UBC Theses and Dissertations

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UBC Theses and Dissertations

Representation of additive and biadditive nonlinear functionals Aulakh , Pritam Singh 1970

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. REPRESENTATION BIADDITIVE  OP A D D I T I V E AND  NONLINEAR  FUNCTIONALS  by  PRITAM B.A.,  A  M„A.,  THESIS  SINGH AULAKH  Panjab .University,  SUBMITTED  IN  T H E REQUIREMENTS  the  F U L F I L M E N T OF  FOR T H E D E G R E E  MASTER  in  PARTIAL  i 9 6 0 , 1962.  OF  OF  ARTS  Department  of  MATHEMATICS  We a c c e p t t h i s t h e s i s required standard  THE UNIVERSITY  as  conforming  OF B R I T I S H  April  1970  to  COLUMBIA  the  In  presenting  this  an a d v a n c e d  degree  the L i b r a r y  shall  I  f u r t h e r agree  for  scholarly  by h i s of  this  written  thesis at  the U n i v e r s i t y  make  tha  permission  of  Mg^  \±J  Columbia  ;?7°  the  requirements  B r i t i s h Columbia,  I agree  r e f e r e n c e and copying of  this  shall  that  not  copying  or  for  that  study. thesis  by t h e Head o f my D e p a r t m e n t  is understood  f i n a n c i a l gain  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  Date  It  permission.  Department  of  for extensive  p u r p o s e s may be g r a n t e d  for  fulfilment of  it freely available for  representatives. thesis  in p a r t i a l  or  publication  be a l l o w e d w i t h o u t my  Supervisor:  E.  E.  Granirer.  ABSTRACT  In integral  this  representation  and b i a d d i t i v e functions measure  and  space  Also of  atoms  theory  thesis  in  for  < p  < «  functions an  integral  measure  atom-free  we a r e  concerned space of  .  to  The  the  type  nonlinear  Urys.ohn  0  functions for  .an a r b i t r a r y  this  a-finite.  extend  the  presence  the  class  our previous  measure  was  considered  associated  representation  consideration  and  when  measure results  atom-free.  on  here.  L/^-spaces,  taking  studied  0-finite  measurable  and  [11 ]  is  additive  of  transformations  operators.  an  finite  under  c h a r a c t e r i z a t i o n : extends space  spaces  complicates  the  representation is  >  obtaining  nonlinear  on f u n c t i o n  essentially  measurable  space  of  is  called  to  class  p  a measure  ;  a  concerned with  on L ^ - s p a c e s ,  functionals  ,  of  functionals  A class; of 1  we a r e  measurable we  the  describe associated  space where  and the  this  TABLE  OF  CONTENTS  SECTION  1:  INTRODUCTION  SECTION  2:  PRELIMINARIES  SECTION  y.  R E P R E S E N T A T I O N OP A D D I T I V E J U N C T I O N A L S ON T H E V E C T O R S P A C E OP R E A L V A L U E D MEASURABLE FUNCTIONS '  SECTION  4:  REPRESENTATION ON  OF A D D I T I V E  FUNCTIONALS  LP-SPACES  SECTION  5:  E X A M P L E S AND COUNTER E X A M P L E S ON REPRESENTATION OF A D D I T I V E FUNCTIONALS  SECTION  6:  REPRESENTATION  OF B I A D D I T I V E  SECTION  7:  REPRESENTATION  OP N O N L I N E A R  TRANSFORMATIONS  BIBLIOGRAPHY  ON  LP-SPACES  FUNCTIONALS  ACKNOWLEDGEMENTS  I am t h a n k f u l to my s u p e r v i s o r , Dr. E.E. .Granirer f o r i n t r o d u c i n g me to t h i s t o p i c  and f o r h i s generous and  v a l u a b l e a s s i s t a n c e d u r i n g the p r e p a r a t i o n o f t h i s t h e s i s . 1  I would a l s o l i k e to thank Dr. Lee E r l e b a c h f o r h i s n i c e suggestions. I am g r a t e f u l f o r the f i n a n c i a l  to the U n i v e r s i t y o f B r i t i s h Columbia  support.  L a s t , but n o t l e a s t , I. wish to thank Miss S a l l y Bate for typing this  thesis.  