The space of Denjoy-Dunford integrable functions is ultrabornological

The space o Denjoy-Dun o d in eg able unc ions is ul abo nological Ped o J. Pa´ul∗ Dedicado a mi amigo Pepe Bone con ca i˜no y admi aci´on. Abs ac Le Xbe a Banach space. We p o e ha he no med and non-comple e space o all unc ions :[a, b]→Xwhich a e in eg able in he sense o Denjoy- Dun o d (in oduced by Go don in 1989) is ul abo nological. 1 In oduc ion Fo unc ions :[a, b]→R he e exis mo e in eg als han ha o Lebesgue’s, al hough maybe no so widesp ead; namely, he in eg als a ached o he names o Denjoy ( wo ypes), Hens ock, Khinchin, Ku zweil, Luzin, McShane, and Pe on. The ela ionships be ween hese in eg als a e by now well unde s ood and hey can be classi ied, oughly speaking, wi h espec o he kind o de i a i es ha can be in eg a ed by means o he co esponding Fundamen al Theo em o Calculus, so ∗This esea ch has been pa ially suppo ed by la Conseje ´ıa de Educaci´on y Ciencia de la Jun a de Andaluc´ıa and by La Di ecci´on Gene al de Ense˜nanza Supe io e In es igaci´on Cien ´ı ica, esea ch p ojec no. PB97-0706. Recei ed by he edi o s Ma ch 2000. Communica ed by F. Bas in. 1991 Ma hema ics Subjec Classi ica ion : P ima y: 46A08. Seconda y: 26A39, 28B05, 46E40, 46G10. Key wo ds and ph ases : Denjoy in eg als. Vec o - alued in eg als. Ul abo nological spaces. Bull. Belg. Ma h. Soc. 8 (2001), 75–82 76 P. J. Pa´ul ha Lebesgue ⇔McShane ⇓6⇑ Denjoy (s ic ) ⇔Hens ock ⇔Ku zweil ⇔Luzin ⇔Pe on ⇓6⇑ Denjoy (gene al) ⇔Khinchin meaning ha he lowe in his a angemen , he ewe condi ions a e imposed on he no ion o de i abili y o a de i a i e unc ion o be in eg able. We e e he eade o he excellen monog aphs on he subjec w i en by Go don [Go3] and Hens ock [H], and also o Saks’s classic [Sa]. All o hese in eg als ha e been ex ended o unc ions :[a, b]→X aking hei alues in a Banach space X, bu in eg als ha we e equi alen in Rmay be di e en in a gene al Banach space X. Fo ins ance, we ha e he in eg als o Bochne , Dun o d, and Pe is [DU, II.2–3], he in eg al o McShane [F1], [F2], [F3], [FM], [Go2] and [Sw], and he in eg als o Denjoy-Dun o d, Denjoy-Pe is, Denjoy- Hens ock, and Ku zweil [A], [F1], [F2], [FM], [Ga], [GaM], [Gi] and [Go1]. F om he poin o iew o Func ional Analysis, each o hese no ions o ec o - alued in eg a ion p oduces i s own linea space o ec o - alued in eg able unc ions oge he wi h a no m de ined on i in a na u al way. Howe e , hese no med spaces a e usually non-comple e; he space o Bochne in eg able unc ions being he excep- ion. Fo una ely, du ing he las yea s mos o hese non-comple e no med spaces o in eg able unc ions ha e been shown o be ul abo nological. Le us ecall a his poin ha an absolu ely con ex and bounded subse B o a locally con ex space Eis said o be a Banach disc i he linea span o B, usually deno ed by EB, endowed wi h he no med opology de ined by he gauge o Bis a Banach space [PB, 3.2]. A locally con ex space Eis said o be ul a- bo nological i e e y absolu ely con ex se ha abso bs he Banach discs o Eis a ze o-neighbo hood. A locally con ex space is ul abo nological i and only i i is he induc i e limi o a amily o Banach spaces [PB, 6.1]. Thei impo ance lies in he ac ha ul abo nological spaces o m a good class o domain spaces o which he ollowing o m o he closed g aph heo em holds [K, §35.2.