A characterization of the Radon-Nikodym property

Pub . Ma . UAB Vol . 29 Ns 1 Ab il 1985 ACHARACTERIZATION OF THE RADON-NIKODYM PROPERTY Osca Blasco de la C uz §l . INTRODUCTION . The aim o his pape is o gi e a new cha ac e iza ion o he Radon-Nikodym p ope y in e ms o ma ingales in X- alued O liczspaces . Le X be a Banachspace andpu Z : , he Lebesgue measu able se s in 00,11 . I is well known (see [l]) : (1 .1) . X has he Radon-Nikodym p ope ywi h espec o [0,1] i and only i e e y boundeduni o mly in eg able ma ingale in LX[0,1] , ( n ,B n ) whe e o(UB n ) _  , is con e gen in  LX[0, 1] . We a e in e es ed in a gene aliza iono his ac . In his pape we shall p o e he ollowing Theo em (1 .2) . Le ~ be a Young unc ion wi h he ¿S 2 -condi ion .  X has he Radon-Nikodym p ope y i and only i e e y bounded ma ingale in  LX  ,  ( n ,B n )  whe e  o(U Bn )  _  ,  is  con- e gen in LX . The de ini ions and he main esul s ela ing o X- alued ma ingales and O liczspaces may be oi :nd in [1] and [2] espec i ely . We a e going o deno e by 0 a Young unc ion, a .nd LX = { : [0,11  -,>-X~ s ongly úcasiu able wi h espec Lebesgue measu e s . . p( ,O) =  0(11 (x)l1)dx < -} . 0 Le ~ be he complemen a y Young unc ion o ~ . We 1 ~~ (x)~~ ~g(x)1dx wi h { :[0,11 -~ X s ongly shallw i e  1 I l l o =  sup { g e Lyá , p(g,~) 1 1} and L o= measu ablewi h  U ilo < W} I is well known ha Lo is a ec o space and ll li~ is a no mon i . Besides, LX = LX i and only i 0 e i ies he lá 2-  _ condi ion .  I is easy o p o e ha he con e gen e and he boundedness in L~ and L. a e equi alen using he X X ollowing ac : (1 .3)  Suppose  ~  e i ies 0 2 -condi ion, i .e . he e exis s  K > 0  and  T 'k 0  such  ha  QS(2 )  4 K« )  o all ~! T hen   Tm+2  See (2] 7 page 158 .,  o a p oo . 2 §2 . PREVIOUS LEMMAS . Lemnia 1 . I ( n , n e N) is a bounded sequence in LX , hcss ( n , n e N) is a boundeduni o mly in eg ablesequence in L 1 X I he eexis s m, helongs o (N . wi i  p( ,0) L' 1/K m P oo . Fo a Young unc ion we ha e (2 .1) ~ - :> °°  as +  , and by (2 .1) we ob- ain li n 11 1 ¢ p( n ,~) + A , whe e . A is a eons an . LX We ha e only  o show ha  j E  Il n(x) Ildx  0  as m(E) -> 0 . Gi en E > 0 , by (2 .1) , he eexis s T > 0 such ha (2 .2)  l  > sc  o  >T  whe e  sup  p( n m  4 c . n Le d = E/2T . I m(E) <ó and deno ing An ={x : j I n (x) 11  :1 T} (1 E  and Bn = {x : ji n (x)11 > T} (1 E we ob ain Lemma 2 . P oo : E 11 n (x)11 = 11 n (x)~~dx + A n + J li (x)jIdx  < B  n  (2 .2) TM(E) + E 2c 0 ~(11 n (x)11)dx<E I 0 e i ies he ~ 2 -condi ion hen he simple unc= : ions a e dense in LX . Gi en e LX, sünce is s Ongly measu able, he e exis s a sequence  ( n  , n e N)  o co un ably alued unc- ions such ha (2 .3)  ~ n(x)- (x)  <  ñ  o almos al l  x e  [0,1] and o all n e N . Suppose n = - 57 - x n m XE whe e m=o n,m x n m e X and  XE  a e he cha ac e is ic unc ions o '  n,m disjoin measu ables se s . Since 2JI n (x)jj < 211 (x)II, ñ a .e . and 0 is a con ex unc ion .2 we ha e 2 n e LX . The e o e, he e is a numbe pn e N such ha Lemma 3 (2 .4)  Jl, .  