(2), 2 2 ) cotais f (s) a.a.. ad with the same agumet the closed ball B 2 =B B cotais f (s) a.a. ad evey s ad adius of B 2 is less tha o equal to. Cotiuig iductively we ca costuct the sequece of 2 B closed balls ( m ) m = such that fo evey m, B m B m+ ad adius of B m is less tha o equal to 2 -m, ad f (s) B m a.a.. ad evey s. Thus by vitue of completeess of Baach space X, thee exists a fuctio f(s), defied o such that {f} is uiquely i I Bm. m= Lemma 0. If the sequece of simple fuctios (f ) is a st-cauchy sequece o Baach space tha exists st- lim ( ) f sdμ. ε Poof. We have that f (s)- f N (s) < a.a. ad a itege N, μ ( ) so f ( s) dμ f ( s) dμ f ( s) f ( s) dμ N N f ( s) f ( s) μ( ) < ε a.a. ad itege N. N We obtai that ( () sdμ ) is a st-cauchy sequece i X by the om, it implies f that the sequece is st-coveget. Let X is a sepaable Baach space. Defiitio. The fuctio f : X is called st- Boche itegable if thee exists a st- Cauchy sequece of simple fuctios (f ) such that : i) statistically coveget a.e. by μ to the fuctio f ;

8 6864 A. Caushi ad A. Tato ii) st lim f( s) fn( s) dμ = 0 a.e. s st-lim f () sdμ is called st- Boche itegal ad deote with (B s ) ( ) f xdμ s This sequece (f ) of simple fuctios is called detemiat of fuctio f. Theoem 2. If (f ) ad (g ) as st-cauchy sequeces ae detemiats of the same fuctio f tha st-lim f( sdμ ) = st-lim g () s dμ Poof. The iequality f (s)-g (s) f (s)-f(s) + f(s)-g (s) shows that (f ) ad (g ) ae equivalet st-lim f-g =0 o fo evey ε>0 f (s) g (s) <ε a.a.. By the defiitio of itegal: f () s ( st lim f () s dμ)) < ε g s st g s dμ < ε ad () ( lim () )) a.a. ad evey s. Coside the diffeece ( st lim f ( s) dμ) ( st lim g ( s) dμ) + () μ () <3ε ( st lim f ( s) dμ) g ( s) dμ) g () s dμ) ( st lim g () s dμ) Ad we have poved the above equality. f sd g sdμ + It easy to watch that usual Bohe itegal is a st-bohe itegal but the evees is ot tue. Example 3. Let (f ) is a sequece defied by fomula p p ( + )( x) fo [3,3 + p[, p=, 2,... f ( x) = 0 o the cotay If we tae x Ρ \[-,], =,2,... the p( p+ ) { : f ( x) 0 wheeve x R\[,] p+ 3 o, we have that st-limf (x) = 0 o Bs f ( ) 0 R\[,] x dμ =, O the othe had, the usual itegal is udefiite

9 tatistical itegal of Bohe type 6865 B ( )( x) dx ( ) ( B + ) + = ± 3. The popety of statistically itegal Theoem 4. If the fuctio f is st-bohe itegable the the fuctio f is also st-bohe itegable. Poof. Followig defiitio of st-itegability, thee exists the sequece of simple fuctios f coveget almost eveywhee ad a.a.. to the fuctio f ad f( s) fn( s) dμ < ε a.a.. We coside the iequality f - f f f. Hece st-lim f (s)=f(s) a.e., it follows that st-lim f (s) = f(s) a.e. Iequality f f dμ f f dμ hows that f is st-itegable. N N The equality st lim f dμ = ( Bs) fdμ,whee (f ) is sequece of simple fuctios detemiat to f, ad the well ow popeties of classical itegal allow us to fomulate the followig popeties of st- Bohe itegal: (I) (Bs) ( α f ( s) + βg( s)) dμ = ( Bs) α f( s) dμ+ ( Bs) β g( s) dμ (II) If A=A A 2, A A 2 = ad (f (s)) is a sequece of simple fuctios detemiat of the fuctio f(s) the f() sdμ= f() sdμ+ f() sdμ A A A2. If we tae the limes of above equality wheeve, we have ( Bs) f( sd ) μ = ( Bs) f( sd ) μ+ ( Bs) f( sd ) μ A A A2 (III) ( Bs) fdμ ( Bs) f dμ This iequality we obtai fom the same iequality fo simple fuctios ad isotoic popety [9]. f dμ f dμ

10 6866 A. Caushi ad A. Tato (IV) Applyig the popety (III) fo the fuctios statistically bouded [5], if f(s) K, we have ( Bs) f dμ ( Bs) f dμ Kμ( ) ε I case whe C is a subset of C such that μ( C) < δ = fo evey ε>0 we get K ( Bs) f dμ <ε. C (V) The iequality fo the simple detemiat fuctios f g a.a. implies f dμ g dμ a.a.. Isotoic popety of itegals gives ( Bs) f dμ ( Bs) g dμ (VI) If A ad A 2 ae subset of σ-algeba Α ad A A 2 the ( Bs) f dμ ( Bs) f dμ A A2 Iequality follows by the popety ( Bs) f dμ = ( Bs) f χ dμ ( Bs) f dμ A A A2 A2 (VII) Chebishev iequality Let f ad f be the st-itegable fuctios. If fo evey ε>0 we have f (s)-f(s) ε fo evey s A ε whee A ε Α ad A ε A the fom popety(v) we have ( Bs) f f dμ ( Bs) f f dμ εμ( A ) A Aε This iequatio implies that fo evey ε ad A μ( x : f f ε} ( Bs) f f dμ ε A ε 4. Covegece theoems of statistically itegal Theoem 5( Fatou) Let (f (s)) be the sequece of st-measuable fuctios statistically coveget almost eveywhee to the fuctio f(s). If fo a.a. ad evey s f (s) f + (s), the