SECTION  1  INTRODUCTION  Some on  an  integral  defined have  been  some  of  the  measure  and  [l]  and  in  sufficient  nonlinear  these  shall  An atom theorems  more  general  conditions  additive  functionals  ditions .  an  shall  precise  is  J> o b t a i n s  also  later  we a l s o  integral  notations defined  on.  the  prove  results necessary  representation  various  continuity  of  con-  /  In of  space  Section  under  We  make  and  need  and  [8],  shall  we w i l l  the  functionals  functions  Mizel  terminology  setting  [l]  generalizations.  a measure  of  for  J.  and  some  the  that  Mizel  F r i e d m a n [4];  results  in  part  and  make  2 we g i v e  essential  this  of  J.  measurable  and b y V .  3  and  Section  The of  Chacon  '[J>]'  and V .  non-linear additive  V.  account  some  of  b y R.  theory.  we p r o v e  Martin  real-valued  results  In  D.  of  and M. K a t z e  a unified  A.  spaces  extended  give  of  representation  on v e c t o r  Friedman  of  results  Section  additive  4, we c o n s t r u c t  functional  on  L  the  (^-spaces  integral.representation for  p >  0  and  XT  then the on  we  establish  integral L  -spaces In  some  necessary  representation for  1  Section  <_ p 5>  of  and  sufficient  non-linear  conditions  additive  functionals  <_ <» . we g i v e  some  examples  on  for  integral  r e p r e s e n t a t i o n s and i n S e c t i o n 6 we e s t a b l i s h analogous i n t e g r a l r e p r e s e n t a t i o n s of n o n l i n e a r b i a d d i t i v e f u n c t i o n a l s . p r e c i s e l y we e s t a b l i s h necessary a. b i a d d i t i v e f u n c t i o n a l  F  p r e s c r i b e d subspaces  c  r e p r e s e n t a t i o n where  M^  and s u f f i c i e n t c o n d i t i o n s f o r  d e f i n e d on the Product. ,  and  More  c M  M^  2  to permit  X-^xXg  of  an i n t e g r a l  denote v e c t o r spaces o f  r e a l - v a l u e d e s s e n t i a l l y bounded measurable f u n c t i o n s on  (X,E,ia).  In S e c t i o n 7 we d e s c r i b e i n t e g r a l r e p r e s e n t a t i o n s f o r a c l a s s of n o n l i n e a r f u n c t i o n a l s and n o n l i n e a r  transformations  p on the spaces  L (X)  y  a - f i n i t e measure space considered and  (1 <_ p £ «) (X,S,|i) .  a s s o c i a t e d w i t h an a r b i t r a r y ^ : • i The c l a s s o f f u n c t i o n a l s  here d i f f e r s from those c o n s i d e r e d  [4] and i t s study i s m a i n l y motivated  with n o n l i n e a r i n t e g r a l equations  /  by i t s c l o s e  i n [11].  extends e a r l i e r r e s u l t s i n [ l ] and [ 2 ] ,  i n [l],  [2],  [3]  connection  Our c h a r a c t e r i z a t i o n  SECTION 2 PRELIMINARIES.  We assume here some o f the u s u a l axioms o f s e t theory [ 2 ] , [3]> [4] and [8] as standard r e f e r e n c e s  and we use [ l ] ,  except f o r some n o t a t i o n s and d e f i n i t i o n s which we' g i v e below. Throughout  t h i s t h e s i s we w i l l use the f o l l o w i n g a b b r e v i a t i o n s  "iff"  f o r the phrase  "V"  i n s t e a d of  "For every"  "3"  for  "There  "e"  for  "belongs t o "  "w.l.o.g."  , r  for •  "w.r.t."  for  "s.t."  for  i f and o n l y i f "  exists"  "without l o s s o f g e n e r a l i t y "  I  "with r e s p e c t  "such t h a t "  to"and  .  We w i l l use the f o l l o w i n g c o n v e c t i o n s  2.1  (J>- denotes . the empty s e t .  2.2 2.3  R denotes the r e a l l i n e . R* = R U-{»} U {-»} the extended r e a l  2.4  I = {1,2,3,  2.5  A~B = {x e A  2.6  ^ i^i=l  2.7  ^ i^ieJ x  }  and  ='  x  =  X  1  x £ B} •  t i> 2>--'» n)x  ^ i x  9  x  i € J}  ' ^\  Let  line.  x  where '  be any s e t .  J "denotes an index sc s  •  .  .....  