(2)]: “I Eand Fa e locally con ex spaces such ha Eis ul abo nological and Fis webbed, hen e e y linea mapping om Ein o Fis con inuous p o ided ha i has closed g aph.”(The de ini ion o webbed space, due o De Wilde, is a bi oo in ol ed o be gi en he e. No e, howe e , ha almos all ele an spaces a e webbed; his is he case o he space o es unc ions and he space o dis ibu ions [K, §35.1].) Going back o he spaces o ec o - alued in eg able unc ions we we e discussing, Gilioli [Gi] p o ed ha he space o Ku zweil in eg able unc ions and he space o eal- alued (i.e., X=R) Denjoy-Khinchin in eg able unc ions a e ul abo nologi- cal. I was p o ed in [DFFP] ha he space o Dun o d in eg able unc ions and he space o Pe is in eg able unc ions (bo h in a gene al σ- ini e measu e space) a e ul abo nological. The heo em o he McShane in eg al was gi en in [DFFP] and ex ended o he in eg al o e Rby Swa z [Sw]. Finally, G´amez [Ga, Cap. 4] ga e a uni ied p oo o in eg als sa is ying some gene al p inciples, bu he ec o - alued The space o Denjoy-Dun o d in eg able unc ions is ul abo nological 77 Denjoy-Dun o d in eg al was explici ly excluded. The p oo s o hese heo ems consis in di e en e sions o du Bois-Reymond, Lebesgue, and Toepli z’s sliding hump echniques which, oughly speaking, can be desc ibed as ollows: o p o e ha an absolu ely con ex se A ha abso bs Banach discs is a ze o-neighbo hood, p oceed by con adic ion and s a by inding, wi h he help o a sui able amily o p ojec ions, a sequence o elemen s in he uni ball (xn), ei he disjoin o wi h dec easing suppo , such ha xn/∈nA.Thenuse he elemen s (xn) o cook up a Banach disc con aining hem and, consequen ly, no abso bed by A. (This echnique wo ks also o some cu ious subspaces o C[a, b], he in e es ed eade is e e ed o [BS] and [Gi]). As we ha e poin ed ou , he e is a gap in he lis , namely, he space o Denjoy- Khinchin in eg able unc ions is known o be ul abo nological in he scala - alued case, bu i is no known whe he he Denjoy-Dun o d ex ension o he ec o - alued case gi en by Go don [Go1] p oduces an ul abo nological space o in eg able unc ions. The pu pose o his no e is o p o e ha he answe is also a i ma i e. 2 Vec o -Valued Ex ensions o he Denjoy-Khinchin In eg al We s a by ecalling he Denjoy-Khinchin in eg al o eal alued unc ions (alias “Denjoy in eg al” [Go1], “Denjoy in eg al in he wide sense” and “Khinchin in eg al” [Go3, Ch. 15]; no e, in pa icula , ha he e minologies in [Go1] and [Go3] do no ma ch exac ly) and i s ela ionship wi h he usual Lebesgue in eg a ion (see [Go1], [Go3] o [Sa] o he p oo s). A e wa ds, we shall desc ibe he ex ensions o he ec o - alued case p oposed by Go don [Go1]. In wha ollows, Xs ands o a Banach space wi h dual X∗and bidual X∗∗. De ini ions Le F:[a, b]→Xbe a unc ion and le Sbe