The space of Denjoy-Dunford integrable functions is ultrabornological

The space of Denjoy-Dunford integrable functions is ultrabornological Pedro J. Pa´ul∗ Dedicado a mi amigo Pepe Bonet con cari˜no y admiraci´on. Abstract Let Xbe a Banach space. We prove that the normed and non-complete space of all functions f:[a, b]→Xwhich are integrable in the sense of Denjoy- Dunford (introduced by Gordon in 1989) is ultrabornological. 1 Introduction For functions f:[a, b]→Rthere exist more integrals than that of Lebesgue’s, although maybe not so widespread; namely, the integrals attached to the names of Denjoy (two types), Henstock, Khinchin, Kurzweil, Luzin, McShane, and Perron. The relationships between these integrals are by now well understood and they can be classified, roughly speaking, with respect to the kind of derivatives that can be integrated by means of the corresponding Fundamental Theorem of Calculus, so ∗This research has been partially supported by la Consejer´ıa de Educaci´on y Ciencia de la Junta de Andaluc´ıa and by La Direcci´on General de Ense˜nanza Superior e Investigaci´on Cient´ıfica, research project no. PB97-0706. Received by the editors March 2000. Communicated by F. Bastin. 1991 Mathematics Subject Classification : Primary: 46A08. Secondary: 26A39, 28B05, 46E40, 46G10. Key words and phrases : Denjoy integrals. Vector-valued integrals. Ultrabornological spaces. Bull. Belg. Math. Soc. 8 (2001), 75–82 76 P. J. Pa´ul that Lebesgue ⇔McShane ⇓6⇑ Denjoy (strict) ⇔Henstock ⇔Kurzweil ⇔Luzin ⇔Perron ⇓6⇑ Denjoy (general) ⇔Khinchin meaning that the lower in this arrangement, the fewer conditions are imposed on the notion of derivability for a derivative function to be integrable. We refer the reader to the excellent monographs on the subject written by Gordon [Go3] and Henstock [H], and also to Saks’s classic [Sa]. All of these integrals have been extended to functions f:[a, b]→Xtaking their values in a Banach space X, but integrals that were equivalent in Rmay be different in a general Banach space X. For instance, we have the integrals of Bochner, Dunford, and Pettis [DU, II.2–3], the integral of McShane [F1], [F2], [F3], [FM], [Go2] and [Sw], and the integrals of Denjoy-Dunford, Denjoy-Pettis, Denjoy- Henstock, and Kurzweil [A], [F1], [F2], [FM], [Ga], [GaM], [Gi] and [Go1]. From the point of view of Functional Analysis, each of these notions of vectorvalued integration produces its own linear space of vector-valued integrable functions together with a norm defined on it in a natural way. However, these normed spaces are usually non-complete; the space of Bochner integrable functions being the exception. Fortunately, during the last years most of these non-complete normed spaces of integrable functions have been shown to be ultrabornological. Let us recall at this point that an absolutely convex and bounded subset B of a locally convex space Eis said to be a Banach disc if the linear span of B, usually denoted by EB, endowed with the normed topology defined by the gauge of Bis a Banach space [PB, 3.2]. A locally convex space Eis said to be ultrabornological if every absolutely convex set that absorbs the Banach discs of Eis a zero-neighborhood. A locally convex space is ultrabornological if and only if it is the inductive limit of a family of Banach spaces [PB, 6.1]. Their importance lies in the fact that ultrabornological spaces form a good class of domain spaces for which the following form of the closed graph theorem holds [K, §35.2.(2)]: “If Eand Fare locally convex spaces such that Eis ultrabornological and Fis webbed, then every linear mapping from Einto Fis continuous provided that it has closed graph.”