ON THE CONVERGENCE THEOREMS OF THE McSHANE INTEGRAL FOR RIESZ-SPACES-VALUED FUNCTIONS DEFINED ON REAL LINE

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1 O The Covegece Theoems... (Muslm Aso) ON THE CONVERGENCE THEOREMS OF THE McSHANE INTEGRAL FOR RIESZ-SPACES-VALUED FUNCTIONS DEFINED ON REAL LINE Muslm Aso, Yosephus D. Sumato, Nov Rustaa Dew 3 ) Mathematcs Depatemet, Uvestas Lampug Jl. Soemat Bodoegoo No. Bada Lampug. Emal : asomath@yahoo.com ) Mathematcs Depatemet, Uvestas Dpoegoo Jl.Pof.H. Soedato,SH,Tembalag,Semaag. Emal :sumato_3@yahoo.com 3) Mathematcs Dept. Uvestas Swaya,Palembag Abstact Ths pape s a patal esult of ou eseachs the ma topc "O The McShae Itegal fo Resz-Spaces-valued Fuctos Defed o the space R ". We have costucted McShae tegal fo Resz-spaces-valued fuctos defed o the space R by a techque volvg double sequeces ad poved some basc popetes whch cocdes wth the McShae Itegal fo Baach-spaces valued fuctos defed o eal le. Futhe, we costuct some covegece theoems volvg ufomly covegece theoems, mootoe covegece theoems ad Fatou s lemma the sese of ths tegal. Keywods : Resz Space, McShae Itegal. INTRODUCTION The ecet esults of covegece theoems Itegal theoy wee the Hestock- Kuzwel tegal ad the McShae tegal. Sce, the Hestock-Kuzwel tegal fo Resz-space-valued fuctos defed o bouded subtevals of the eal le ad wth espect to opeato-valued measues was vestgated by Reca(989,99) ad Reca ad Babelova(996), wth espect to D - covegece (that s a kd of covegece whch the -techque s eplaced by a techque volvg double sequeces, see Reca ad Neubu(997)), wth espect to the ode covegece, see Boccuto(998) ad Boccuto ad Reca(004) wth espect to the ode covegece but the Hestock-Kuzwel tegal fo Resz-space-valued fuctos was defed o ubouded subtevals of the eal le ad futhe, Aso (007) have also costucted Hestock- Kuzwel tegal fo Resz-space-valued fuctos defed o Eucldea spaces R ad 5

2 Vol. 3, No., Desembe 007: 55 McShae tegal fo Resz-spaces-valued fuctos defed o the eal le have bee costucted by a techque volvg double sequeces (Sumato ad Aso007), the the covegece theoems ae also be sgfcat to be vestgated. The ma goal of ths pape s to cotuct some covegece theoems the sese the McShae tegal fo Resz-spaces-valued fuctos defed o the eal le.. PRELIMINARY Let N be the set of all stctly postve teges, R the set of the eal umbes, set of all stctly postve eal umbes. Let... a a b a b b., R be the ab s called teval o cell. a b, a, b R, a b, 0,,,...,. A collecto of tevals a b s called oovelappg f the teos ae dot. A addtve fucto o a set a, b R s a fucto F defed o the famly of all subtevals of ab, such that F B C F B F C fo each pa of oovelappg tevals B, C a, b. A oegatve addtve fucto o a set ab, s called legth o, ab, defed by, P A, x, A, x,..., A, x be a collecto of pas of A, x, whee A A oovelappg tevals, A A ad, a b b a. Let,,..., A ae A O x x, whee O x, x s ope ball wth cete x ad adus x,,,...,. We say that P s fe McShae Patto o A. The eal vecto space L (wth elemets E, E,... ) s called a odeed vecto space f L s patally odeed whch satsfy : E E E h E h fo evey hl ad E 0 ke 0 fo evey k 0 R. If, addto, L s lattce wth espect to the patal odeg, the L s called a Resz space. Fo example, wse addto ad scala multplcato. ad R wth the famla coodate by coodatewse odeg,.e., fo x x,..., x, y y,..., y, we defe, x y wheeve x y k, the R s a Resz space. k k,,..., 6