E  _  0(211 n (x) 11)dx < m=pn n,m We conside he simple unc ion By (2 .3) and (2 .4) 1 n pn gn =  x  . n,m X m=o E n,m J o ~1 I (x) - gn (x)j - j)dx ¿ 2  Jo 0(21 j (x) - n (x) 1 jdx 1 21 (2~~ n (x) - g n (x)dx  2 0(ñ)  + ñ 0 Since « ) -> 0 as 0+ he p oo is inished . Le (BT , T e 1) be a amily o sub-a- ields o  . Suppose ~ .wi h á2 -condi ion . I n con e gs o in LX hen E( n /B T ) con- e gs o E( /B T ) uni o mly in BT , whe e E( ./B Z ) deno es he condi ional expec a ion ela i e o Bz . P oo I may be p o ed , wi h-a sligh , modi ica ion in he a gumen in [lj,pa :g° 122 ha i B is a sub-a- ield o hen p(E(G/B)M  p(g,0) o all g e Lo . Now, gi en e > 0 , le m o be a numbe such ha max  (~ m +2  ,  m )  < c  whe e  K,T  a e he  cons an s  in 2 o  Ko he 0 2 -condi ion . Sinc .e  1 1 n - 1 1 ~ (n -) hen p( n - ,O) -> 0 (n -> -) so he e is a numbe n o such ha i n > n o we ha e p( n- ,O) < m . This implies,by Ko (1 .3) and he i s esul in he p oo , ha 1 JE( n - /B T )110 < e  o  n ~ n o and i is ue o all Te I . 93 . PROOF OF THE THEOREM (1 .2) . Suppose X has he Radon-Nikodym p ope y and le ( n ,B n ) be a bounded ma ingale in LX wi h a((JB n ) _ By lemma 1 and (1 .1), he e is a unc ion in LX such ha n -~ in LX and n = E( /B n ) as i may be seen in [1] . Since he con e gen eo ma ingales in LX implies he con e gen ealmos e e ywhe e, we ob ain,using he con inui y o 0 ha « ll n (x)il) -~ « li (x)l1) a .e . and by Fa ou's Lemma . 1« 11 (x)11)dx  l~  lim  in  j ¢(11 n (x)11 dx  ~=  M The e o e belongs o LX J which coincides wi h LX . We shallp o e ha n --~ in LX . F om Lemma 2, we see ha gi en e > 0 , he e exís s a numbe m o and a sequence o simple unc ions such ha (3 .1)  lis m - il a <  e/2  o  m  >  mo and usingLemma 3 wi h Bn , he e is m 1 in !N such ha (3 .2)  11E( -sm/BnM O .  <e/2  o  m ~ m l and o all n Since  o(U Bn )  = T-  , we can ake he unc ions  s m on measu able se s om lJ B n . I s m = Em i G Bn  o i = 1, . . .,p and in his case E(s m /B n ) = s m o n = n o . The e o e i n !> n o , by (3 .1) and (3 .2) To p o e he con e se we a e going o use he cha ac e- iza ion o he Radon-Nikodym p ope y in e mso ope a o s : Fo e e y T :L 1 [0,1]  ~p X he e exis s a unc ion in Bn he o- ixed gene a ed by he dyadicin e also leng h 2 n  ,  i .e .  Bn =  o(l n  i  '  i  =  0, . . .,2n-1)  whe e Le m be a ixed numbe such ha m > max (mo ,m l ) . le  n o be 'a numbe such ha ji - n110 `  lI - s m 11 0  +  lis m- nil~  = = jj -s m 11 0 + 11E(sm - /B n )jj < e/2 + c/2 = e 1 such ha T(q) _  9P(x) (x)dx o ~o e L*1 [0, 1]  (see [j, page (i3) . Le  T :L~ [0,1] -,oX be a boundedo , pe a o . We conside ii±l In,i = `2n . 2n) 2 n _1 Le  n =   2n T (XI  _)XI  .  I is easy o p o e  ha i=o  n,i n,i E( n+1 /B n )  = n and ob iously  o(U Bn ) = E-  . Since n 2ni 1I n 11  =  Z1  2  1 T(XI  x,  ,  i is clea ha i=o  n,i n,i 11 n (x)jj 11 JITII o all x e [0,1] . Then p( n ,¢) 11 0(11TI1)  o all n, and we can ind a unc ion in LX such ha n -~ in LX . This is equi alen o « li n ID) --  « ji il)  in  L 1 and he e o e he e is a subsequence  ~ (l i n (x) 1 J)  ->  (  1 (x) 11)  ái :e .  Hencé . k (x)~~)  ¡TI 1)  a .e .  and belongs o To conclude he p oo , we mus only p o e ha . 1 (3 .3)  T (s)  _~  s(x) (x)dx  o all simple unc ion on 0 U B n -measu able se s .  -Fi s , .we,Shall .p o e ha (3 .4)  n =  E( /B n ) . I E is a B n -measu able se , 1 n (x)dx =í E n+k (x)dx o k~,l lE and hen i is su icen o p o e ha (x)dx -~~ (x)dx as n-*oo .I is clea om he Holde S E n  '  E inequali y i 11 n  - (x)  -  (x) 11  dx~  11 n -  l1XEU 'F om  (3 .4) ~ J I  (x)dx =~I  n (x)dx = T(XIn,i) n,i  n,i and by linea i y we ob ain (3 .3) and inish he p oo . SPAIN REFERENCES [1] J . DIESTEL and J .J . UHL (1977) . Vec o Measu es . Ame . Ma h . Soc . Ma hema ica l Su eys, 15 . [2]  A .  KUFNER e al .  (1977) .  Func ion spaces .  Noo dho In e na ional Publishing . Rebu ee . 9 de 'no embne dei 1984 Depa amen o de Teo ía de Funciones Facul ad de Ciencias Za agoza