 Advances in Pure Mathematics, 2013, 3, 653-659 Published Online November 2013 (http://www.scirp.org/journal/apm) http://dx.doi.org/10.4236/apm.2013.38087 Open Access APM On the Differentiability of Vector Valued Additive Set Functions Mangatiana A. Robdera, Dintle Kagiso Department of Mathematics, University of Botswana, Gaborone, Botswana Email: robdera@yahoo.com, dintlek.kagiso@gmail.com Received October 7, 2013; revised November 8, 2013; accepted November 15, 2013 Copyright © 2013 Mangatiana A. Robdera, Dintle Kagiso. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT The Lebesgue-Nikodým Theorem states that for a Lebesgue measure :2 0,, an additive set function which is :F-absolutely continuous is the integral of a Lebegsue integrable a measurable function ; that is, for all measurable sets :f,A d.AFAf Such a property is not shared by vector valued set functions. We introduce a suitable definition of the integral that will extend the above property to the vector valued case in its full generality. We also discuss a further extension of the Fundamental Theorem of Calculus for additive set functions with values in an infinite dimensional normed space. Keywords: Vector Valued Additive Set Function; Lebesgue-Radon-Nikodým Theorem; Fundamental Theorem of Calculus 1. Introduction This note can be considered as a continuation of the work started by the first author in . Throughout the paper is a 2-ring of subsets of a nonempty set . A vector measure is a countably additive set function  defined on taking values in a Banach space X and such that . We denote by 0,MX the set of all X-valued vector measures defined on . Let be a nonnegative scalar measure. If a function :0, :fX is -Lebesgue-Bochner in-tegrable, then it is easily seen that the function :FX defined by dAFAfFA is a vector measure. It readily follows from the property of the Lebesgue-Bochner integral that whenever Such a property is expressed by saying that 0A0.F is absolutely continuous with respect to . Definition 1. A set function :FXf is said to be λ-Lebesgue-Bochner differentiable if there exists a λ- Lebesgue-Bochner integrable function such :,that for every ,Ad.AFAf Such a function f is called the -Radon-Nikodým derivative of .F It easily follows from the property of the Lebesgue-Bochner integral that if f is a -Ra- don-Nikodým derivative of F then so is any function g such that  :0xfxgx.  A repre- sentative of the class of such functions is denoted by ddF. The classical Lebesgue-Nikodým theorem states that for an additive real valued set function (not necessarily countably additive) :F, -absolutely continuity implies -Lebesgue differentiability (see for example ). The analogue of the Lebesgue-Nikodým theorem does not extend to the vector valued case in general. For instance, it is a well known fact that the 10,1L-valued vector measure defined by A1A for every Borel subset A of 0,1, is absolutely continuous with respect to the Lebesgue measure  on 0,1 but fails to be - Lebesgue-Bochner differentiable (see for example ). In this note, we reactualize the approach to the defini-tion of the integral first introduced in  in order to ob-tain a Lebesgue-Nikodým type theorem on the differen-tiability of Banach space valued additive set function. The exposition will be organized as follows. In Section 2, we introduce a new approach to integration theory that does not require the elaborate machinery of Lebesgue measure theory, and at the same time significantly sim- M. A. ROBDERA, D. KAGISO 654 plifies the approach to gauge integral. In Section 3, we shall see that the space of classes of integrable functions (in the sense of the definition of integrability in Section 2) can be naturally given a Banach space structure. In Sec-tion 4, we state and prove our main result which can be seen as an extended vector valued version of the Lebes-gue-Nikodým theorem. The fifth section is devoted to some extension of the Fundamental Theorem of Calcu-lus. 2. Extended Notion of Integral We begin by recalling the definition of limit in its most general form, that is, the Moore-Smith limit, also known as the net limit. A nonempty set is said to be directed by a binary relation , if has the following properties: D 1) for ,,xyz D if xy and , then yzxz (transitivity); 2) for ,,xyDy there exists such that and (upper bound property). zDzxzGiven a set ,X a net of elements of X is an X-valued function defined on a directed set ,D. The notion of convergence can then be defined whenever the set X is a metric space, with distance function . dA net :, ,fDXd is said to be convergent if there exists an element xX denoted by ,:limDxf such that for every >0, there exists 0D such that for every 0, d,,Next we introduce the notion of size-function that will generalize the usual notion of measure in integral theory. In the following, is a nonempty set. The power set of , that is, the set of all subsets of will be de-noted by  2.Definition 2. Let By a size-function, we mean a set-function 2.: 0, that satisfies the following conditions:  0;  AB whenever AB in  (monotoni- city);  nnnnAA for every sequence nnA in such that (countable subadditiv-ity). nnAObviously, any measure defined on a -ring is a size-function. The length function defined on the - ring generated by the bounded intervals in is another example of size function. It is clear from the definition that an outer-measure is a size-function defined on 2. In fact, an arbitrary size-function :0, defined on a -ring  can be extended to an outer-measure defined on the whole of the power set 2 as follows. Proposition 3. Let :0, be a size-function. For every define ,A*1inf:, .n nnnAIAInI Then η* is an outer-measure such that EE* for every .E Definition 4. Let  be an arbitrary subset of 2, and let .A A -subpartition of PA is any fi- nite collection ,;,,1,2,ii iIIAI in with the following properties: 1) 0, there exists such that for every 0,PA ,,,PA we have 0PP,d0, and let such that for >0N,>mnN in , sup :.nmnm ffffPP A (2) In particular, if we consider the subpartition ,AA , then for ,>mnN in , <.nmff A Since <,A we infer that the sequence nnf is Cauchy in .X Since X is a Banach Open Access APM M. A. ROBDERA, D. KAGISO 656 space, we can define a function .limnnff On the other hand, since ,,nmff AX,, there exist such that ,nmPP Ad< whenever,nfnAPf PPn d< whenever.mfmmAPf PP Combining these inequality with (2), it follows that for ,>mnN in and for every we have ,nmPPP dd dd<3.nmnm fnffAA AfmAnmffPfPf P This proves that the sequence dnAnf is Cauchy in ,X and thus converges to say .aXNow since for each ,A ,limnnff there exists >NN such that for ,>mnN in , .nmffA It follows that for ,: 1,,iiPIt ikA, and for every we have ,max:1,,:itmnNikN,P 1.nmkffi nimiiPIftft If we let we obtain ,m.nffP Since d,flimmmAa there exists >PNN such that d N. Thus for ,>,nm Nd<3.nnmfffffmAPa Pfa P Since ε > 0 is arbitrary, this shows that ,,fAX and that d.faAIt goes without saying that when the -equivalent functions taking values in a Banach space X are iden-tified, the restriction of the seminorm  defines a norm on ,, ,IAX and gives the space ,, ,IAX the structure of a Banach space. 4. Extended Lebesgue-NikoDým Theorem Again, we fix a (real or complex) Banach space .X Let :2 0, : a size-function. We say that a set function FX is -absolutely continuous on  if for every >0, there exists >0 such that 0. By F, we-absolutely continuity of can find P and Q so refined that \<3.JQJQIFIAIAIPFIAF A (4) For such and there exists P ,Q 0RA such that for 0,RR  d<, d<33QP FQAAFand .PFRF R (5) It follows from (4), (5) that for 0,RR dd.PQ PFF FPRRQAQFAIP JQRFFRFIAF IA   (6) Open Access APM M. A. ROBDERA, D. KAGISO 657This proves our claim. By Theorem 9, there exists ,,fX such that the net PPF converges seminormto f with respect to the . Thus given >0, there exists a tagged that for 1RP subpartition suPA1ch sup PfFR:< .3RRA (7) other hosuOn theand, it follows frm (3) that there exists 2PA ch that for R  2P .3PFRFA (8) Finally, by definition of the integral, there ex ists 3A such that for 3RP Pdf.3fAR (9) Combining (7), (8), and (9), we have for3 12RPPP    ddffFA fR.APPAAfFFRRRF The desired result follows since >0 is arbitrary chosen. The proof is complete. As a direct corollary of our main Theorem 10, we do have the following extended version of the vectoued Radon-Nikodým Theorem. Corolla ry 11. Let be a σ-algebra of a set r val and :X is a countably additive vemeasure that is absolutely continuous with respect nite measure ctor to a fi:0,. Then there