Int. Journl of Mth. Anlysis, Vol. 8, 2014, no. 19, 901-906 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/10.12988/ijm.2014.4367 The Bochner Integrl nd the Wek Property (N) Besnik Bush Memetj University of Vlor, Science Nturl Fculty Informtics Deprtment, Vlor, Albni Copyright c 2014 Besnik Bush Memetj. This is n open ccess rticle distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited. Abstrct The Russin mthemticin N. N. Luzin recognized the importnce of the condition which now bers his nme nd is lso referred to s property (N). We present descriptive chrcteriztions of Bochner integrl in terms of the wek property (N). Mthemtics Subject Clssifiction: Primry 28B05, 26A45, 26A46 ; Secondry 46G10, 26A48, 58C07 Keywords: Bochner integrl, Bnch spce, wek property (N), strong bounded vrition 1. Introduction nd Preliminries It is known tht Bnch spce-vlued function F : [0, 1] X is the primitive of Bochner integrble function f :[0, 1] X if nd only it F is sac nd F (t) =f(t) lmost everywhere on [0, 1], Theorem 7.4.15 in [11]. For more informtion bout Bochner integrl we refer to [2], [6], [1] nd [11]. In this pper, we present descriptive chrcteriztions of Bochner integrl in terms of the wek property (N), see Theorem 2.3 nd Theorem 2.5. Let X be rel Bnch spce with its norm.. We denote by B(x, ε) the open bll with center x nd rdius ε>0 nd by X the topologicl dul to X. We denote by I the fmily of ll non-degenerte closed subintervls of [0, 1], by λ the Lebesgue mesure on [0, 1] nd by λ the outer Lebesgue mesure. The intervls I,J I re sid to be nonoverlpping if int(i) int(j) =, where int(i) denotes the interior of I. If point function F :[0, 1] X is given, then we define the intervl function F : I X by F ([u, v]) = F (v) F (u), for ll [u, v] I.
902 Besnik Bush Memetj Assume tht n intervl [, b] [0, 1] nd function F :[0, 1] X re given. A finite collection {I i I: i =1, 2,..., m} of pirwise nonoverlpping intervls is sid to be prtition of [, b], if m i=1i i =[, b]. We denote by Π [,b] the fmily of ll prtitions of [, b]. Let us define the totl vrition V b F of F on [, b] s follows b V F = sup{ F (I) : π Π [,b] }. I π If we hve V 1 F < +, then F is sid to be of strongly bounded vrition 0 (sbv ). Ifc (, b) nd F is sbv, then (1.1) b V F = c V F + b V c F. The lst equlity ws proven for rel vlued functions in [10], Theorem VIII.3.5, but the proof works lso for vector vlued functions, it is enough to chnge the bsolute vlue with the norm.. The function F is sid to be strongly bsolutely continuous (sac) if for ech ε>0there exists η ε > 0 such tht for ech finite collection {I i I: i =1,...,n} of pirwise nonoverlpping intervls, we hve m i=1 F (I i ) <ε, whenever m i=1 λ(i i) <η. The function F is sid to be wek sac if for ech x X the function x F is sac. The function F is sid to be wek continuous on [0, 1] if for ech x X the function x F is continuous on [0, 1]. A function f :[0, 1] X is sid to be Pettis integrble if x f is Lebesgue integrble, for ll x X, nd for ech Lthere is vector ν f () X such tht x (ν f ()) = (L) (x f)dλ for ll x X. The set function ν f : L X is clled the indefinite Pettis integrl of f. For informtion bout Pettis integrl we refer to [7], [8], [9] nd [12]. We now extend the helpful property (N), due to N. N. Luzin, from rel vlued functions to Bnch spce-vlued functions. The function F :[0, 1] X is sid to stisfy the wek property (N) if for ech x X the function x F hs the property (N), i.e. for ech L, we hve λ() =0 λ((x F )()) = 0. 2. The Min Results The min results re Theorem 2.3 nd Theorem 2.5. Let us strt with few uxiliry lemms. Lemm 2.1. Let F :[0, 1] X be function. If F is sbv nd wek continuous on [0, 1], then F is continuous on [0, 1].