2l  2.8  P(X) = a l l subsets o f  2.9  A  2.10  A~ = X~A .  denotes the c l o s u r e o f  Let  2.11 2.12  f  X .'  f  A ,  be a f u n c t i o n and  A  A <= X .  1  be a s e t . '  1  f [ A ] = {y": y = f ( x )  f o r some  x e A}  [ A ] = {x : y = f ( x )  f o r some  ye  _ 1  A}  U n l e s s otherwise mentioned, i n t h i s s e c t i o n , be an a r b i t r a r y nonempty c l a s s E,P,A,B,....  2.13  w i t h elements  . w i l l be subsets o f  A. nonempty c l a s s  S  X  x,y,....  will  and  X .  o f subsets o f  a r i n g i f EUP , E-F e T, whenever  E,F e E  X  i s called  and i s c a l l e d a  CD  a-ring i f  U E n=l  € S  whenever  E  n  2.14  Definition.  A ring  a l g e b r a i f X € -T,  Remark.  Let of  Let  that c o n t a i n  £  X  i s c a l l e d an X  i s called  7  be an a r b i t r a r y  f a m i l y o f subsets o f  X .  o f a l l a-algebras of subsets  6 ..  §(£) " i s a l s o of subsets o f  X  and a a - r i n g . T, o f subsets o f  denote the i n t e r s e c t i o n X  V n e I .  Y, o f subsets o f  a cr-algebra i f X e S .  2.15  e S  n  a a - a l g e b r a and i s the s m a l l e s t cx-algebra  containing  £ .  2.16  Definition.  If  X  i s a t o p o l o g i c a l space, l e t B(X)  the s m a l l e s t a - a l g e b r a of subsets of open s e t . of  (R(X)  Then the members o f  that c o n t a i n s  every  a r e c a l l e d the; B o r e l se  X .  2.17  Definition.  a f u n c t i o n on if W  X  b  If  M  » c P ( x ) , a set f u n c t i o n  to  R*  w r i t t e n as  V A,B e ii , ADB = «J> , we have i s any f a m i l y of subsets o f  2.19  Definition.  cp : # -*  cp  R*  on  H ' ±£ l  is additi|  u(AUB) = u'(A) + \x(B)  wheri  X .  An a d d i t i v e s e t f u n c t i o n  p : P(X) -  [0,«]  i s c a l l e d an o u t e r measure i f the f o l l o w i n g h o l d : . (i) (ii)  and  |a (i) (ii)  n (<|>)- = 0  and  u(E) <  n(E-t)  E iel  I whenever  E  c  1  U E. c X , iel 1  i s c a l l e d a measure i f f |i((j>) = 0 n(E)  and  S |i(E. ) iel . 1  E_ j  n  Ej = >  2.20 D e f i n i t i o n .  , i  whenever /  4= j  E =  U E, .<=X lei  .  We w i l l use caratheodory outer measure \i  f o r which we have: A set  A c j  |i(E) = |i(ETlA) +  and  1  i s immeasurable i f f V . E c X ,  ^(EnA~)  .  2.21 D e f i n i t i o n . by  M  If  the s e t  |i i s a measure on  (A : A c X  2.22 D e f i n i t i o n .  and  A  I f f o r every  X , then we denote^  i s (^-measurable].  A £ M  , |i(A) < «  then  u  is  c a l l e d a f i n i t e measure. '  2.23 D e f i n i t i o n . { ^ ^ 1  sequence  ii(A  n  )  .<  M  v  co  n  o  e  on  X  f  I  2.24 D e f i n i t i o n .  i s called a-finite i f V . A e M s e f c s  l  n  ,j  s  u  c  h  X  nonempty c l a s s u  Let  a  t  A  c  ^ -^n  a  n  A c X  then the measure  d  \  take  |a by  H (E)  of measurable  A .  X = (X,S,|i)  subsets o f  X  =.u(AnE^  A  as a measure  Is a nonempty c l a s s o f elements and  E  is a  which i s a a - a l g e b r a  i s a measure on. ( X , E ) .  2.25 D e f i n i t i o n .  Let  (X,5Vu)  i s c a l l e d an atom i f u(A) ={= 0 either  h  J  i s c a l l e d the, r e s t r i c t i o n of  space where  t  .  In f u t u r e we w i l l  and  M  , 3 a,  u(B) = 0  or  A space every subset of "N 2.26 Lemma.  Let  be a measure space. and i f B e  E , BCA  A e S then|  u(B") = u(A) .  ( X , E , u )  X  /  w i l l be c a l l e d an atomic space i f  t h a t belongs to  (X,S,|i)  S  i s an atom.  be a f i n i t e measure space and  E e E  7.  0 < n(E) < »  s.t. neither  sets i s an atom, then  E  E  n o r any o f i t s u-measurable sub-  c o n t a i n s subsets  of a r b i t r a r i l y  small p o s i t i v e measure. Proof:  Since  E  F c E ,' F € S Hence one o f  does n o t c o n t a i n any atoms, there  s . t . E = P U (E-F)  F  and  u(F^) _< •||i(E) .  