(The definition of webbed space, due to De Wilde, is a bit too involved to be given here. Note, however, that almost all relevant spaces are webbed; this is the case of the space of test functions and the space of distributions [K, §35.1].) Going back to the spaces of vector-valued integrable functions we were discussing, Gilioli [Gi] proved that the space of Kurzweil integrable functions and the space of real-valued (i.e., X=R) Denjoy-Khinchin integrable functions are ultrabornological. It was proved in [DFFP] that the space of Dunford integrable functions and the space of Pettis integrable functions (both in a general σ-finite measure space) are ultrabornological. The theorem for the McShane integral was given in [DFFP] and extended to the integral over Rby Swartz [Sw]. Finally, G´amez [Ga, Cap. 4] gave a unified proof for integrals satisfying some general principles, but the vector-valued The space of Denjoy-Dunford integrable functions is ultrabornological 77 Denjoy-Dunford integral was explicitly excluded. The proofs of these theorems consist in different versions of du Bois-Reymond, Lebesgue, and Toeplitz’s sliding hump techniques which, roughly speaking, can be described as follows: to prove that an absolutely convex set Athat absorbs Banach discs is a zero-neighborhood, proceed by contradiction and start by finding, with the help of a suitable family of projections, a sequence of elements in the unit ball (xn), either disjoint or with decreasing support, such that xn/∈nA.Thenusethe elements (xn) to cook up a Banach disc containing them and, consequently, not absorbed by A. (This technique works also for some curious subspaces of C[a, b], the interested reader is referred to [BS] and [Gi]). As we have pointed out, there is a gap in the list, namely, the space of Denjoy- Khinchin integrable functions is known to be ultrabornological in the scalar-valued case, but it is not known whether the Denjoy-Dunford extension to the vectorvalued case given by Gordon [Go1] produces an ultrabornological space of integrable functions. The purpose of this note is to prove that the answer is also affirmative. 2 Vector-Valued Extensions of the Denjoy-Khinchin Integral We start by recalling the Denjoy-Khinchin integral for real valued functions (alias “Denjoy integral” [Go1], “Denjoy integral in the wide sense” and “Khinchin integral” [Go3, Ch. 15]; note, in particular, that the terminologies in [Go1] and [Go3] do not match exactly) and its relationship with the usual Lebesgue integration (see [Go1], [Go3] or [Sa] for the proofs). Afterwards, we shall describe the extensions to the vector-valued case proposed by Gordon [Go1]. In what follows, Xstands for a Banach space with dual X∗and bidual X∗∗. Definitions Let F:[a, b]→Xbe a function and let Sbe a subset of [a, b]. (1) The function Fis said to be absolutely continuous on Sif for each ε>0 there exists δ>0 such that PikF(di)−F(ci)k<εwhenever {[ci,d i]}is a finite collection of nonoverlapping intervals having endpoints in Sand such that P(di−ci)<δ. The set of all absolutely continuous functions on Swill be denoted by AC(S, X). (2) The function Fis said to be absolutely continuous in the generalized sense on Sif Fis continuous on Sand Scan be expressed as a countable union S=SnSnsuch that F∈AC(Sn,X) for all n∈N. The set of all functions which are absolutely continuous in the generalized sense on Swill be denoted by ACG(S, X). (3) A point t∈(a, b)issaidtobeapoint of density of a set S⊂[a, b]if lim h→0+ µ∗(S∩[t−h, t +h]) 2h=1 where µ∗stands for the outer Lebesgue measure on R. (4) The function Fis said to be approximately derivable at a point t∈(a, b)with approximate derivative F0 ap(t)∈Xif there exists a measurable set S⊂[a, b] 78 P. J. Pa´ul that has tas a point of density and such that lim s→t s∈S F(s)−F(t) s−t=F0 ap(t). As it is well known, the Fundamental Theorem of Calculus for the Lebesgue integral tells us that a function f:[a, b]→Ris Lebesgue integrable in [a, b]ifand only if there exists F∈AC([a, b],R) such that F0=falmost everywhere on [a, b]. The Denjoy-Khinchin integral is obtained by replacing AC by ACG and “derivative” by “approximate derivative” in this theorem. Definitions A function f:[a, b]→Ris said to be Denjoy-Khinchin integrable on [a, b] if there exists F∈ACG([a, b],R) such that F0 ap =falmost everywhere on [a, b]inwhichcasetheDenjoy-Khinchin integral of fon [a, b] is defined by Zb af:= F(b)−F(a) (there is no ambiguity here because if F∈ACG([a, b],R), then Fis approximately derivable almost everywhere and, moreover, if F0 ap = 0 almost everywhere, then Fis constant on [a, b]). It is worth noting that every Denjoy-Khinchin integrable function is measurable. The function fis said to be Denjoy-Khinchin integrable on a set S⊂[a, b]if fχSis Denjoy-Khinchin integrable on [a, b]inwhichcaseRSf:= Rb afχS. We denote by DK([a, b],R) the linear space of all Denjoy-Khinchin integrable (classes of almost everywhere equal) functions on [a, b]. This space is topologized in a natural way using the scalar Alexiewicz norm k·kAdefined by kfkA:= sup nRd cf:c, d ∈[a, b],c<d o. This is equivalent to the norm defined by kfk∗ A:= kFk∞where F∈ACG([a, b], R) is such that F0 ap =falmost everywhere on [a, b]andF(a) = 0; indeed kfk∗ A≤ kfkA≤2kfk∗ A.ThespaceDK([a, b],R) endowed with the scalar Alexiewicz norm is a non-complete, ultrabornological normed space [Ga], [Gi]. Bochner, Dunford and Pettis integrals are the best known extensions of Lebesgue integral to functions f:[a, b]→X(see [DU, II.2–3], for instance). Based on these, the Denjoy-Bochner, Denjoy-Dunford and Denjoy-Pettis integrals were defined by Gordon [Go1] as follows. Definitions A function f:[a, b]→Xis said to be Denjoy-Bochner integrable on [a, b] if there exists F∈ACG([a, b],X) such that F0 ap =falmost everywhere on [a, b]inwhichcasetheDenjoy-Bochner integral of fon [a, b] is defined by Rb af:= F(b)−F(a) (there is no ambiguity here because, again, if F∈ACG([a, b],X)is such that F0 ap = 0 almost everywhere on [a, b], then Fis constant on [a, b]). A function f:[a, b]→Xis said to be Denjoy-Dunford integrable on [a, b]ifthe composition t→hx∗,f(t)iis Denjoy-Khinchin integrable for every x∗∈X∗.G´amez and Mendoza [GaM, Thm. 3] proved that for every interval [c, d]⊂[a, b]thereisan element Rd cf∈X∗∗ called the Denjoy-Dunford integral of fon [c, d] such that Dx∗,Rd cfE=Zd chx∗,fi=Zb ahx∗,fiχ[c,d]. The space of Denjoy-Dunford integrable functions is ultrabornological 79 (N. B. The existence of these elements Rd cf∈X∗∗ wasassumedbyGordonasapart of his original definition of Denjoy-Dunford integral [Go1, Def. 25].) A function f:[a, b]→Xis said to be Denjoy-Pettis integrable on [a, b]ifitis Denjoy-Dunford integrable and the Denjoy-Dunford integral Rd cfis in Xfor every [c, d]⊂[a, b]. As in the scalar case, these spaces can be topologized in a natural way using the vectorial Alexiewicz norm k·kAdefined by kfkA:=supn  Rd cf  :c, d ∈[a, b],c<d o =supnRd chx∗,fi:c, d ∈[a, b],c<d;x∗∈X∗,kx∗k≤1o followed by the appropriate quotient, namely, two functions f,g:[a, b]→Xare in the same equivalence class if hx∗,f −gi= 0 almost everywhere for every x∗∈X∗. None of these spaces is complete, but G´amez [Ga, 4.2.13] proved, by means of a clever adaptation of Gilioli’s technique [Gi], that the spaces of Denjoy-Bochner and Denjoy-Pettis integrable functions are ultrabornological; the Denjoy-Dunford case remaining open. Theorem. The space DD([a, b],X)of Denjoy-Dunford integrable functions endowed with the vectorial Alexiewicz norm is ultrabornological. Proof. The sliding hump technique that we shall employ is, essentially, the one we used in [DFFP, Thm. 1]. However, the measure theoretic framework of that paper is too general for our purposes here, so we need to make some major adjustments, mainly contained in the Lemma below, to adapt it to the case at hand. Let Abe an absolutely convex subset of DD([a, b],X) that absorbs all Banach discs. To prove that Ais a zero-neighborhood, we have to check that Aabsorbs the unit ball Bof DD([a, b],X). Assume, on the contrary, that Adoes not absorb B.Letcbe the midpoint of [a, b]andwriteB=χ[a,c]B+χ[c,d]B,thenAdoes not absorb χ[a,c]Bor Adoes not absorb χ[c,d]B.LetI1be a half of I0=[a, b]such that Adoes not absorb χI1B. The obvious inductive procedure tells us that we can find a sequence of nested intervals (In) such that for every