3 O The Covegece Theoems... (Muslm Aso) Defto. (Zaae,996) : A Resz space L s sad to be Dedekd complete f evey oempty subset of L, bouded fom above, has supemum L. Defto. (Reca, 998) : A bouded double sequece a,, o D -sequece f, fo each N, a 0, that s a, a, N ad L s called egulato a. 0 Defto.3 (Boccuto ad Reca, 004) : Gve a sequece L. Sequece s sad to be D -covegece to a elemet L f thee exst a egulato a, satsfyg the followg codto : fo evey mappg : N N, deoted by N N, thee exsts a tege 0 such that a, fo all 0. I ths case, the otato s deoted by D lm. Defto.4 (Boccuto ad Reca, 004) : A Resz Space L s sad to be weakly dstbutve f fo evey D - sequece a, the a 0. Thoughout the pape, we shall always assume that L s Dedekd complete weakly dstbutve Resz space. Hee ae some ecet esults (Sumato ad Aso007): Defto 3. : A fucto f : a, b R L s sad to be McShae tegable deoted by f M a, b, L,, f thee exsts a elemet that fo evey N N we ca fd a fucto : a, b R such that E L ad D -sequece a L such,, k P f x I E f x I E a fo evey -fe M patto P a, b, x I, x, I, x,..., I, x o, k ab. We ote that the McShae tegal wth espect to s well- defed, that s thee exsts at most oe elemet E, satsfyg Defto 3. ad ths case we have M fdx E. ab, 7

4 Vol. 3, No., Desembe 007: 55 Futhemoe we have some fudametal popetes of M A, L,. Theoem 3.3 : If f, f M A, L, ad k, k R, the M kf kf dx k M fdx k M fdx. a, b a, b a, b kf kf M A, L, ad Theoem 3.4 : If f, g M a, b, L, ad f x g x fo evey x A, the M fdx M gdx. a, b a, b Defto 3.5 (Elemetay Set): A set a, b R whch s uo of fte cells s called a elemetay set. Evey elemetay set ca be segmeted to o-ovelappg cells. If A ad A ae elemetay sets the A A ad A \ elemetay set ca be costucted though the followg theoem. A ae also elemetay sets. Itegato o Teoema 3.6 : Let A ad A be o-ovelappg tevals R ad A A A. If f M A L ad f M A L, the,,,,,, f M A L ad Coolay 3.7 : Gve a elemetay set M fdx M fdx M fdx AA A A A A R. A fucto : f A L s sad to be McShae tegable o A, deoted by f M A, L,, f f M A, L, fo evey whee p A A ad,,..., fucto f o A s A A A s ay oovelappg tevals of A. The McShae tegal of M fdx M fdx A A The Cauchy cteo was gve the followg theoem. Theoem 3.8 : A fucto f : a, b L s McShae tegable f ad oly f thee exsts a D sequece a L such that, fo evey N N we ca fd a fucto, 8

5 O The Covegece Theoems... (Muslm Aso) a b R ad fo evey fe M- patto P a, b, x ad,, :, ab,, we have P f x I P f x I a., We povde a esult about McShae tegablty o subcells. P a b x o Theoem 3.9 : Let a, b R. If f M A, L,, the f M B, L,, fo evey teval B a, b. By usg Theoem 3.9, we defed pmtf fucto of McShae tegable fucto f o a cell a, b R as follows. Defto 3.0 : If f M a, b, L, ad I a, b s a collecto of all subcells, the a fucto F : I a, b L satsfyg fo evey teval J I a, b. F J M fdx ad F J 0 ab, I A s called Pmtf of McShae tegable fucto f o MAIN RESULTS Covegece of a sequece of fuctos s always assocated wth ts lmt ad fucto popety the sequece. Let f be a sequece fucto defed o a, b R. The ths sequece s sad to be coveget to a fucto f o ab, f fo evey x a, b. A sequece of fuctos deceasgly mooto f f x f x o lm f x f x f s called ceasgly mooto o f x f x fo evey, espectvely. Let f : a, b L fo evey N. The a sequece fucto x a b, f s 9

