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TRU Mathematics 19−2 (1983)

A卿ZATION.OF.㎜㎜R I唖肌

      BY THE METH【)D OF RANKED SPACES

Yoshiko TAGUCHI

(Received October 3, 1983)      Introduction. A generalization of the integral for real valued functions by the method of ranked spaces was prqposed by Professgr K.㎞㎎i in 1956 (【2], [5]). The generalizpd integra1, called the E.R. integral by him, has been investigated by many researdhers ([1], 【9L  I10D.  In 1968, Professor S.

撫㎞ishi dwe1帆㎞ther the theoηof the E.R. integral in a series of

papers ([4D,Where she showed, c《加Pleting.the original idea of Professor KUnugi, that actually the space jしof E.R. integrable functions is a completion in the sense of ranked spaces,of the spaceε’pf si㎎〕1e (or stq)) functions suitably def血迦g preneighborhoods and ranks onε. Wbreover, it was shom that the integraユ1(f)of an E.R. integrable function f∈↓L is given by the limit 1‡m I(fi) 砲ere {fi} is a Caudhy sequence inε converging to f.        エ      In this paper, we follow the w《)rk of ProfessoT Nakanishi replacing the reals by a B鎚ch space X i.e.,we study v㏄tor valued integrals by the method of ral止ed spaces. The spaceε  of simple fmctions is defined as usua1・ In order to define suitable preneighborhoods and ranl(s onε,we separate the conditions of Professor Nakanishi into two,㎜ely, convergences a・e.’andl convergences with respect to a suitable quasi−senli¶b㎝川川(i.e. a s㎝i− mm which satisfies a modified tria㎎1e inequality as appeared in a paper by Amemiya−Airida [1D. One adva皿tage to do so is that our theory gives usual Bodhner integral and.E.R. type Bodhner integral by suita1)1y cbOosing the

quasi−S㎝i−no㎝川川.

In a later paper (.[6】, [7]), Professor Nakanishi fOrmuユated a general theory of cc呵pletio血s of rahked spaces and showed again that,、fbr real valued case, the ranked spaceε of simple functions has a special type (may be called a ρ一〕 completion Z Whi(h has one−one correspondence with the space of E.R. integra1)1e fmctions. In・our case, if we choose for the quasi−s㎝i−mnn ll川ll the id・ntiC・11y・e・・fun・ti・n・1, th・n・the c・mP1・ti・n i・th・SPaceフπ・f・B・・imer measurable functions so that for any other quasi−semi−no㎝1旧ll satisfying ・Ui・ah・e c・ndi・⊇・he c。mP・・ti・n LHI川㎝b・・ea・i・ed・・…Ul・SPace・f

141

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142

Y.TAG[JCHI 1π and it is actually a p−c㎝Pletion in the sehse of Professor.Na㎞ishi. th・血・・g・a・・㈹・f・fU・g・i・n. ・fgAll ll卜㎜y b・ca・・ed・lll lil一血・・g・ab…i・ always defined by the limit

      iP i(fi)    −

.血・h・B・n・Ch space X,ぬ・・e.{fi}i・a伽・hy・eq・e㏄・i・ε111111(・h・・anked spaceεwith preneighborhoods and ranks using l旧ll〕. As a b)?Toduct of our 麺tr(》duction of(岬si−selU−m㎝, we can show that the one defined by lllfll』 = ‖1〔f)11 is maximal in the sense that fbr any 川 111 satisfying ・己・ab・e c・nditi・n・WhiCh・ar…蝋f・r Q・・th・・ry・Lll川ll i・c・nt・ined・in

£1旧ll、…p・・七i・曲・・・…切・恥dmer血・・g醐・fm・ti・鵬are㎜ぬ・・

integrablg 〔i・e・ lll lll3−integrable) and actually the class of maximal illtegrable fUnctionS is bigger.      We outline the「content of eadl section:We start with preliminaTies on 士anked spaces, their completion and vector valued functions in I. In I I, we d・fi・・th・h・nogen・・u・・ranked・㏄・・r・pac・εlll. ll1・㎞9・n・・us・eans.・h・・

紐ypren・ig吻血・ni・V〔f)・f f・ε1旧ll i・th・t郷・…f・V(・)・f・n・fb・

血・0・ig桓・・聖e space L川lll i・m…d・Ced・Where・f・・ead f・如ll・・

preneighboτhood V〔f, A, l ll‖1,ε)is defined by using a set A祠hich is

蹴三蒜㌻lll㌃艦:P:e右{llllぱ㌔el:㌢,.

c・nrP・・ti・n.・fεlll lll・・he spaceπ・f・B・chn・・m…urab・・、fun・ti・鵬i・

i・t・・血・ed・・an・・x・mP・・6f Lm田…f皿・・…血・血・・ry・f麺・・g・a・i・

carried out, verifying eadh conditions fbr the case of the政)dhner integrable, the E.R。 type Bodhner integral and the ma⊇d皿al integral giving an exa皿ple showing that maximal is biggeτthan E.R. t)?e.      The author would li】(e to e即ress her sincere gratitude to Professors S.Na㎞ishi飢d Y.Naga㎞ra fbr their valuable criticisms and suggestions. which impτoved greatly the presentation of this paper.      1. Preliminaries (【5】, 【6], 【7】, [12]).    −1.Ranked spaces.   −Aspace E is called a ranked space(of indicatorωo)if, fbr every point XeE, there is associated a non−e㎎)ty family「レ/てx〕 consisting of sψsets of E ・nd, f・r百ery n・n−n・g・tiv・int・ger n, there is ass・ciated・a・sUl・famj[1y垢 ・fヅ・{γ〔x);x・E}S・ti・fying.th・f・11・wing・・頑ti・鵬・

㈹ω

F・revery皿剖由er U〔x)・fγこx〕,x・eU(x). F。t everY P・int x・f E, ev・ry・ma・ibe・U〔x)・fγ〔x)and ev・ry non−negative integer皿, there are another integer皿and a ma・iber・V〔x)・f P’rm s・idh that皿・nand V(x)ζU(x).

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143

(正NERALIZATION OF THE BOCHNER INTEGRAL     ㎞b・rs・fγ∼x)a・e called p・eneighb・rh・・dS・fx・r Of cent・・x姻 頑tten U〔x),V(x),・t・. Pren・ighbO・h(x・お・f x be1・㎎土㎎.t・ち訂e 5・id t。 be of rank n and written U(x,n〕, V(x,n), etc.(;」stαnarily, we define that , U(x,0) = E fbτ all x∈Eピ .These prenei.ghbOrhoods・・are often written si珂ply U,V, etc.     Asequence{Ui}.={U〔xi, ni〕}of preneighb《》τhoods of center xi is called ㎞damental if it satisfies the following conditions:.      (f・1) ・U1,U21⊃…  ⊃Ui⊃・…      (f.2)  n1<112 く ... < ni く .....      〔f・3) For eadh i・there is a j suCll that j ≧i・xj =xj+1 and nj 〈nj+1・

A㎞dan㎝tal sequence{Ui}fb四hidl xi=xfor all i is called a fUndamental

sequence of center x. The Ilotati㎝f.s. is an abbreViation. fbr the woτd fUndainental sequence or seqμences−      Sequences. of preneighborhoocls are sometimes written u, v, etc. For t肘o sequences u.= {Ui}and v.= {Vi}, u>vmeans that, for every Ui, there is a Vj ・u・虫th・t Ui⊃Vj and・the equivalence u∼v m・・n・ that u>v飢d v>u・      Asequence{xi}in a ranked space E is said to be a Cauchy.・・secfuence if there exists a fU皿damental sequence u= {Ui}, called. a defi卿seqμence,. so that fbr every j, there exists a皿i。 suCh that xi∈Uj for.all i≧io. A(;audhy sequence is said to be r−comrergent or r−cornverges to x and we write x=r−1im xi if its defining se(luence is・.a f輌ental sequencg. o£Center x∈E.      Aran](ed space E is said to satisfy the separation property{r−Tl)if, fbr every.x of E and for every.ftmdamental sequence u={U〔x, ni〕}of center x,

th・inter・㏄ti・nハ・・QU(…i〕・{・}〔a set・・邸t血9・f th・e1㎝・・t・

alo】ie〕.      Aranked space E is sa工d to be c㎝互)1ete. if, for every fundam即tal sequence ・・{U(・i・・i)}・th・麺terseCti・n(・・?U(・i,・i〕キφ・      2・Homogeneous ranked v㏄toT spaces・      Let E be a vector space over the scalar field 〃て 〔=此orC.) and let E be a・rank。d・pace ass・・i・t・d厄th th・f・・nili…fp・㎝・i輌・h・。d・.rJ/1(x〕ai・d・」(n, w,itt。n・1・・a・(E,γ1・〕,77n). th・・anked・pace(E,γてx〕,殉i・ca11・d ahσmogeneous ranked space if the fbllo輌g condition〔SI)is satisfied.