11 tatistical itegal of Bohe type 6867 st lim f dμ = ( Bs) f ( s) dμ Poof. We peset evey fuctio f as a limes of o deceasig sequece of simple fuctios f coveget to f a.a.. Fo example as such sequece we ca use ( ) ( )2 0,,...,2 h fo h f s < h+ h= f = 2 fo f ( s) 2 Now, we costuct the simple fuctios g (s) = max{ f ( s),..., f ( s ) } We ote that f () s f() s fo evey ad almost all also f (s) f (s) fo all <. We obtai the iequalities () f () s g () s f() s By the st-itegal popeties we have (2) ( Bs) f ( s) dμ ( Bs) g ( s) dμ ( Bs) f ( s) dμ By vitue of isotoy ad iequality () we get f ( s) st lim g f( s). The limes show that fo almost all the sequece (g (s)) is a odeceasig sequece bouded by above fom the fuctio f(s). Taig i accout that f (s) f(s) almost evey s ad almost all m we get g (s) f(s) almost evey s ad almost all. Hece the sequece ( f ) is also sequece of simple fuctios by the defiitio of st-itegal we wite f ( s) dμ f dμ st lim f dμ so. st lim f dμ = ( Bs) f ( s) dμ Theoem 6. Let f(x) be the fuctio with value i sepaable Baach space ad st-measuable by a pobability measue. If fo almost all s holds the iequality f(s) g(s), whee g(s) is a fuctio statistically itegable the the fuctio f(s) is statistically itegable.

12 6868 A. Caushi ad A. Tato Poof. We deote f m the sequece of simple detemiat fuctios that coveges statistically to the fuctio f. It holds m m f f + f f f + ε m < f + f 2 so m f 2 f 2g almost all ad m. ad m m m m f ( s) fn ( s) f ( s) + fn ( s) < 4 g( s) By the popety (V) of st-itegal, we have m m ( Bs) f ( s) f ( s) dμ ( Bs)4 g( s) dμ N a.a.. If we tae the measuable set C such that μ(c)<δ i vitue of popety (IV) fo the st-itegal, we get gsdμ ( ) < ε. o, we pove C C m m f f dμ < ε N m Hece the sequece ( f ) coveges almost eveywhee o ad a.a. to the m m fuctio f which implies that measue of set B={s : f ( s) f λ, λ>0} is zeo. m Let \B be the set {s : f ( s) f m N <λ}. By the popety (II) of st-itegal f f dμ = f f dμ+ f f dμ N N N B \ B <ε + λ μ( ) o, we have st- lim f( s) fn( s) dμ = 0. The poof is completed if we substitute the fuctio g(s) with f(s). Theoem 7. ( Lebesgue) Let (f (s)) the sequece of st-measuable fuctios with value i sepaable Baach space coveget to the fuctio f(x) almost eveywhee ad a.a.. If fo the fuctios f (s) holds a.a.. the iequatio f (s) g(s),whee g(s) is st- itegable fuctio, the ( Bs) f( s) dμ = st lim f ( s) dμ. N

13 tatistical itegal of Bohe type 6869 Poof. By the theoem 6, the fuctios f ae st-itegable. The iequality f (x) g(s) ad covegece f (s) f(s) almost eveywhee ad a.a. implies iequality f(s) g(s) a.e. This meas that the fuctio f(s) is st-itegable ad except this the iequality 0 f (s)-f(s) 2g(s) holds a.e. This poves that the fuctio f (s)-f(s) is st-itegable a.e. st lim f ( s) f ( s) dμ = 0 Iequality f ( sd ) μ f( sd ) μ f( s) f( s) dμ yields that sequece ((Bs) f ( sdμ ) ) is st-coveget moeve uifomly to the st-itegal (Bs) f ( sdμ ). s Refeeces [] J. Coo, M. Gaichev, ad V. Kadets, A chaacteizatio of Baach spaces with sepaable duals via wea statistical covegece, Joual of Mathematical Aalysis ad Applicatios, vol. 244,o., pp , 989. [2] H. Fast, u la covegece statistique, Colloquium Mathematicum, vol. 2, pp , 95. [3] J. A. Fidy, O statistical covegece, Aalysis, vol. 5, o. 4, pp , 985. [4] J. A. Fidy, tatistical limit poits, Poceedigs of the Ameica Mathematical ociety, vol. 8, o. 4, pp , 993. [5] J. A. Fidy, C. Oha, tatistical limit supeio ad limit ifeio, Poc. Ame. Math. oc., 25,. 2(997) [6] A.Göha, M. Gügö, O poitëise statistical covegece, Idia Joual of pue ad applicatio mathematics, 33(9) : , 2002.

14 6870 A. Caushi ad A. Tato [7] I. J. choebeg, The itegability of cetai fuctios ad elated summability methods, The Ameica Mathematical Mothly, vol. 66, o. 5, pp , 959. [8]. chwabi, Y. Guoju, Topics i Baach space itegatio, eies i Aalysis vol. 0. Wold cietific Publishig Co. igapoe [9] H. teihaus, u la covegece odiaie et la covegece asymptotique, Colloquium Mathematicum, vol. 2, pp , 95. [0] B. C. Tipathy, O statistically coveget sequece, Bulleti of the Calutta Mathematical ociety, vol.90,.4.pp ,998 [] A. Zygmud, Tigoometic eies, Cambidge Uivesity Pess, Cambidge, UK, 979. Received: eptembe, 202