6 Vol. 3, No., Desembe 007: 55 coveget to a fucto f at, a L such that fo evey N N, we ca fd fo evey 0. A sequece fucto oly f sequece sequece fucto thee exsts such that x a b f ad oly f thee exsts 0, f x f x a N, such that D -sequece f s coveget to a fucto f o ab, f ad f x s coveget to a fucto f x fo evey x a, b. A f s ufomly coveget to a fucto f o A f ad oly f f D -sequece fo evey 0, x a, b. b L such that fo evey N N, we ca fd,,, f x f x b N, 0 Now, we gve some covegece theoems fo McShae Itegable fuctos wth values L. Theoem 4..(Ufomly Covegece Theoem). Let a, b R ad f,,, M a b L. If sequece of fuctos fucto f o A, the f Poof: A sequece fucto M a, b, L, ad lm M fdx M f dx a, b a, b f s coveget ufomly to a f s ufomly coveget to a fucto f o ab, f ad oly f f thee exsts D -sequece a L such that fo evey N N, we ca fd 0 N, such that fo evey 0, x a, b.,, I f x f x a (.) 0

7 O The Covegece Theoems... (Muslm Aso) A fucto f M a, b, L, f thee exsts N N we ca fd a fucto : A R such that k D -sequece b,, k I I P f x I M f d b L such that fo evey fo evey -fe McShae patto P a, b, x I, x, I, x,..., I, x o,,,,,, ad f P x a b P x a b ae - fe McShae patto o ab,, the Based o (.) ad (.3), we have ab. P f x I P f x I c (.3) m P f x I P f x I P f x I P f x I whee m m P f x I P f x I m P f x I P f x I a a I c I I I d (.4) d a c.ths shows that f HK a, b, L,. Futhemoe, sce f M a, b, L,, the thee exsts D -sequece b L such that fo evey N N we ca fd a fucto ':AR such that k,, k A A f x I HK f d b fo evey -fe McShae patto P ' A, x o A. By takg x x x fo evey x A ad f P A, x s abtay - fe McShae m ', patto o A, the we have

8 Vol. 3, No., Desembe 007: 55 M f d M f d M f d P f I A A A whee lm P f I P f I P f I M f d b a A A c A A e A, (.6) e a b c. Thus, we have poved that M fdx M f dx. a, b a, b Theoem 4. (Mooto Covegece Theoem). Let f a, b, L,. If sequece of mooto fuctos o ab, ad lm M f dx exsts, the f M a, b, L, ad ab, lm M fdx M f dx a, b a, b Poof: We ust eed to pove fo sequece Based o assumpto, we ca fd a, b R be a cell ad f s coveget to a fucto f f of ceasgly mooto o ab,. EL so that lm M f dx E. Sce ab, f of ceasgly mooto o ab,, t follows that ths sequece, M f dxs ab, ceasgly mooto ad E as ts least uppe boud. Hece, thee exsts D -sequece a Sce L such that fo evey N N,thee exsts,, N, such that f 0, we obta 0 E H f dx a (.) ab, f s coveget to a fucto f o ab,, the thee exsts D -sequece b L such that fo evey N N ad x a, b thee exsts m0 m0, x such that f