He。e。ft。。,鳩d。n。t。 th・緬1i。・巧0)飢d拓ハ」/r(o)by♂and・if・

respectively(n∈N, N={0,1,2,...}).      〔SI)m・ry・menibe・V(x〕・fγω…md・V(x, n)・f P/Tn ar・。f th・f・m        x.+V・f・r・some V e♂and・x・Vfbτ・・me V e ifn.respectiv・1y. If th・㎞9㎝・・us・.rank・d・pace〔E,夕「(x〕,Pt7n〕・ati・fi・5 th・f・11・Wing conditions (SII〕 ∼(SIV), it becomes a Ta唾ed vector space.called a homogeneous

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144

Y.TAG㏄HI

ranked vector spaCe, sO that theΦeratiohs of additionσ

multiplication by scalafs P:K−×E÷Eare』r−continuous

【12】p.133〕.         . ..’  .t’ .        “ ” 〔SII)’ (SIII〕 〔SIV〕

:ExE÷Eand

(【8]p.180,p.181, th・・y・t㎝. y・C・eN)sati・fi・・th・fO11㎝迦、氾血ti。nS,

fo、)血er・exi・t・輌・曲血t if u、・4h, u、・ゴk副

   ・h・k≧…血・n,血er・.exi・t・a・・ゴj允・・鴫≦・,

      suCll that U1+U2ζW.  .      一    ズ. .        1≡’ ωF・・wery Vξ3m, th・。e・exi・tS、an h・n(。)’・m。u、h       th・tifU、,U、⊂V,U、∈♂h,U、∈♂k飢dh, k≧・,       血㎝U・+U・⊂V・and tihere exi・…W・♂j允・・㎝・j      with m.〈j≦nsudl that Ul+U2 c:W.ζV.1 』   − The・yst㎝♂n〔n∈N)i・功・・rb・nt,廿iat iS,.fb・ea・虫x∈E, there・ exi・t・af.・. u。・{Ui}顧thUie∂血〔i〕.砲・rem〔i)÷+・6・as i・→+。。su(虫that eadh Ui absolbs the point xうi.e., there exists a’ Xi d /K・頑廿rat・X∈λ咋「曲enev《)r lλ1≧1λi.1. ha・虫・U∈♂・(・e旬i・ci・c1・d, i.ε;, if・・U・and 1λ1≦1,.the・ .λX∈U..@‘「・L.      ・    .     で      3.Ranked vector subspaces『.  ・       』  ・・      L・t〔E,うてx〕,tVrn). b・a(h㎝・g・n・・us)’rank・d・veCt・t・pace田蛆1・t A b・

・v㏄t・rs・b・p㏄・・t E. If鳩d・fin・th・f皿1iesγ〔P;A〕−ahdシ繭旬

         シ0;A)・・.{Vtp)n A;VO〕∈γ6)}㊤d〕 1 . 

and

         γ両一{VωnA;Vω∈ヅn, P・eA}〔h醐,

th・n the ten助〔A, P/TfP;A〕,〆竺㈹)b㏄(加es a(h㎝・gen・・US)r砿ed.vect・r ・pace.㎞er・6fシ6;A)..鋤・fγfn〔A〕・.ar・d・n・ted・・U(P;旬,v〔P;A〕,・et・. and U(P,n;A〕,V〔P,n;A〕, etc.     血・r皿・dv㏄t・r・pace〔A;夕6;A),L/rn〔A〕)i・ca11・d a・ank・d vect・r ・ub・pace・f〔E,γ〔x),‘)/7n)if th・f・ll・wing・O・diti・h:・      (RVS)F・r・v・ワf…㎞・ntq1・餌・㎝C・{Ui}={U(Pi;mi;A〕}血A・there        existξaimdamental−sequence{Vi}= {V口)i,ni)}in E sudl that,        允rs㎝e i。e’N, mi・ni紐d VinA・Ui’砲6heve・i≧i。∴ is s〒atisfied.  』.       .     .、     4. 〔麺)1etionS of ran](ed spaces..       ご     For a fundamenta1 Sequence u={Ui}in a ranked space E, the・intersection

…0・…d…t輌θ(・〕・F・・加・f輌・n・…e…nc・・卿・・

uρvm・ans廿田tth・・e exi・t・aw・頑血tw・u紐dw・v:B>・ M飢d咋・

・we denote the s6ts of all funda血erital sequenCes.in E al1(10f lall fundamental sequence§ Of center・P∈E in E resp㏄tively.・If E is a homogenec uns ranked.「 vedtor space, then the fol10wing condition5:  .’ 』   ’・ 『  . .  ’−  三 ・.’

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145

G日団ERALIZATION OF THE BO〔HNER INTEGRAL

     (M・)Pξθ〔y〕;μ・Mde..⇒㌔・悔;Vp・u・.・、㌧「. 二

     (M3〕 U,、v二∈M★;uハv± φ→ UPV.  .  .       .三   .    ....・    . are satisfied(【5]p.316, Prop.30). Furthermore, if E. satエsfies the cgndition 〔M3〕, the relation u p v becomes an equivalence relation on M★([5]pご・313, Prqp.

24〕・ ,、. 、  、.  .・. tt ..、  .

     Let E be a皿incomplete ranked space.1 A ra皿ked space F, associated with M/Ttr)〕 and M/7n, is said to l)e a completion of E if it satisfies the foiiσwinざー「

conditions

     (CI〕      〔CII)    、(CIIJ) If B satisfies there exists a completion of E space satisflソing the separation property Th. 3.13,       ,      5.Vector valued functions,  、       ・      L      Let 〔S,.」4,p)be a non−atomic, finite measure space satisfying p(S)=1. dnd it is㎞σwn that the following condition(St)holds:      〔S・)F・r.『any A・・誹;P〔A〕・0飢d㎝yエeal number r;0・r≦1.i(A),ther・        exists a set BζAsudh that B∈メand O<p(B)≦r.

L・tXbe a㎞・h・SPace with・th…mh旧い蛤X−va1・・d麺・ti・n f.d・f頑

・nS・d・mt・dby f・S◆X・is ca1↓・d a・imp1・fm・ti・n if it ta㎏s°ply.a finite number of values on S, i.e., it has the following、 e】「pression:For a

f皿te sUbset{x・∈X;’= P・・…}c}.°f X・飢d輌’te fam’ly°f輌11y

垣・」・in・・et{Si∈メ・iy、 Si・S・i……・k}・f(・)・ii、・i XSi!・∼・ぬere XSi(i=1・…・k)i・血e dla「acte「istic f皿cti°n°f Si・     .The integra1 〔over S) of the si㎎)1e function f is defined as the value:

謡1、ご・1・:。三。皇盤孟lf蓋。lll、1・麟dS∴A蕊二n

・・∫Afdu・r IA(f).      Byε,we denote the family of all X−valued siコ哩)1e functions. By a quasi−se皿i−no㎝ll川‖onε,we mean a real valued fUnctional onε

satisfying the fOllo砿㎎conditions(i)∼(iii):

(i〕 〔ii) 〔iii〕 (C工) ∼ (CIII〕.:  [the ranked space F is comPlete.  [[hle ranked space E is q fanked subspace of the ranked space F.  Eis dense in the ranked space F as a sUl)set of F, that is, for W』・yp・F飢df・t・v・ry.Vω・γ6),・・h・v・. V(P)n−E ¥ ’¢.     the conditi㎝〔M1), 〔M3〕 and the separatl’ on property  (r−Ti),        .、It fOllows that a homogeneous ranked vector       (r−Ti〕 has a completion ([6] P.416, [5]p.313). For any f∈ ε, 川f川 ≧ 0. For any scalar λ∈Kand f∈ ε , “1λf Hl= 1λ1 11iflll.・ th。。e i、 a.PO、i・i。。 number・K≧…u・h血・, fb・any .f and gξ E), 川f+glll≦K(川f川 + 川glll).