The Bochner integrl nd the wek property (N) 903 Proof. Let us define the function Φ : [0, 1] [0, + ) by Φ(0) = 0 nd Φ(t) = t V F for ll t (0, 1]. Since Φ increses on [0, 1], we obtin by Theorem VIII.1.1 0 in [10] tht the set Z of discontinuity points of Φ is t most countble, nd since F (t + h) F (t) Φ(t + h) Φ(t) it follows tht ech continuity point of Φ is continuity point of F. Fix n rbitrry point t Z. We re going to prove tht the equlity (2.1) lim F (t + h) =F (t +0) X h 0 + holds true. First we will show tht given n rbitrry sequence (h n ) of nonnegtive rel numbers which converges to zero, the sequence (F (t+h n )) converges to n element x X. Indeed, since (Φ(t + h n )) is Cuchy sequence nd F (t + h n ) F (t + h m ) Φ(t + h n ) Φ(t + h m ), the sequence (F (t+h n )) is Cuchy. Further, the sequence (F (t+h n )) converges to n element x X. Secondly, let (h n ) be nother sequence of nonnegtive rel numbers which converges to zero. Then, the sequence (F (t+h n)) converges to n element x X, nd since F is wek continuous, we obtin x (x x )=0, for ll x X. The lst result together with Hhn-Bnch Theorem yields tht x = x. This mens tht (2.1) holds true. By the sme mnner s bove it cn be proved tht lim h 0 + F (t h) =F (t 0) X. We now ssume by contrdiction tht F (t+0) F (t 0). Then, by pplying Hhn-Bnch Theorem gin, there exists x X such tht (x F )(t +0) (x F )(t 0). But tht contrdicts with continuity of x F. Consequently, F (t+0) = F (t 0). By the sme mnner s bove, we get F (t+0) = F (t 0) = F (t). Since t is rbitrry, the lst result yields tht F is continuous on [0, 1] nd the proof is finished. Lemm 2.2. Let F :[0, 1] X be function which is differentible lmost everywhere on [0, 1]. IfFissBV nd wek sac, then F is sac. Proof. Since F :[0, 1] X is differentible lmost everywhere on [0, 1], there exists subset Z [0, 1] with λ(z) = 0 such tht F (t) exists for ll t [0, 1] \ Z. Thus, we cn define the function f :[0, 1] X s follows { F f(t) = (t) t [0, 1] \ Z. 0 t Z Since F is sbv, it cn be shown in the sme mnner s in the proof of Lemm 2.1 in [4] tht f is Pettis integrble nd the countble dditive vector mesure ν : L X defined by ν() = (P ) fdλ, for ll L, is of bounded vrition. We now prove tht f is strongly mesurble. By virtue of Lemm 2.1, F is continuous on [0, 1], nd becuse this the set {F (t) : t [0, 1]} X is compct
904 Besnik Bush Memetj nd therefore seprble. If Y X is the closed liner subspce spnned by the set {F (t) :t [0, 1]}, then Y is seprble. Since F (t + h) F (t) Y for ll t [0, 1] (h 0,t+ h [0, 1]), h we obtin tht f(t) Y, for lmost ll t [0, 1]. This mens tht f is essentilly seprble vlued. Hence, by Theorem II.1.2 in [1], the function f is strongly mesurble. Therefore, we obtin by Theorem 4.1 nd Remrk 4.1 in [8] tht ν () = f(t) dλ for ech L, nd since ν is of bounded vrition, the function f(.) is Lebesgue integrble on [0, 1]. Further, we obtin by Theorem II.2.2 in [1] tht f is Bochner integrble. Since the Bochner nd Pettis integrls coincide whenever they coexist, we hve ν() =(B) fdλ, for every L. Finlly, since ν(i) =F (I), for ll I I, we obtin by Theorem 7.4.15 in [11] tht F is sac nd the proof is finished. Let us now present the first result. Theorem 2.3. Let F : [0, 1] X be function. Then the following re equivlent. (i) F is the primitive of Bochner integrble function f, i.e., F (I) =(B) fdλ for ll I I, (ii) F is sac nd F (t) =f(t) lmost everywhere on [0, 1], (iii) F is sbv,weksac nd F (t) =f(t) lmost everywhere on [0, 1], (iv) F is sbv nd wek continuous on [0, 1], stisfies the wek property (N) nd F (t) =f(t) lmost everywhere on [0, 1]. Proof. By Theorem 7.4.15 in [11], the equivlence (i) (ii) holds. By Bnch- Zrecki Theorem 1.1 in [4], we obtin (iii) (iv). Note tht