a subse o [a, b]. (1) The unc ion Fis said o be absolu ely con inuous on Si o each ε>0 he e exis s δ>0 such ha PikF(di)−F(ci)k<εwhene e {[ci,d i]}is a ini e collec ion o nono e lapping in e als ha ing endpoin s in Sand such ha P(di−ci)<δ. The se o all absolu ely con inuous unc ions on Swill be deno ed by AC(S, X). (2) The unc ion Fis said o be absolu ely con inuous in he gene alized sense on Si Fis con inuous on Sand Scan be exp essed as a coun able union S=SnSnsuch ha F∈AC(Sn,X) o all n∈N. The se o all unc ions which a e absolu ely con inuous in he gene alized sense on Swill be deno ed by ACG(S, X). (3) A poin ∈(a, b)issaid obeapoin o densi y o a se S⊂[a, b]i lim h→0+ µ∗(S∩[ −h, +h]) 2h=1 whe e µ∗s ands o he ou e Lebesgue measu e on R. (4) The unc ion Fis said o be app oxima ely de i able a a poin ∈(a, b)wi h app oxima e de i a i e F0 ap( )∈Xi he e exis s a measu able se S⊂[a, b] 78 P. J. Pa´ul ha has as a poin o densi y and such ha lim s→ s∈S F(s)−F( ) s− =F0 ap( ). As i is well known, he Fundamen al Theo em o Calculus o he Lebesgue in eg al ells us ha a unc ion :[a, b]→Ris Lebesgue in eg able in [a, b]i and only i he e exis s F∈AC([a, b],R) such ha F0= almos e e ywhe e on [a, b]. The Denjoy-Khinchin in eg al is ob ained by eplacing AC by ACG and “de i a i e” by “app oxima e de i a i e” in his heo em. De ini ions A unc ion :[a, b]→Ris said o be Denjoy-Khinchin in eg able on [a, b] i he e exis s F∈ACG([a, b],R) such ha F0 ap = almos e e ywhe e on [a, b]inwhichcase heDenjoy-Khinchin in eg al o on [a, b] is de ined by Zb a := F(b)−F(a) ( he e is no ambigui y he e because i F∈ACG([a, b],R), hen Fis app oxima ely de i able almos e e ywhe e and, mo eo e , i F0 ap = 0 almos e e ywhe e, hen Fis cons an on [a, b]). I is wo h no ing ha e e y Denjoy-Khinchin in eg able unc ion is measu able. The unc ion is said o be Denjoy-Khinchin in eg able on a se S⊂[a, b]i χSis Denjoy-Khinchin in eg able on [a, b]inwhichcaseRS := Rb a χS. We deno e by DK([a, b],R) he linea space o all Denjoy-Khinchin in eg able (classes o almos e e ywhe e equal) unc ions on [a, b]. This space is opologized in a na u al way using he scala Alexiewicz no m k·kAde ined by k kA:= sup nRd c :c, d ∈[a, b],c<d o. This is equi alen o he no m de ined by k k∗ A:= kFk∞whe e F∈ACG([a, b], R) is such ha F0 ap = almos e e ywhe e on [a, b]andF(a) = 0; indeed k k∗ A≤ k kA≤2k k∗ A.ThespaceDK([a, b],R) endowed wi h he scala Alexiewicz no m is a non-comple e, ul abo nological no med space [Ga], [Gi]. Bochne , Dun o d and Pe is in eg als a e he bes known ex ensions o Lebesgue in eg al o unc ions :[a, b]→X(see [DU, II.2–3], o ins ance). Based on hese, he Denjoy-Bochne , Denjoy-Dun o d and Denjoy-Pe is in eg als we e de ined by Go don [Go1] as ollows. De ini ions A unc ion :[a, b]→Xis said o be Denjoy-Bochne in eg able on [a, b] i he e exis s F∈ACG([a, b],X) such ha F0 ap = almos e e ywhe e on [a, b]inwhichcase heDenjoy-Bochne in eg al o on [a, b] is de ined by Rb a := F(b)−F(a) ( he e is no ambigui y he e because, again, i F∈ACG([a, b],X)is such ha F0 ap = 0 almos e e ywhe e on [a, b], hen Fis cons an on [a, b]). A unc ion :[a, b]→Xis said o be Denjoy-Dun o d in eg able on [a, b]i he composi ion →hx∗, ( )iis Denjoy-Khinchin in eg able o e e y x∗∈X∗.G´amez and Mendoza [GaM, Thm. 3] p o ed ha o e e y in e al [c, d]⊂[a, b] he eisan elemen Rd c ∈X∗∗ called he Denjoy-Dun o d in eg al o on [c, d] such ha Dx∗,Rd c E=Zd chx∗, i=Zb ahx∗, iχ[c,d]. The space o Denjoy-Dun o d in eg able unc ions is ul abo nological 79 (N. B. The exis ence o hese elemen s Rd c ∈X∗∗ wasassumedbyGo donasapa o his o iginal de ini ion o Denjoy-Dun o d in eg al [Go1, De . 25].) A unc ion :[a, b]→Xis said o be Denjoy-Pe is in eg able on [a, b]i i is Denjoy-Dun o d in eg able and he Denjoy-Dun o d in eg al Rd c is in X o e e y [c, d]⊂[a, b]. As in he scala case, hese spaces can be opologized in a na u al way using he ec o ial Alexiewicz no m k·kAde ined by k kA:=supn  Rd c   :c, d ∈[a, b],c<d o =supnRd chx∗, i:c, d ∈[a, b],c<d;x∗∈X∗,kx∗k≤1o ollowed by he app op ia e quo ien , namely, wo unc ions ,g:[a, b]→Xa e in he same equi alence class i hx∗, −gi= 0 almos e e ywhe e o e e y x∗∈X∗. None o hese spaces is comple e, bu G´amez [Ga, 4.2.13] p o ed, by means o a cle e adap a ion o Gilioli’s echnique [Gi], ha he spaces o Denjoy-Bochne and Denjoy-Pe is in eg able unc ions a e ul abo nological; he Denjoy-Dun o d case emaining open. Theo em. The space DD([a, b],X)o Denjoy-Dun o d in eg able unc ions en- dowed wi h he ec o ial Alexiewicz no m is ul abo nological. P oo . The sliding hump echnique ha we shall employ is, essen ially, he one we used in [DFFP, Thm. 1]. Howe e , he measu e heo e ic amewo k o ha pape is oo gene al o ou pu poses he e, so we need o make some majo adjus men s, mainly con ained in he Lemma below, o adap i o he case a hand. Le Abe an absolu ely con ex subse o DD([a, b],X) ha abso bs all Banach discs. To p o e ha Ais a ze o-neighbo hood, we ha e o check ha Aabso bs he uni ball Bo DD([a, b],X). Assume, on he con a y, ha Adoes no abso b B.Le cbe he midpoin o [a, b]andw i eB=χ[a,c]B+χ[c,d]B, henAdoes no abso b χ[a,c]Bo Adoes no abso b χ[c,d]B.Le I1be a hal o I0=[a, b]such ha Adoes no abso b χI1B. The ob ious induc i e p ocedu e ells us ha we can ind a sequence o nes ed in e als (In) such ha o e e y n∈N he se A does no abso b χInBand leng h(In)=2 −n(b−a). Deno e by 0 he unique poin belonging o all o he in e als (In). Fo e e y n∈N, ake a unc ion nin ( he se o equi alence classes) χInB, hence k nkA≤1, such ha n( 0)=0and n/∈nA. Now ake a sequence (αn)∈`1.I is a poin in [a, b] such ha 6= 0, hen he se ies Pnαn n( ) con ains only a ini e numbe o non-ze o e ms because he suppo s o he unc ions ndec ease o { 0}, so we may de ine a unc ion :[a, b]→Xby ( ):=Pnαn n( ). We claim —i will be p o ed in he Lemma below— ha o e e y x∗∈X∗ he scala unc ion gi en by hx∗, ( )i=X n αnhx∗, n( )i( ∈[a, b]) is Denjoy-Khinchin in eg able on [a, b] and ha he se ies Pnαnhx∗, nicon e ges o hx∗, iin he scala Alexiewicz no m. This implies ha is Denjoy-Dun o d in eg able. Le us also see ha Pnαn ncon e ges