9 O The Covegece Theoems... (Muslm Aso) 0 m, we have, f x f x b (.) Sce f M a, b, L, fo evey, t follows that thee exsts D -sequece c such that fo evey N N, we ca fd a postve fucto : A 0, such that fo evey -fe McShae patto,, P a b x o ab,, we have,, c P f x I F A (.3) Takg postve fucto : A 0, whee x x fo evey x a, b ad m, x m, x max 0, m0, x. Hece, f P a, b, x s -fe McShae patto o ab,, the L P f x I E f x I E k k m, x f x I f x I m, x m, x f x I M f dx I m, x M f dx E b k c ( I ) a I k f k I (.4) whee, f a b c. Ths shows that f M a, b, L, ad lm Futhemoe, Let Theoem 4.3. Let the f f M a, b, L,. M f d M fdx E. a, b a, b f f,f,... f f f be fmum of sequece a, b R be a cell. If f, g M a, b, L, ad f. f g fo evey, 3

10 Vol. 3, No., Desembe 007: 55 Poof: Let h x m f x, f x,..., f x be defed fo evey x a, b. It follows that h x g x fo evey x a, b. Thus, we have a deceasgly bouded sequece h whee f fo evey, the that f h x g x be ts boud fo evey x a, b. Sce f M a,, L, h M a, b, L, ad sce k g h f fo evey k, t follows k M gd M h dx M f dx (3.) a, b a, b a, b ad lm M h dx exsts, say E lm M h dx. Thus, ab, ab, fo evey k. Ad, fom Theoem, ths shows that k M gdx a M f dx a, b a, b f f M a, b, L,. Theoem 4.4. (Fatou s Lemma). Let a, b R ad f M a, b, L, fo evey. If k f lm f f ad f lm f M k f dx, the f M a, b, L, ad k ab, Poof: Let lm M fdx M f dx a, b a, b h x k f f be defed fo evey x a, b. It follows that k ceasgly sequece o A ad h x f x fo evey x a, b. Thus, h s ad a, b a, b h s coveget to f. Sce lm covegece Theoem, we have lm M h dx lm M f dx (4.) f a, b, L, ad ab, lm lm f M h dx exsts, the ude Mooto M fdx M h dx M f dx. k a, b a, b a, b Coolay 4.5.Let a, b R be a cell, If lm f f the f, g M a, b, L, ad f a, b, L, ad M fdx lm M f dx. a, b a, b g f g fo evey. 4

11 O The Covegece Theoems... (Muslm Aso) CONCLUDING REMARKS The covegece theoems fo the McShae tegal fo Resz-space-valued fuctos defed o eal le have bee bult by geealzg the esults the McShae tegal fo bouded-sequece-space-valued fuctos defed o eal le ad by usg by a techque volvg double sequeces. REFERENCES Aso M. (007). O The Hestock-Kuzwel Itegal fo Value Resz Spaces Defed o Eucldea Spaces, Poceedg of Natoal Sema o Reseach,Math. ad Math. Ed.,5 Augt. 007, FMIPA, UNY, Yogyakata. Boccuto, A. (998). Dffeetal ad tegal calculus Resz Spaces, Tata Moutas Math. Publ.,4, Boccuto, A ad Reca, B. (004). O The Hestock-Kuzwel Itegal fo Resz-Space- Valued Fuctos Defed o Ubouded Itevals, Chech. Math. Joual, 54,3, Pfeffe, W.F. (993). The Rema Appoach to Itegato, Cambdge Uvesty Pess. Reca, B. (989). O the Kuzwel Itegal fo Fuctos wth Values Odeed Spaces I, Acta Math. Uv. Comea , Reca, B. (99). O Opeato Valued Measues Lattce odeed Goups, Att. Sem. Mat. Fs. Uv. Modea, 40, Reca, B ad Neubu, T. (997). Itegal, Measue ad Odeg, Kluwe Academc Publshes, Batslava. Reca, B ad Vabelova, M. (996). O The Kuzwel tegal fo Fuctos wth Values Odeed Spaces III, Tata Moutas Math. Publ.,8, Sumato, Y.D.,ad AsoM. (007). O The McShae Itegal Fo Resz Space Valued Fuuctos Defed o Real Le, Poceedg of Natoal Sema o Math. Ad Math. Educato, Nop. 007, UNY, Yogyakata. Zaae, A.(997). Itoducto to Opeato Theoy Resz Spaces, Spge Velag. 5