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146

Y.TAGUCHI

By 7π, we denote the family of all X−valued measuぽable㎞ctions f :S今X (in the sense of Bbdbne1), d遭t is, there exists a sequence {fi∈ ε; i=1,2,...} such that, fOr alnK)st all s of S,          f(s〕=聖fi(s〕血x・ Hereafter, we identify two functions f and g血(土are identical almost every血ere.     II・Ra血ked v㏄tor spaces and their cc服Pletions.

    …㎝・9・…US・ani・ed・vect・呼㏄・ε川川・

    For eveτγf∈εand fbr every n∈N, we associate a nqn−enrPty family

巧f〕・fp・㎝・ig吻血一・ff翻an・h一卿tyf麺・yγ4n・fp。㎝。ig吻。h。。dS

・f蜘knd・fin・dinth・f・11・wing・w・y. F・r ・ny fe e,Aeメ;U(A)・O

and  ε > 0, we put         V〔f,A, 川 川, ε〕=f+V(0, A, 川 川, ε〕,

孤dV(°・A・1旧ll・ε)={

P∈ε;e?・㍗ll「(s)‖≦ε・川「lll≦ε}

        Mf)・{V(f, A,1旧ll,。);A∈メ,ε>0}. For every n∈N, we def垣e         V〔f・A・・〕=V(f・A蝋1・〔、6・。)…〔S一旬≦〔,6・・

and

        %・{V(f,A,n);f∈ε,A∈」4}.. th・t・m・ワ〔ε・夕ff)・1・rn)i・w・itt・・ε1旧ll・麺・y・・     In the following discussion, we assume that quasi−s㎝i−nom川川also

satisfies the follo砿㎎condition(T):    .   .

    (T)F・r孤yε>0,A・メ.孤d・∈ε・ati・fyi。gμω・0孤d

        e㌘.㍗ll・〔・〕ll<1・血ere exi・t・…’・ε・・Ch・h…即P・・に・・         e誓。㍗ll「(s)+「’(s)ll=1・・?。・gmP|1・’(・〕ll≦1酬1・’lll<ε・     Le㎜a 1. 1.et ε and ε, be positive numbers and A and Al be me而)ers of

メ.・f、≧ε1孤d、(A−A・〕.。,,h。n  r

        V〔f,A, 川 川, ε)っV〔f, A・, lll川, ε,).     P…f.F・r組y g∈V(f, A・)lll川,・,),聡・;an・find・㎝・・.∈ε・u(血

that

        9=f+「・・;こ1響ll・〔・)ll≦ε’・Ill・川≦・’・

S血ce娘一Aり=°・we』ve e誓.㍗ll・〔・)ll≦ε’・b°…ver・血e c・nditi・n

ε,@≦ ε implies that      g         ess s訓r〔s)ll≦εand lHrlH≦ε.       s∈A

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147

Hence,          9∈V(f,A, lll 川, ε).     1…2. v(f,A, ll1川, ε)⊃v(g, B, 川 lll,η)i顕pliesε≧nand μ(A−B) = 0.       N

    P…f.S卿se・⑭… F㎜the c・㎡i・i・・(T〕apP・yi㎎f・・昔・・,

A−Be●4 and r〔s)三…0, there exists a㎞ctioll r・eεSudh that          ・卿・・’・A−…管。・㌍11・’(・)ll・・飢・lll・’lll・貴・ Putti㎎ r = 3εr., we have          r∈ε and g+r∈v(g, B, 111 111,η) −v(f, A, 川 ll1,ε〕. In fact,          ll1・111=3・川・’lll<…誉。spll「〔s)11=°〈n・

晒・d・伽・i・n・fes・en・・砿・卿一,』・・…・・,血・re融・tS・・et

c∈メ;c⊂A−B,μ(c)・o・・㎞d砲tfb・a㎞。・t・11・∈c, ll・・(・)11≧1一ε、. Then, for allnost all s∈…C,          ll・(・)1ド3・ll・’〔・)ll≧3ε一3εε・・3ε一号

and

         ll〔9◆r〕(s〕−f(s)ll ≧ ll r(s)ll − 119(s〕−f(s)ll        ≧3ε一号一ε・2ε一号・≧・・ Hence,          ・㌘,㍗11(・・r)〔・)−f(・)ll≧妾ε〉ε・ This would imply          g+r∈V〔g,B, 田 川, n) −V(f, A, 川 川,ε). This is a cOIltradiction.      Next, supposeε<ηand 1.t〔A−B) = 0. Since V(f, A, lll lll,ε)⊃V〔g, B,川 lll,n), there is all r ∈ ε such that          9=f+r・・?、㍗ll・〔・)ll≦・・ll剛≦・・

T・k・・a・舳erδ・画血tε<δ・n. F㎜・he c・㎡i・i・n(T)・卯・y血g fb・2,

A飢d}r;

髢 1(1㌔s:s、yr’∈εs t

         ・讐。・iyp 11・’(・)ll≦・・lll・‘川・号・ Then,          ・㌘。㍗ll・(・)+δ・’〔・)ll=δ〉ε・lllδ「’川く・

and

         ・警。・iYP11δ・’ll=e㌘。・響llδ・’ll≦δ<・・ These would i叩ply

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148

Y.TAGUCHI

         g+δr,=f◆r+δr・∈v(g,B, ll1 川,n) −v(f, A, 川 111,ε). This is a contradiction.

    L−3・ε111111i・a㎞・。9・・《・anked・pacρ・

     Proof. From the definiti㎝of V(f, A, 川 lll,ε〕, it is eno㎎h to show th且t the conditions 〔A) alM1 〔a) are satisfied・・〔A):Eveτy V(f, A, lll lll,ε) contains f. (a):For everγV(f, A, lll lll,ε)a刀d every n∈N,’we take m∈N and V(f, B, m〕        1<ε,AζB,B・メ鋤μ(S.B)≦1価。nμ〔S−A)≦

such that m>叫

       (5K)2m        〔5K)2m   1        ,put B=A. Otherwise, put B=Sor put B=AUB,;Bl⊂S−A, (5K)2m ・(S−AuB’)≦

k、S・・S・Ch・・et・B・・x・・勒血e c・㎡・・i・n(Sつ)・by・一・・

we have V(f, A, 川 ll1,ε)⊃V(f, B, m〕      Lαma 4. Every V(0, A, n);A∈”t’,n∈i N is circled.      Proof. For any r∈V(0, A, n)and anyλ∈〃r sudh that lλ1≦1,          噺‖・・〔・)‖≦1・1・管。㍗ll・〔・)ll≦〔、き・。

and

         lll・・ll国・1111・lll≦(,き・パ Hence,          λr∈V〔0,A, n〕.     1.e晒na 5. For eadh f∈ε,there exists a f.s. v= {Vi}ロ{V〔0, Ai, ni)} such that each Vi absorbs the e1㎝ent f, i.e., there exists aλi∈〃r such that f∈λiVi.     Proof. For eadh si皿ple f血ction f, the fUIIK㎞ental sequence v= {V〔0, S, i)}

meets the ab°ve「輌「㎝ent・In fact・1etλ=蝋e等.sSXP 11 f(s〕IL川f川〕孤d

t・k・λie /k・u・虫廿坦t lλil・λ〔5・)2if・・e紬i, th・n,          esξ。㍗ll吉f〔・〕ll≦1走1…(、#tl{i・

and

         lll吉・1‖≦1走1川・川・(,3・,,・ Hence,          f∈λiVi fOr eadh i.