o in he ec o ial Alexiewicz 80 P. J. Pa´ul no m. This ollows om he con e gence in he scala Alexiewicz no m by no ing ha i c<da e poin s in [a, b]andx∗is in he uni ball o X∗, hen Zd c*x∗, − m X n=1 αn n+ = ∞ X n=m+1 αnZd chx∗, ni ≤ ∞ X n=m+1 |αn|Zd chx∗, ni ≤ ∞ X n=m+1 |αn|k nkA≤ ∞ X n=m+1 |αn|−→ m→∞ 0. Since he se ies Pnαn ncon e ges o e e y sequence (αn)∈`1,i ollows,asi is well known, ha he se C:= {Pnαn n:k(αn)k1≤1}is a Banach disc in DD([a, b],X) which is no abso bed by Abecause n∈Cbu n/∈nA o all n∈N. A con adic ion.  Lemma. Le (In)be a sequence o nes ed in e als in [a, b]wi h a unique common poin 0.Le ( n)be a sequence o Denjoy-Khinchin in eg able unc ions such ha nis suppo ed in Inand n( 0)=0 o each n∈N.I Pnk nkA<∞, hen he unc ion de ined by ( )=Pn n( )is Denjoy-Khinchin in eg able and he se ies P ncon e ges o in he scala Alexiewicz no m. P oo . We may assume ha 0=b(i no , eason as ollows in [a, 0]andsym- me ically in [ 0,b]) so ha we may w i e In=[cn,b]whe e(cn) is inc easing and limncn=b. Fi s no e, as abo e, ha is well-de ined because o e e y ∈[a, b) he e is only a ini e numbe o non-ze o e ms in he se ies Pn n( ). Now, o each n∈N ake he p imi i e Fn∈ACG([a, b],R) such ha (Fn)0 ap = nand Fn(a)=0. Since PnkFnk∞<∞because kFnk∞=k nk∗ A≤k nkA, i ollows ha he se- ies PnFncon e ges in C([a, b]) o a con inuous unc ion F. We shall p o e ha F∈ACG([a, b],R)and ha F0 ap = almos e e ywhe e on [a, b]. This will show ha is Denjoy-Khinchin in eg able on [a, b]. No e ha i k≥n hen Ikand [a, cn) a e disjoin . The e o e, kis ze o in [a, cn) so ha Fkmus be cons an in his in e al. Using ha Fkis con inuous and ha Fk(a) = 0, i ollows ha Fk=0in[a, cn]. This ells us, on he one hand, ha Fis ze o in [a, c1], and on he o he hand, ha F=n P k=1 Fkin [cn,c n+1] o each n∈N. The e o e, we ha e ha Fχ[cn,cn+1 ]∈ACG([cn,c n+1],R) o eachn∈Nand his p o es ha F∈ACG([a, b],R). The same equali y F=n P k=1 Fkin [cn,c n+1]also shows ha F0 ap = n X k=1 (Fk)0 ap = n X k=1 k= almos e e ywhe e on [cn,c n+1], so ha F0 ap = almos e e ywhe e on [a, b]. Finally,      − m X n=1 m    A ≤2     − m X n=1 m     ∗ A =2     F− m X n=1 Fn    ∞ ≤2 ∞ X n=m+1 kFnk∞−→ m→∞ 0, which shows ha he se ies Pn ncon e ges o in he scala Alexiewicz no m.  The space o Denjoy-Dun o d in eg able unc ions is ul abo nological 81 Bibliog aphy [A] A. Alexiewicz, On Denjoy in eg als o abs ac unc ions, C. R. Soc. Sci. Le . Cl. III Sci. Ma h. Phys. 41 (1948), 97–129; [BS] A. Be na d and S. J. Sidney, Some ul abo nological no med unc ion spaces, A ch. Ma h. (Basel) 69 (1997), 409–417; [DFFP] S. D´ıaz, A. Fe n´andez, M. Flo encio, and P. J. Pa´ul, Awideclasso ul- abo nological spaces o measu able unc ions, J. Ma h. Anal. Appl. 190 (1995), 697–713; [DU] J. Dies el and J. J. Uhl J ., Vec o Measu es, Ma hema ical Su eys and Monog aphs, ol. 15, Ame ican Ma hema ical Socie y, P o idence, RI, 1977; [F1] D. H. F emlin, The Hens ock and McShane in eg als o ec o - alued unc- ions, Illinois J. Ma h. 38 (1994), 471–479; [F2] D. H. 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