if F is sac, then F is sbv, wek sac. This yields (ii) (iii). Finlly, we obtin by Lemm 2.2 tht (iii) (ii) nd the proof is finished. Corollry 2.4. If F : [0, 1] X is BV. nd wek sac, then F is the primitive of Bochner integrble function f (for the notion of BV. we refer to [4] ). Proof. Since F is BV., we obtin by Theorem 2.6 in [3] tht F is differentible lmost everywhere on [0, 1]. Hence, by (iii) in Theorem 2.3, F is the primitive of Bochner integrble function f nd the proof is finished. We now extend Theorem 2.3 to Frechet spce X with topologicl dul X. We refer to [5] for the notions used in the following theorem. The function F :[0, 1] X is sid to be wek sac if for ech x X the function x F is
The Bochner integrl nd the wek property (N) 905 sac; F is sid to be wek continuous on [0, 1] if for ech x X the function x F is continuous on [0, 1]; F is sid to stisfy the wek property (N) if for ech x X nd L, we hve λ() =0 λ((x F )())=0. We sy tht F is sbv if F is sbv with respect to ech p k. Theorem 2.5. Let X be Frechet spce nd let F :[0, 1] X be function. Then the following re equivlent. (i) F is the primitive of Bochner integrble function f, (ii) F is sbv nd wek continuous on [0, 1], stisfies the wek property (N) nd F (t) =f(t) lmost everywhere on [0, 1]. Proof. (i) (ii) Assume tht F is the primitive of Bochner integrble function f. Then, for ech k N, φ k f is Bochner integrble with the primitive φ k F. Hence, by Theorem 2.3, ech function φ k F is sbv nd wek continuous on [0, 1], stisfies the wek property (N) nd (φ k F ) (t) = (φ k F )(t) lmost everywhere on [0, 1]. Thus, for ech k N, there is subset Z k [0, 1] with λ(z k ) = 0 such tht (φ k F ) (t) =(φ k F )(t) for ll t [0, 1] \ Z k. We hve lso tht X = { x k φ k : k N, x k X k }, where X nd X k re the topologicl dul of the spce X nd X k, respectively. Therefore, F is sbv nd wek continuous on [0, 1], stisfies the wek property (N) nd F (t) =f(t) for ll t [0, 1] \ + k=1 Z k. (ii) (i) Assume tht (ii) holds. Then, ech function φ k F is sbv nd wek continuous on [0, 1], stisfies the wek property (N) nd (φ k F ) (t) = (φ k F )(t) lmost everywhere on [0, 1]. Hence, by virtue of Theorem 2.3, ech function φ k f is Bochner integrble with the primitive φ k F. Thus, f is Bochner integrble with the primitive F nd the proof is finished. References [1] J. Diestel nd J. J. Uhl, Vector Mesures, Mth. Surveys, vol. 15, Amer. Mth. Soc., Providence, RI, 1977. [2] N. Dunford nd J. T. Schwrtz, Liner Opertors, Prt I: Generl Theory, Interscience, New York, (1958). [3] S. B. Klij, The differentibility of Bnch spce-vlued functions of bounded vrition, Montsh. Mth., 173(3), (2014), pp 343 359. [4] S. B. Klij, A Bnch-Zrecki Theorem for functions with vlues in Bnch spces, Montsh. Mth., DOI 10.1007 / s00605-014-0609-3. [5] B. B. Memetj, The Rdon Nikodym property nd the limit verge rnge in Frechet spces, Int. Journl of Mth. Anlysis, 7(56), (2013), pp. 2781 2786. [6] J. Mikusinki, The Bochner Integrl, Acdemic Press, New York, (1978). [7] K. Musil, Vitli nd Lebesgue convergence theorems for Pettis integrl in loclly convex spces, Atti Semin. Mt. Fis. Univ. Moden, 35 (1987), 159-165. [8] K. Musil, Topics in the theory of Pettis integrtion, Rend. Ist. Mth. Univ. Trieste, 23 (1991), 177-262.
906 Besnik Bush Memetj [9] K. Musil, Pettis integrl, in:hndbook of Mesure Theory Vol.I,.Pp (ed.), Amsterdm: North-Hollnd., (2002), 531-586. [10] I. P. Ntnson, Theory of Functions of Rel Vrible, 2nd. rev. ed., Ungr, New York, (1961). [11] Š. Schwbik nd G. Ye, Topics in Bnch Spce Integrtion, Series in Rel Anlysis, vol. 10, World Scientific, Hckensck, NJ, 2005. [12] M. Tlgrnd, Pettis Integrl nd Mesure Theory, Mem. Am Mth. Soc. No. 307 (1984). Received: Mrch 11, 2014