    蛭Lεlll lll i・a㎞9・・e・田融・d・㏄t・・space・

    Proof.  Lemmas 3 and 4 show that the conditions 〔SI) and 〔SIV) are satisfied. The condition (SIII〕 fbllows from lamnas 5 and 4. So, it is enough to shOw that the condition (SII) is also satisfied. 〔SII)(b1〕: For any n, h, k∈Nsudh that h, k≧n>1and V1=V(0, A, h), V、・V(0,B, k),we can・in。・・e a・・n一印pty set C・AnB∈メ,・㎜ber兄∈N

・・一・・≦㎜(n−…’〕f…’・a鞠’㎎・(S−C〕・

k、。}……V〔・・…〕・

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GENERALIZATION OF THE BOCHNER INIEGRAL In fact, for any rleVl, r2∈:V2,          ・・9ige11(・・+r・)(・)llエe誓.S,UPII「・(s)+「・〔s)ll        ≦・㌘。・t・・11「・〔S)ll ’ es,S。ScXP ll「・(S)ll        ≦・㌘。㍗ll・・(S〕ll+e誓。S駅ll「・(S)il

       ≦ 1 . 1 ≦ 2 < 1 ≦ 1

       −〔5.〕2h〔5。〕2k−(5・)2n(5・〕2〔n−1)(5・)2兄’ similarly,          川r1+r、川≦K川r、 lll・111r,lll)

       ≦・.K≦ 1 ≦1 .

       一(ik〕2h(5。)2k(5・)2〔n’1)        〔5K)22 Wbreover,          ・(S−・)・〔、。}・…〔、;… These imply rl+r2 GW, that is, Vl+V2⊂W. 〔SII)〔b2): For any V(0, A, m), ta】(e n∈Nsudh that n≧m+2. For h, keN sudh h, k≧nand Vi・V(0, B, h),V2・V〔0, C, k)(V(0, A, m〕, it follows that Vl+V2⊂V. In fact, h, k≧mand, by lenna 2,μ(A−B)=Oand p(A−C〕 = 0. Hence, the fbllowing inequality          1.t(A−Bnc) ≦μ(A−B) +p〔A−C) iirplies          P(A−Bnc) = 0 . For any rleVI and r2∈V2,          eSS SmPH ri〔S〕・r、(S)ll≦eSS SXp 11 r・(S)11 ・ eSS SUp 11 r・〔S〕ll          seBAC      s∈Bnc        s∈Bnc      、        ≦・㌘。㍗ll・・〔・)ll・e慧叩ll・・(・)ll・(、5・h・〔、き・・        く   1   〈   1   <  1        (5・)2(n−1)(5・)2(m+1)(5・)㎞

and

         1“r1+r・111≦K(川Klll・川r・lll〕        〈   K   <   1   ≦  1  .        (5・)2〔・’1〕(5・)2〔・“2)        (5K〕2m That is,          Vl+V2⊂V.      Next, take the set B n C 〔this is non一㎝畦)ty〕, the nuriber£=m+1 and the set W・V(0, B n C,兄). then frcm the above inequalities, V1◆V2ζWand from lenma 1, WくV.      The next 1臼鵬, the Egoroff,s theor㎝in the vector valued case, will be needed in the proof of the fbllowing Prop・ 2 and else血ere・      Lerma E…ff). L・t{fi}be a sequ・nc・in m. If{fi}・・順rges t・a

149

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150

Y.TAG㏄HI

fm・ti・・f・.e.・・S, th・・, f・・町、>0, th。。e・。xi。t, a。et A∈メ,曲血t μ(S−A) ≦εand{fi}converges to f unifbnnly on A.      Proof・ If {fi}converges to a ftmction f a.e. on S, then their nρms ll fi l l converges to ‖fll a.e. on S. Hence fbτ any(Bodmex)measurable function f, its norm ll f l l is a real valued measurable function on S. Cbnsequently, any proof fOr real valued functions using fbr jnstance          ・j・9.{・∈・・ll fi(・)−f〔・〕11≦・}・メ・…       1=コ carrles over to o町vector valued case. ㎞・k・in・ur・anked・paceεll川ll, ・ny dec・ea・i・g・eq・・㏄・

u=

@{V〔9k, Ak, nk)}satisfying nk+1 −nk≧ 2 can be modified to a f.s. v={V(h兄,B兄, m兄)}sudl that      i){h兄}孤d{gk}are identical as a set・      ii)  {B兄} and {Ak} are identical as a family of sets・      iii)V>U.       ・ In fact, fr㎝          V〔gk, Ak, nk)⊃V(9k+1, Ak+1, nk+1〕 it follows that          V(gk, Ak, nk−1)⊃V(9k+1, A1(+1, nk+1−2)⊃V(9k+1, Ak+1, nk+1−1〕 .

    迦⊥L・t{fi}be a・明・・砲桓εlll lll・th・・eq・㎝ce{fi}i・da・Chy

in £111111 if孤d・虹y if i・・ati・fies th・f…㎝i㎎…c・㎡i・i。品・      (i) There exists a measura工)1e function f∈π Whidh is the a1瓜)st          every曲ere liJnit of {fi} on S.      (ii〕 The quasi−semi−norms of fi−fj tends to O as i・ j tend to infinity・     Proof. (⇒〕 By the definition of a Caudhy sequence, there exists a f.s. u= {V(gk, Ak, ii!lc)}satisfyi㎎ that for any 1(, there is a ko sudll that          fi∈V(9k, Ak, mk)fbr i≧ko, i.e.,

{欝f

G:i二fl臼)ll≦〔・きk㎞” ’、

So, we can take a set Nk⊂Ak sudl that          μ(Nk)=0,        ・叩ll fi(・)−fj(・)ll≦1f・・i, j≧k。. Let A= Let M=

..・eAk一咋    (5・)mk

U (Ak 一 Nk). Th・n・・hav・P(S−A)・・. k=1          co 〔s−A〕U(UNk). Then we still haveμ(M)=o. Now,1et s{…M, then         k=1

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151

G日旺ERALIZATION OF THE BOCHNER INTEGRAL

  ’s

秩E1{㌶〒ls篭:d:漂ζ・.・・

Since・w・・h…Akと』⊂…,出s s㎞・th・…he・e・exi・t・a・皿・・f{fi(・)}

i・X,’ 堰D・.,d・f㎞91i・fi(・}・.f(・),鳩』f・πdnd・fi・f・….・・S・       ・ .1◆◎o  ・       . Hence the co亘d匝tion (i) is satisfied.     .       .      The condit工On (ii) follows frc冊the folloWing inequality:

      1“fi−fj・lll≦〔認Kが…j≧…   .

     〔←〕 First,1et{fi}be a sequence cOnvergi㎎a・9・on S to a fmction f.Fr㎝the condition(i), and血e above le唖(Egoroff), fbr餌y k,、 there exists・a、set均くsatisfyi㎎ that       ・      .        ,  .

      ・〔S−Bk)・て、き・・   .

al品[fi李fun直fb1凪y on Bk. Let A1(=BI U...UBk、 Then we have

A1(A2⊂_.⊂Ak(...,

      ・(S−A・)≦(、5・k and fi÷funifbnnly on Ak・       ・

T・㎞gint・acc・mt・1・・the c・頑ti・・〔ii)川fi.−fj lll÷0・・i・j→輪・鳩

ca⊇・・se a sUbseq・・nce{9£k}・f{fi}satisfying .    ・

      兄・〈£・〈…<21(<…・     .      ・

      lll・㌦一fjl‖≦(、6・・鋼≧・k

      。葦乳‖・・k(・)−f」〔・)ll≦〔、6・・f・rj≧£k・’  −

F・㎝・h・.E醐・kp・ec㏄屯㎎・h・p・叩・2・・h・deCrea・i・g・明…ce u={V(9兄k・Ak・2k)} can be modified to a f.s. v= {V(hk, Bk, mk)}so that v is a definiコg sequence

fb・{fi}, i…,{fi}i・da・d・y麺ε川川・  .