exists an ,,fX (not necessarily Bochner integrable) such that dAAf for all . Proof. We notice that the vector measure A is an ad-ion and that the outer-meas the Carathéodory extension of the measure ditive set functurtained bye ob defines a size-function on. The desired ult isained by applying Theoo the set nction 2rem 10 tresfu obt  and the ction size-fun. Remark. Note again that in the above Collary 11, the density function orf is not a Radon-Nikodým de- rivative. It is also important to realize that in contrast with the classical vector valued Radon-Nikodým Theo- rem, the boundedness of the variation of the set function  is not required here. It is a well known fact that if the -density function f of a vector measur dAfA for every .A Recall that a Banach spce is said to have the Radon- Nikodým Property if ey vecto measure :aver rX of bounded variation that is absolutely contrespect to a finite measure don-Nikodým derivative. Wellary. inuous with has a Ra-on with the :0,  end th sectiisfollowing immediate coroCorollary 12. A Banach space X has the Radon- Nikodým Property if and only if for every -algebra  of a set , for every vector measure :X of bounded variation that is absolous with re- spect to a finite measure :0,, the utelontiy cnu-deriva- tive of  is a Radon-Nikodým derivative. 5. Extended Fundamental Theorem of Calculus In this section, we give conditions under which a given vector valued function (noteasurale) is the derivati of a given finitely additive set-f necessarily mbveunction. Our integral thresult can be compared to that of Lu and Pee in . It clearly follows from our definition of the at ,,Xif and only if ,,fAXfe exists and is Bochner inte-grable with respect to , then  must have bounded variation and its variation is given by  for every subsets A of . This gives rise to a set function :FX: dAFAf for all 2.A It im- mediately follows from the properties of the integral that such a set function is finitely additive. Such a set function F is called the -indefinite integral of the funct .f Of coble function ion urse, an integraf is always the -derivative its of-indefinite integral. The next result gives a necessary and sufficient condi-tion under which an additivon e set functiF defined on 2 is the -indefinite integral of an integrable function :fX. In what follows the size-function  is consideredo be finite, that is, t<. Theore13. Let m  be a nonempty set and :2 0, b a finite size-functon. Then the following ste iatements are equivalent for an additive set function :2FX and a function :.fX ) 1,,fX and F is the -indefinite inte-gral of .f 2) For every >0, there exists P  con-sisting of elements of  :, :IFIIfI  every ,PPsuch that for  0 and let Proof. For the neceP as in 2. Then for PP Open Access APM M. A. ROBDERA, D. KAGISO 658      \\2.ttttffiiiiIPttIP IPtIPFtIPtiiIPPF PI ftFIIftIFIIft  Since FIFI I ft>0 is arbitrary, this shows that ,,fX and Conversely, assume that d.fF ,,fX. Let .kGiven ε > 0 there exists a η-subpartition :1kEkf ,kP such that for every ,,kPP  21.2ttt kIPFIt k I fFor each ,n let   if 0 nk1 otherwise.knfEf Then for ,1,nkkPP we have  .2ttttn tIP IPttIPIft IkFII ftk  On taking the limit as , we infer that n.2ttIftIP It then follows that .22tIPfttttttIP IPtFIFII ftI  The proof is complete. Let now assume that is a topological space. We say that a size-function  :2of 0, . For a fiis regular if it is no sxed n-zero on open set, let  the set of all neiω. Tghborhoods of hen  ng e followiis directed by inclusionce thdefinition. Definition 14. Let We introdu.  be a topological space and :2 0, a re size-function. Let gular2 contain the topology of . A set function :FX is said to be -differentiable at a point , if ,mF exists in X. In other words, :FliX is -differentiable at a point , if there exists a vector fX such that for every >0, there exists open neighborhood ,U such that for every ,U ,UU  <.UU fUF We call the set ,: tslimFFexis  the domain of differentiability of .F By the uniqueness ofspondence net limit, the corre ,limF nc denoted by Fdefines a fution on F and called the -derivative of .F is compact. Let 2 Now assume that con-taining the topology of , and :2FX an \,Eaddi-tive set function such that F with E0. Given >0, let U be an open set such that EU and 