n∈Nthe set A does not absorb χInBand length(In)=2 −n(b−a). Denote by t0the unique point belonging to all of the intervals (In). For every n∈N, take a function fnin (the set of equivalence classes) χInB, hence kfnkA≤1, such that fn(t0)=0andfn/∈nA. Now take a sequence (αn)∈`1.Iftis a point in [a, b] such that t6=t0, then the series Pnαnfn(t) contains only a finite number of non-zero terms because the supports of the functions fndecrease to {t0}, so we may define a function f:[a, b]→Xby f(t):=Pnαnfn(t). We claim —it will be proved in the Lemma below— that for every x∗∈X∗the scalar function given by hx∗,f(t)i=X n αnhx∗,f n(t)i(t∈[a, b]) is Denjoy-Khinchin integrable on [a, b] and that the series Pnαnhx∗,f niconverges to hx∗,fiin the scalar Alexiewicz norm. This implies that fis Denjoy-Dunford integrable. Let us also see that Pnαnfnconverges to fin the vectorial Alexiewicz 80 P. J. Pa´ul norm. This follows from the convergence in the scalar Alexiewicz norm by noting that if c<dare points in [a, b]andx∗is in the unit ball of X∗,then Zd c*x∗,f − m X n=1 αnfn+ = ∞ X n=m+1 αnZd chx∗,f ni ≤ ∞ X n=m+1 |αn|Zd chx∗,f ni ≤ ∞ X n=m+1 |αn|kfnkA≤ ∞ X n=m+1 |αn|−→ m→∞ 0. Since the series Pnαnfnconverges for every sequence (αn)∈`1,itfollows,asit is well known, that the set C:= {Pnαnfn:k(αn)k1≤1}is a Banach disc in DD([a, b],X) which is not absorbed by Abecause fn∈Cbut fn/∈nA for all n∈N. A contradiction.  Lemma. Let (In)be a sequence of nested intervals in [a, b]with a unique common point t0.Let(fn)be a sequence of Denjoy-Khinchin integrable functions such that fnis supported in Inand fn(t0)=0for each n∈N.If PnkfnkA<∞, then the function defined by f(t)=Pnfn(t)is Denjoy-Khinchin integrable and the series Pfnconverges to fin the scalar Alexiewicz norm. Proof. We may assume that t0=b(if not, reason as follows in [a, t0]andsymmetrically in [t0,b]) so that we may write In=[cn,b]where(cn) is increasing and limncn=b. First note, as above, that fis well-defined because for every t∈[a, b) there is only a finite number of non-zero terms in the series Pnfn(t). Now, for each n∈Ntake the primitive Fn∈ACG([a, b],R) such that (Fn)0 ap =fnand Fn(a)=0. Since PnkFnk∞<∞because kFnk∞=kfnk∗ A≤kfnkA, it follows that the series PnFnconverges in C([a, b]) to a continuous function F. We shall prove that F∈ACG([a, b],R)andthatF0 ap =falmost everywhere on [a, b]. This will show that fis Denjoy-Khinchin integrable on [a, b]. Note that if k≥nthen Ikand [a, cn) are disjoint. Therefore, fkis zero in [a, cn) so that Fkmust be constant in this interval. Using that Fkis continuous and that Fk(a) = 0, it follows that Fk=0in[a, cn]. This tells us, on the one hand, that Fis zero in [a, c1], and on the other hand, that F=n P k=1 Fkin [cn,c n+1]foreach n∈N. Therefore, we have that Fχ[cn,cn+1 ]∈ACG([cn,c n+1],R)foreachn∈Nand this proves that F∈ACG([a, b],R). The same equality F=n P k=1 Fkin [cn,c n+1]also shows that F0 ap = n X k=1 (Fk)0 ap = n X k=1 fk=falmost everywhere on [cn,c n+1], so that F0 ap =falmost everywhere on [a, b]. Finally,      f− m X n=1 fm    A ≤2     f− m X n=1 fm     ∗ A =2     F− m X n=1 Fn    ∞ ≤2 ∞ X n=m+1 kFnk∞−→ m→∞ 0, which shows that the series Pnfnconverges to fin the scalar Alexiewicz norm.  The space of Denjoy-Dunford integrable functions is ultrabornological 81 Bibliography [A] A. Alexiewicz, On Denjoy integrals of abstract functions, C. R. Soc. Sci. Lett. Cl. III Sci. Math. Phys. 41 (1948), 97–129; [BS] A. Bernard and S. J. Sidney, Some ultrabornological normed function spaces, Arch. Math. (Basel) 69 (1997), 409–417; [DFFP] S. D´ıaz, A. Fern´andez, M. Florencio, and P. J. Pa´ul, Awideclassofultrabornological spaces of measurable functions, J. Math. Anal. Appl. 190 (1995), 697–713; [DU] J. Diestel and J. J. Uhl Jr., Vector Measures, Mathematical Surveys and Monographs, vol. 15, American Mathematical Society, Providence, RI, 1977; [F1] D. H. Fremlin, The Henstock and McShane integrals of vector-valued functions, Illinois J. 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