     Corollary・ ε川Msatisfies the seParation property(r−T・)・      Proof.  It is enough to show that, fOr ally fundamental sequence v={V(0,Ai, i〕}, the intersection n v= ∩V〔0, Ai, i)has only one element       i O. This fbllows easily from the,above prop. 2 applying for the sequence {f’}

G.{㌫証。dvect。r。pace L川川飢・,he c。mP・。ti。。。fε川 ・・血・

 sense of S. Na㎞shi.          N       by 9川田…den。・・血・.f・・nily“・f・11(㎞輌・おu「飾1e㎞cti°ns f

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152

Y.TAGUCHI

・曲峻血ere訂・伽・旬・equencei{fi・桓ε1旧lr・輌聖・、・f・・e・

      ε制i・ave・t・r・ub・p・c・・f.冗・伽

onS.

      One can verify easily that

the quasi−s㎝i−mm川川,we prqpose the fb11㎝血g condition(E):

〔E〕

;e㌔Sl:・三㌶゜1。°;lt罐蕊郷認eriξ川‖l

aC卿・eq・㎝ce血ε川・川nt・h・麺fi・f・…(i…,f・N

the唖th「esp・・t・t・th・ex・ended・q{fa・i一迦i−n・叫川fi− Remark. If the qUa’Si−se頑一norm l旧ll is a s㎝i¶om, i.e.,

to

      if{f.}is       1     εm川〕・ flll→0〔i→+◎。).    if it satisfies the triangle inequality with K=1:       川f+glll≦ 川f川 + lllglH, th・n the c・nditi・n(E)iS aut・ma七ically satiSfi・d. in facr, (遠u(ky『sequence, then we have 川fi − fj l l l→・0 〔i, j →・+。」) lhi・p・・perty・nd th・f・11・wi・g in・・1・・lity・1川fi川一川fj l川 jJ理)1y that there exists a unique finite limit of {川fi川}, hence,       N

・uas’迦㊨n°m°f f=1許fi∈εm川… b・d・f’・・d by川f川

血・therm・r・・f・r・a・fiXed・・{fi−fn}is a…C・uChy in・.8川lll          li皿 (fi − fn〕 = f − fh ,       1 so that          l妻皿 lllfi − fnll【 = 川f − fn ill .       ユ       It fOllows that 川f − fh l l l→ 0 〔n→+c。〕.      In the seque1, we sirppose that the condition (E)       パ d・f拠iti。・・f・ank・!・tru・tur・・nε川卜

    D・f・L・tf∈ε川川・A・et Aξ ・4  .

to f, abbfeviated as ad. w.r. to f,

血ε川川・ud・th・t fi・f・unif・mly・吐・・i・和,

By the lemma a皿admissible Set A G 54

We associate

following way and the ternary      For eaCh f∈ we define          ン     ’  V(f,A,    if{fi}is a

fr㎝the definition.

≦川fi−f」川

  the =’

hlllfi川・

and

       is satisfied. For the       we need the following notion:      is said to be admissible with respect if there exists a CauChy sequence {fi} (Eg・r・ff)・f・r ・ny f∈E)111111 ・nd・・ny e>・,・her・exi・t・       with respect to f such thatμ(S−A〕 ≦ ε. ふ       N      ダ

εIH川with t2e fami1姪sγて旦andγirn defihed in the

.  (ε川lil・γt(f)・』CV7n)i・d…t・d by£川・

ε川川・A・メ・Ai・ad・.・…t・f…d・㈹….and..ε〉・,

m  川, ε〕 = {9∈」

and

         籔f〕・{マ(f,A, lll     ’      パ F・r eaCh・L)EN ・nd ea・血f∈εll      .  V〔f,A, n)=V(f, A,

N

εlll川・・㌘。・㍗ll f(・〕一・〔・)ll≦・・ llif−glll≦ε} ,。);Aξメ。㏄h・h。tAi,ad.。.。.・。f,ε>0}. 111 ・ W・ ’d・fin・ 旧ll・(、6・.f) ・h・・e

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153

GENE−OF皿㎜1唖鍋

・(S−A)≦.1 (Ai。 ad. w. r. t。 f     〔5K)2n・   ・〕鋤       パ      パ       N          γ三・{V(f・A・∋;f∈ε1川II}・      We can show easily the fb110wing Propqsition.

     工Llll l‖i・a・辿・d・㏄t…pace・th・t・is・th・騨・ti・ns・f

additionσ:ExE÷Eand multiplication by scalarsμ:」κxE÷Eare

r−contlnUOUS.

     工Th・・ranked・vect…pace Llll lll i・the c卿・・ti…fεII川

in the sense of S・Nakanishi with Tespect to the equivalence・relation u p v, の 1・e・,      『      1〕Lllf lll㎝b・id・ntified・ith the eq・i・alence c1田ses・f・11 f・…         i・ε1旧11 ・ith resp・・t t・th・re1・ti・n・ρ・・      2)Ll川is c卿1・te as a・飢k・d・pace・

     3)ε1旧ll i・d・鵬・血血・・ank・d・pace L川川お・・ab・et。f L川1

        飢dε酬i・ar麺・d・由・p㏄・・f Ll川1・

Reca・・戯fb・伽・f・・,…inε川’

?E・p・iff there改i・t・athi垣f…

・i・ε1旧ll・u由血… u・・and・・v・加・h・㎜re,血εll川ll,・h・

relation uρvis an equivalence one (1.4.).

     Pr・・f. We br・ve 1)∼3)in three steps:      ・

Proof of 1〕 We will show that there is a one−to−one correspOndence bet脚een

the e・㎝・n…f Ll旧ll皿d・he・・㎝・n…f血・輌i・i…i血・e・pect t・th・

・e1・ti・nρ・・f…mε川ll卜血t i・・fb・⑭f・・… {V(fi・Ai・・i)}孤d

v={V(gi, Bi, ni〕}, uρviff耳m fi=1妻m gi a. e.        1        1      〔⇒)Fr㎝the hypothesis, there is a third f.s. w・{v〔hi, ci,111川,兄i)} sudh that w>uand w>v・ Hence we have li皿hi=1敦n fi a・e. on S and       1       1 14Jn hi= 1↓m gi a.e. on S. It fbllows that 1‡m fi=1藪皿gi a.e. on S. The     ’  1       1      1       1 f皿・t’・nf・・

if・・f・・…n・’・孤・・㎝・…fL側・『

     (←) we take the sequence w={v(hi, ci,兄i〕}曲ere hi=fi, ci=Ai n Bi and兄i=Min(車i, ni) − 2. We have       μ〔S−Ci)≦μ(S−Ai〕 +μ(S−Bi)       ≦  1  +  1   ≦   2   <  1   ノ       (5・〕2mi〔5・)拠i (5・)2(£i+2)(5・)2兄i

th・n鵬ca・ea・ily・bOw血t w i・af…血ε川川・It i・e・・㎎h切・㎞・血t

w>uand w>v. The first half w>uis. evident by the definition of w and’ 1emma 1. The latter half w>vwill be sh(㎜in the fbllowing way.       パ

     L・tf=1㍗f・(=1評・・)・・…nS・輌i・諭・1㎝・・t・fεlll川・舟w・

for each V(hi, Ci,鳥i)and j≧i, we have

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154’ Y.TAGUC HI       e:㌔㍗‖f・〔・)−fj〔・)ll≦(、;孟、・‖1・・司lll・〔,.}・、・・       e:没ll・・(s)−9j〔・)ll≦〔、.;・・、、Olllgi−9jlll.≦(、。i・・、・・ Sh℃e・川烏・f川÷. O as j÷袖by the co血dition(E)l fOr s㎝e sufficiently iarge number j (>i〕,we have       .  』 、・        ’ .   .    ’    .         、lllf’“f川≦K(|llfゴfj・il1+lllfj−fllll≦(、5迦ゴ(5。y・、=〔9:27fifzm、 and similarly    ・      、  . 、、      ,

       lllgi−f川≦〔、S/・、・ 』    ・

usi㎎th・d㎝t麺uity。f th6 h・rn 1川1,w・have−.’

      e認ll f・(・)』f〔・)ll≦(,。}・、. ・ .

and

       ・e:響ll・・(・)−f(・〕ll≦〔、。}・・、・ These i町)1y that          。τえ、酬、ll f・〔・)’9・(・)ll≦(,。y・、・(,。}・。、〈、(詰・・i

and

      lllfi−9・川≦・〔  2K  +  2K〔5K)2mi(5。)2・i〕≦2。(、≒・・i・ Hence f<)r any g∈V(gi, Bi,’ni),we have       e謬II・“f’II≦eミ饗ll 9−9・ll+eミを6野ll gi−f・ll

       ≦ 1 . 1 ・ 1 . 1

       −(5・)2ni 2・(5・)2兄i=2・〔5・)2兄i 2・(5。)2兄i       1       −’   (5K)2兄i      ・

and

      川9−fi川≦・(川9−9illl・IIIgi−fi川)        ≦・〔(、。}・・i+2。(、i〕・・i)≦(、。i・・、・ 1・e・,       ・.         gεV(hi, Ci,兄i) (we have put hi=fi〕. Then, V(gi, Biクni)(V〔hi, Ci,兄i)and hence w>vas required.

’t・・f・f・〕・・垣・・t・sh㎝蹴・・…・・…i・{i(・・・・・…〕}・・£lll川,

       n{i≒ φ. From the definition of pr飢eighborhoods in

the intersection

       Ll1口|卜 the sequence{fi}is convergent a.e. on S. Let f=1im fi a.e.       1

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155’ GENI]RALIZATION OF THE BO(;HNER INTEGRAL       lv       N      ロ     Fi・・t…nt・1・bO・甑・fξεll川1卜F・㎝・h・d・fini・i・n・f Y〔fi・Ai・n’i)・ there exists a Caudly sequenc『{ξij}inε川1t l ll. sudh that .己          1i ,f・j.=f・輌皿y°n Ai・ so that, for eadh i, there exists a lj(i) sudh that, fCr j ≧ j(i),          ll・・(・〕−fi」〔・)11≦〔、.}・。、.・唾・ 』 恥・ea・虫i・鴨d・fine gi s・廿田t・砲・n・ver fi = fi・il∫(i’=0・1・・…幻飢d fi+k≒ fi+k+1,          gi+i’.=gi+k=fi+k j(i+k)  (i, = 0,1,…  ,10 ・

th・軸e・・eq・・nce・u・{V〔9i・Ai,・i)}bec㎝・・af…i・εlll ll卜・・血t th・ ’

・eq・enc・{9i}.i・.伽Ch・ in .em ill ・nd h・眠th・・e…i・t・a皿ea・疏・e・..・ function g.砲ich is the, l a1皿ost..eve蜘ere limit of{gi}. For any s∈Ai,、   『 we have    .     ‘1    .         .. ’,       ll 9〔・)−f(・〕ll≦‖9(・)−9i(・〕1.1・ll gi(s〕−fi.k(・)ll・ll fi.k(・)」f(・)‖        ・・ll 9(・〕−gi(・)1同l fi・k j(i・k)〔・)−fi・k(・〕ll+ll fi・k(・〕−f(・〕1「       tO(i→・ぬ〕. Since S一ΨAi.is a null set, we have f=g=..1im gi a.e. Hence we have     岬   1       .    l

feε川川・一

@.. 一’    . .  ・

     Next, we will show that fぐV(fi, Ai, ni〕 fbr a11.i. For each i,.there       ピ        Nis.a j >isudh血t fj =fj÷1 and nj <nj+1 since u={V〔fi, Ai, ni)} is a  . f.s. Taki㎎this j and any sufficiently large k>j+1, we have the following

in・q・・1iti…       .’、 、

?ソllfl〔・)・fk〔・)ll≦〔、言・巧、・

and       .          .川fj−fllHllfj・・−f川≦・〔川fj・・−fi,胴ll・fk二fllD       ≦・川fj・1−fklll・K2(lllfk−9k川・川9k−fllD・

・征血・㎜f・,・ince 9kヒi・an・・㎝・n・・舳・⑭・・eq−{与j}i・ε制

      N 曲・se a・…」lllii−・i・fk・εlll lll飢d・h・・叫・㎝ce{gk}血εll川ll・・百erg・・

・…t・frε川川,囑ing the c・nditi。・(E〕・w・㎝曲・・se a k・・1・・g・血t

the fbllo砿㎎蜘o inequalities are satisfied:       ‖1・・−9・川≦(、。〕;・j.、 . and       』

      .lllgk−f川≦(、。)i円.、・  . −

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156

Y.TAGUCHI

Fr㎝this we have

      lllfj−fHl MDreover,         N Hence f∈V(fj,Aj,nj). Proof of 3) evident since any preneighborhood of rank n V〔f

N

V(f,A, n)∩      We will show that For any f∈ w.r. tof, i÷+。。,fi So that there exists a j l‖fj−f川≦ε,1.e., fjeV(f, A, lll lll,ε〕. Hence we have V(f, A,川 lll, キφ. ・鋼・k・…h・d・fi・i・i・n・f p・・n・ig吻血鋤・V(f, A, lll川,・)i・Llll if we c㎞・se釦y A∈4耐h・ut the rest・i・ti・n・’A・i・ ・d. w.。. t。 f・,

        ≦K.・1 .K・. 2

        −       (5K〕2nj+1       (5・)2巧・1         ・、、。、〔巳5K)・・」・5、K、iii〕・。j(・・j・・≧nj・・)         ≦1. 1 +  1         − 2   (5K)2nj   2(5K)2nj        1       ■       〔5K)2nj by the continuity of the mrm l l ll,we have

・ミ㌔㍗ll巧(・)−f(・)ll≦(、.}・・ゴ ・

              Since j>i, we conclude that f(E V(fi, Ai, ni). th…tter h・・f・ε1旧ll i・ar麺・d・Ub・pace・・ Lll川II i・        ,.A,・)i・εlll川i・    εlll ll陣・・eAi・ad・・・・…f・       ε川』・・』・・th・・rank・d・pace Ll旧1卜 ・E.川lll孤蜘y V〔f・A・川lll・・)・us血g th・fact th・t A i・ad・

th・re・・xi・t・a(辿⇒・閾・・㏄・{fi}i・ε川lll・ud血t,・・

÷funifoエ凪y on A and then川fi−f川→Oby the condition(E).       su血血t es漂ll fj(・)’f(・)ll≦・孤d        ∼      ∼        ε)n ・・暇y・P・ssib・・t・・b・血・εIII IIIi・d・ns・血

S=[−1,1],X=訴こand川lll

since

      ・㌘。・㌍ll・〔・)ll・伶f・・an…ε・

曲ve e?。・9・・HII−−9(・〕ll≧・欝〔ll}

Hence we have ge▽(},s,・)・ 3.Examples. LIII III・ is the identically zero f皿ctional onε ij E)111   111 ε川

ε川川

       川・        1t ls not For example, 1et        ,that

川・w・ta㎏areal valued

 (=m). Sis not Hl=φ・In fact・ ll−II 9〔S〕ll) =+co.     by・8・,鳩’d・m・・血・加・。9・n…s・rank・d vect…paceε1旧ll・血…th・ quasi−s㎝i−nom l旧il is given by the identically zero㎞ctiona1. Hence K

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157

GENERALIZATION OF THE BOCHNER Im EGRAL in(iii〕、of the def皿tion of the quasi−s㎝i−norm may be taken as 1. Foτthis

quasi−semi−no叫the conditions(T)and(E)are trivially satisfied.町the

・・f血・i㎝,E・・舵.ム,・・m・・㎎ Llll川血血・cas・,・・血e c卿・・ti・n

of εo in the sense of S.. Nakanishi with. respect to the equivalence    . relation u p v (Prqp. 4). R㎝・rk.(りπ,γ〔f).γ7。〕,・bb・evi・t・a・・M。, i・・n・・f th・ex・叩1es・f ・㎝P1・t・ rank・q. /paces which・・㎜・t b・d・fin・d旬町鵬tTヰc吻t・)v… T・ prove this, we ren旧rk first that the convergence of seqUences in?πo aghbes with the Convergence of the sequences inπ,‘that is, almost eveエγwhere convergence(see prgp.2), hence it is eno㎎h to show that the aimost everywhere conve「9ence is mt a convergence by any metric. It will be shσm in the  . following way:      Su甲pose there were a metric d WhiCh wDuld give the comrergence a1ハnost ever殖ere. Let{fi}b・any sequenc6 in M砲i・in c6・v・rges in・measure t・a

㎞・ti・n・f∈π・Tak・飢y・Ub・equ・・c・{fik}・f{fi}・by・p・・perty・f

c°田e「gence in皿easu「e・血e「e exists a sUl’sequence{fikj}°f{fik}whiCh converges almost eyerywhere to f. ’皿1is fact would i㎎)1y that the original sequence {fi}converges a1皿ost everywhere to f since the conveエgence ih a metric space is always ㌔convergence, that is, any sequence {fi} converges to f if and °泣yif・f°「a”y subsequence{fik}°f{fi}・血e「e叙ists a sUbs因ue㏄e{fikj}

°f{f・k}s㏄h血tf・kj ’f as j’伶・

      Ihis contradicts        the fact that the convergence in measu[re does not always imply the almost everywhere cornrergence.       III. Theory of int6grations −Various exa珂Ples.       .   1       S・・far,・・ha…c・n・id・司・ranked・vect…paceεlll lll鋤i・・c㎝P・・ti・n ’L川川麺.th・・en・e・f S・㎞i・hi・W・・n・w・d・・a lthe・ry・f int・g・ati・ns・n

£lll lll b⊇・n th・p・血砲・・血t・f・r f・Ll1川ll・ith・倣hy・equenc・

{f・}血

Dεlll lll・・□tf・÷f ’n L川1・血・麺t姻1(f)’・2・f皿b1・

by the linit:

¥豊1(fi)・Recall・血at・f°「as迦1e㎞rti°n f=i≡、xi XAi・

       k it・血t・g・a11(f〕w・・ d・fi・ed・by・1(f〕=i≧1・ip(Ai〕・       For this purpose,1we propose a condition on the quasi−s㎝直一皿oτm lll 川in addition to the conditions (T〕 and 〔E):       (U〕 Tlie integra11:ε→Xis continu()us with respect to the quasi−        s㎝i−nom 111川,i..e., fbr anyε>0, there exists・δ>Osuch        that lllf川くδi町)1ies ll I(f)ll.〈ε.  Ihen we have the assertion that if the quasi−s6mi−nor皿 lll lll satisfies the

・・㎡i・i・・(u),血・血・・9r…(f)・・ε㎝b・ex・・典・・th・卿・・ti・・乙1旧ll

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158

Y.TAGUCHI

  bγthe 1迦itl .  .  ・ tt 』 ・’∴「・ 」: ・

      1(f〕=1}匝1(fi〕−∵1 ’.‘∴ % . ,・ ”

1” this ・egti・?n・we・give・庖・手・u…x…rP1…f・μSi−・e血二・・輌曲i・虫.・ati・巧thr ・・㎡i・i・mS〔U〕,〔T)md〔E)・.…h・f皿・PX・・EP・・3,.泌. sh㎝血・.⑰ere. exi・t・ aquasi−se血一n°m川lll・曲・・e c・頑・ti・nエ1旧ll,c・…m・a・・、c卿・・ti・n・    ・し制曲・・e・quasi−se血一P・恥1旧H・a・i・fy・he c㎝di・i㎝・〔U),(・)鋤d、(E)・

In pa「ticular・th・B・chner int・9r・1〔』rP1・1)・血・B・R・t)Pe一㎞・eゆt・騨1

  (th(卿1e 2)are c・ntained in・ur㎜(迦al integra1・f the ex餌唄1e 3.        1・ε1旧ll、皿・・C川川、([・]〕,        This. exat呼・i・ares己t・f P・・ξ・S・N・k・ni・}亘・.砲i・h、is sli餉tly mdifi・d   fbr our convenience.

       A・.a f皿cti㎝・1・nε,聡t・k・’11日ll、d・fm・d by    .

      川fHh・∫Sll f(・)Il d・飴・・ad・f∈ε.   rll lll、i・a・tU・11y…rTn・・n’ε.H・nc・the c。㎡iti。n(E〕is sati。fi。d.   Since, for any f eε,       ll・(・〕『1ドll SS・(・〕・・ll・Lll・〔・)ll・P= ill・lll、,、   this no㎝ 川  llll satisfies・the condition (U). Furthermore this no㎝ l l l lll, ・ati・fie・the c・㎡iti㎝(T).1・f・Ct, f・rε>0, Aξメ, r∈ε。ati。fying μω・0・nd ess s叩ll・(・〕ll・1,・虫・・se B←4・・廿iat B⊂A,。(。)−x∈X  .      s∈…A   identically on B and O<1.1〔B)くε〔Using the condition(Sりand recalling the   definition of si㎎)1e functions〕. If x= 0, dhoose x°∈…Xwith ll x l ll= 1 then :1{1〕、:.x’XB(s) ts the 「equ’「ement・If・≒°・th−・’(1〕=〔ll≒1.−1)xふXB〔・元

’in,。㌫ls還、灘.㌻cl細;鑓獅la:eご鷲:,,

㌻蜘ξ貰芸㌶1漂y。sl三:gc;,iif}1,;、三!ll」!・蒜1,,

・・tf・πb・B・dm・・m・・g・ab・…h・n・h・・e exi・t・aC・・血y・e⑯・・{fi}

mε・su・力血t fi・f・…bef血・gi∈)π by 『 一

’.

Xi(・)・

o:’〔s)1::::1:::に::::::1:. .

 rep1

B蕊ll・歴麦.” ・

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GENERAI、IZATION OF THE BOCHNER INrEGRAL.

159

then hi “f.・・…nd ll hi(・)ll≦211 f(・)ll。.。. h。㏄6          111hi−f川・・1,ll・・(・〕−f(・)ll・dp +・,一、 i・e・・{h・}’s a C・u・hy.・eq・・nc・血ε川川・孤…εlll lll1・4旧II、.

     2・εlll lll・即d−£1川ll・」   .  .’・『

     恥d・fi・・th・t㎜cati・n[f〔・〕]k(k・0){b・any・f∈ε。・fb11㎝。、・

      [・r)]k・監1:)1遮lllli,: 1.、.

F・reach f・ε…t川flll・・ll fllβ・1国1,  ’  ..

where.       ll fllβ=iwa ky{・∈S・ll f(・)ll≧k}

and

    、1国1・=認ll∫,[・〔・〕]k釦1卜 『.

川lll、 i・a qua・i(一sem1)一・・m・・ε・㎞ilarly,h。wn 。。 in・th。 real・v。lued、;as。 ([1]p.315,Le㎜). Exとctly as. S. Nakmishi did for the real valued case 〔[4]III・P・9°8・Le㎜14〕・w・・can・や・血t th・q田・i〔−s繊)一・・m.制・ satisfies the condition (U〕.   −      Tb show that the condition 〔E) is satisfied, the similar considerations will be done as in the real valued case 〔see [1] p.136〕. The condition (T) is shown to be satisfied in exactly same way as for the case of”川  l II ,.

     脆甫11・加wth・t th・却er・・f Ll1川‖、are趣acteri・ed i・血e space

of measurable functions M by the followi㎎加o conditions’:.  .』      〔A1) ]Ψ{s∈S; ll f(s)ll ≧k}= o〔1〕 〔k÷+。。).      (A・)是∫、[f(・〕]kdU・x’・t・’・X砲ere∫、[・〔・)]㌔…th・一・       integral of the trullcated function [f(s)]k for k > 0.      國 Theref・re・it i・皿t皿・1 t・・:all th・meniber・・f Qll lll・E・R・−tYIS・蜘mer

integrable fmctions.

     ・・fact・if fψ旧ll、・・h・re exi・t・aC・ud・y・明・・nce{fi}i・ε川」II 2 su(血that fi÷fa. e. and lllfi−flll2÷0. 1.et us prove first f satiSfies (A1). For any siiil[)1e function fi∈・ε,       .         『          k1」{s(きs; 11 fi〔s〕ll≧k} =o      .       ” fbr large k, hence 〔A1〕holds. We have always       {s∈S;ll g〔s)+h〔s)Il≧k+兄}(:{s∈S;ll g(s)ll≧k}U{sξS;ll h〔s)ll≧兄} so that  』『          P{s』∈…S;‖9〔s〕+h(s〕ll≧k+£}.      ・      「       ≦μ{・ξS;ll 9〔・〕ll≧k}・μ{・∈S;llh(・)‖≧兄}. .’・

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160

Y.TAGUCHI

Applying this inequality, we have          μ{s∈S;ll f(s)ll≧2k}       ≦P{・∈s;ll f(・〕−fi〔・〕ll≧k}・u{・∈S;ll fi(・)‖≧k},

hence

   . 2kp{s∈S・ll f(・〕ll≧2k}≦2‖f−fillβ・2k・{・∈S;II・fi〔・〕ll.≧k}・ This proves      .       1im lq」{s∈S;‖f(s)‖≧k}=0.       k・Hec The satisfaction of (A2) by su(血an f is straightfbrward (see an argument in the real case [1] p.136).      Comrersely, 1et f∈…ητ satisfy (A1) and (A2). By 〔A2〕, for any k> 0, the truncation [f〔s)]1(is Bod皿er i皿tegrable and we prove       川f− [f]klll2→O as k→・+。。 (see [1]p.137 fOr the real valued case). From the discussion of Bochner jntegrable functions in exa㎎1e 1, for eadl k,

・・c・n・血・・=・・eq・・㏄・{…}血ε川川・s輌・fik・[f]k〔・・和)

血£川lll、・

     F・・岨yb・und・d・bO・hner麺t・g・ab1・fm・ti・n g・ ・C{1川‖、, w・h訂・       ll∫、[・〔・)剛≦∫、II・(・)ll・・

and

         ky{・・S・ll 9(・)ll≧k}≦・∫ll 9〔・)11・d・       ・       S so that       川g川2≦2111g川1 Consequently, fr㎝       川fik−[f]k川、≦2111fik−[f]kl.ll、今0(as i◆栢〕,

鳩鋼Ch…e fiki∈ε.s・that

         fiki÷f ・・. e・〔i→袖)       ’

and

      lllfiki−f川・→°〔i→輪〕・   ∼

…ce{…、}・・aCW・叫uence血εlll川、飢d f∈ε川lll、・411川、・

     3・εIll川・飢d均旧ll・・   .  ・

     F・r孤yf∈ε,1・t ll剛1・・ll I(f)ll.・F・㎝th・v・ry d・finiti・n, th・ condition (U) is satisfied withδ = ε:       川flll3−〈ε ⇒ |l I(f〕ll <ε .

恥・i11・h㎝血t諭y・pace£ill川,f・・砲i・in・th・ qua・i−s㎝i−・。m伯ll.

・a・i・fies the c・ndi・i・n〔U)・i・…if・II・ fi・f血£lll川・・h・、麺・・9r・・

・ffi・d・f皿bl・by I(f〕=ig ..1(fエ〕・i・c・n匂i面垣・u・・pace身ll川・・

      〔quasi−)semi−mm川       川3will give the maximal theory of integrals         the That is,

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161

GENERALIZATION OF THE BO〔HNER INTEGRAL based°杣e c・㎎1・tiρngf・綱.・pac・1「εllll lll・『H・re・.・n岬q皿・i−)・r厄一 ・gPa.1旧ll3, the cO頑ti・ns④飢d.(T)qre・・atiSfエ・d.(R㎝・・k(IL2..)、}瞬1・ 1) and we s迦ose that,’also on.the. quasi−semi−noym lll ll札 the lconditionS (E) and (T) are satisfies.  .   .       ・    ..  .  ”      L・t川川br・q皿・i−s㎝i−mm w垣由・ati・fi6・. the c・nditi6・〔u),㊤飢d

〔T)・T・k・・f’∈Lll1‖1・BY th・d・麺tighジ臨e・exi・已(辿嘩卿・㏄・{fi}

血ε川・・由・h・三fi・f・・…nS紐d¶lfi−f川・・おi・樋・・by・h・

P・叩・4,fi÷f垣ε・孤d by the c・nditi・n〔U),」I I(fi}−1(f〕1い0, i.・.,

川fi−flll・‘・・堅i−・.血泣i・・fi・f桓「Llll lll,・r f・£川lll.、・舳

P・・v・・th・niaximality pf Llll ’1“、○ ・:  ’  .. ・一

     恥wi11・h。w紐叙卿1・。f Teal w1・ed・f皿・ti・yS嚇虫.・・e・・t E・R・ . integrable but magd皿al integrable.     ・       .t・・  ∫

     r・・tS・卜…p・晒・・rd・皿輌・99u・…r…dX・ne・

伽・id・・th・fm・ti・n f(・)=言・・Fi・・t f i・n・t E・R・血t・grabl…ince f・IOes− not satisfy CAi):      .         .        .       kμ{・・s;lf〔・)1≧k}・・2≠>o・・k・・fO. 、

0・th・・ther㎞we c血血・OS・. a ・equenc6・f・麺1・fm・ti・品{fh}・u・血d皿t

        ・fh〔−s)=−fh〔s) and       .      .・       ㌔(・〕・÷ ・・…n[一・,・】. Fr㎝ the dhoice of fh,       .       ・〔・。)・Sl・n(・)dS・・.       −1 so that

    、 lim I(島)=0い

       n Th・…h・W・・th・・血・㎞・ti・n÷∈£lll川,・      ・・t・us−k血・f・・血・f皿・ti・・f〔・)・÷,       川f川・・譜{・∈S・lf〔・)1≧k}・iY。吋、[f(・)]k(ip ll        =2<柚. That is, the space       {f∈π; 川f川2 < +oo} is actually bigger than the space of E.R. integrable functions in this case.        REFERENCES [1]  1. Amemiya and T. Ando: On the class of functiolls integrable in a      certain generalized sense, J. Fac. Sci.,Hokkaido Uhiv., 18〔1965),      128−140.

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162

Y.・TAGUCHI [2] [3] 【4] 【5] [6] 【7】 [8] [9] 【10] [11] [12] 113】 K;㎞㎎i:Application de la m6thode deS espaces ra㎎6S a la th60rie de 1,iIltegration, 1, Proc. Jqpan Acad., 32(1956) 215−220. .K.㎞㎎i:Sur une g6n6ralisation de 1,桓t6grale・Fundamental and APPlied Aspects of M已theraatics, the Reseaエ℃h lnstitute of APPlied Electricity, Hbkkaido University, 7(1959), 1−30. S. N麺sht:On generalized integrals, 1 − III, Proc. Japan Acad., 44(1968), 133−138,.225−230, 904−909. S.・Nakanishi:

㎞㎎i,Math.

S.Nakanishi: no.4(1978), S.Nakanishi: no・6〔]979), S.Nakanishi: The method of ranl(ed spaces propOsed by Professor Kinj iro Japonica 23, no.3〔1978), 291−323. Co呵pletions ofl ranked spaces, 1, Math・Japonica 23, 409−422.. {md Borel sets, Math.

S、Na㎞ishi:

㎞ctions飢d the Rad㎝一Nikod)m theor㎝

H.Okano:Sur

J., 11(1959),

H.0㎞。:Sur 1,麺毎ale

41〔1965〕, 17−22.

H.0㎞o:知fessor㎞j iro㎞㎎i・s work and its developments on the

theory of integration, Math. Japonica 23, no.3(1978〕, 325−331. Y.Tらgud亘:Fin工te dimensional r血ked vector spaces, TRU Mathematics,. 18−2(1982), 131−137.   Co叫)1etions of ranked spaces, II, Math・ Japol亘ca 23, 629−646二   Main spaces in distribution theory tTeated as ranked spaces          Japonica 26, Nb●2〔1981), 179−201.   Boc㎞er integral fOr certain ran1(ed−vector−space−valued       (to appear)・   1es int6grales (E.R.〕 et ses apPlicqtions, Osaka Math・   187−212.       〔A) et 1.integrale 〔IL R.), Pr㏄..Jal)an Acad., SCIEN(E UNIVERSITY OF TOKYO

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