and  2.27  w(P') > 0 , n ( E - F ) > 0  E-F , c a l l i t F.^ , s a t i s f i e s  Now s i n c e  F^  i s n o t an atom, c o n t i n u i n g  w i t h the above method fo•decomposition, subsets  exists  i t f o l l o w s that  E has  o f a r b i t r a r i l y s m a l l p o s i t i v e measure.  Lemma.  If  (X,E,|_i)  i s a f i n i t e measure space then  there 00  i s a countable  family  then  i s atom f r e e and  X  C = X~A  {A.}  o f atoms'of X = AUC  X  s.t. i f A =  U A.  i s a decomposition o f  i n t o atomic and atom-free p a r t s .  Proof:  Since  X  i s a f i n i t e measure space,  i s o f f i n i t e measure.  By i d e n t i f y i n g  any atom  A c X  the u-almost equal atoms  we get that t h e . d i f f e r e n t atoms a r e d i s j o i n t . number o f d i f f e r e n t atoms c o n t a i n e d i n . X  So the t o t a l  i s a t most  countable  00  say C  An,A_,... .  Let A =  U A.  and  i s atom-free and the decomposition  2.28 D e f i n i t i o n .  x""= (X,S,u)  v a l u e p r o p e r t y "If  V  0 <_ a <_ |a(S) 3  S e £  C = X~A  clearly  i s n-almost unique.  has the s t r o n g i n t e r d e d i a t e  and V r e a l number  A e E , A cr S  then  s.t.  \x{k)  a ,  = a .  .>  2.29 Theorem.  A measure space  X  has  the s t r o n g  Intermediate  v a l u e p r o p e r t y I f f I t i s atom f r e e . [For Proof, see "Set F u n c t i o n s " , by Hahn and Chapter 1,  §5.6.']  •'  (X,E,u).  Definition.  2.30  has  p r o p e r t y i f f V r e a l number A c X ,  2.31  u(A)  s.t.  X .  0 <_ a <_ |i(X) , 3  |  A e E ,  = a .  Let  Then  the weak i n t e r m e d i a t e v a l u e  a ,  Theorem. (Z. N e h a r i ) .  measure space. of  Rosenthal  Let  —  m  2 —  has  ( X , S , n )  (X,E,|-i) ***  ^  be a countable ^  e  e  m  e  a  s  u  r  e  s  discrete  °? atoms  the weak i n t e r m e d i a t e v a l u e p r o p e r t y  CO  i f f • m_  <  !  E  m,  ,  ^ ~k=n+l * 2.32  f  .  Definition.  : X - R*  finite :  set  s e X  x^  2 . 3 5 Lemma.  •  c e R*  = {Q  j  then  , {x : f (x). < c} e E  X f E  i s  c  a  l  l  e  d  ^  /  the c h a r a c t e r -  '  x  i s E  A  measurable i f f E e E . function  c^,c ,...,c 2  ,  !  be a measure space,  function.  Definition.  (s  (X,E,n)  i s measurable i f f V  2.34 Theorem.  •.  _  Let  2.33 D e f i n i t i o n , / istic  = 1,2,3,.'..  n  fCs)  =,c^}  Let  f  n  of  f  values is  e E  : X -*  : X  R*  R*  w h i c h has  and-''for w h i c h  called  a  simple  only  f~ (c ) 1  i  a = j.  function.  be measurable, then there e x i s t  a sequence  {f }  o f simple f u n c t i o n s s . t .  f  f .  If  f>  n+1 \i V n [For p r o o f see T a y l o r  2.36  Definition.  ]  Two r e a l v a l u e d f u n c t i o n s  measurable i f V B o r e l s e t  S  on  (-«.,'») , f  f , g are equi- 1  (S)  and  g (S) - 1  are measurable and have equal measure.  •2.37 D e f i n i t i o n . Let  f  Let  (X,E,|j)  be a f i n i t e measure  be a bounded measurable f u n c t i o n w i t h  sup f ( x ) = M . x  §(A)  space.  i n f f(x) = m , x  n £ y  S(A)  mu(X) <_ §(A) </ S(A) <_ M |i(X) .  Then  Q  sup §(A) = i n f S(A) = f f"d(j = f-f"du . A  A  ;  X  Before we f i n i s h t h i s s e c t i o n , we prove some lemmas N  which we s h a l l need l a t e r on i n Chapter 5 and f o r p r o o f s we shall follow [1].  j  1 0  2.38  Lemma.  L e t m^,m,m^,... 00  numbers s . t . n i > 2 . Let  ,n=l,2,...  Em, k=n+l  1 1  K  o'^,c' ,...  and d ^ d ^ d ^ , . . .  2  - 1 , 0  of r e a l numbers having v a l u e s 00  be a sequence o f p o s i t i v e  2  be two sequences  or 1 .  Then  |.  CO  E cm i=l 1  1  '  = E d.m. i=l 1  Proof:  1  1  Let  i  e N-j_  i  eN  2  •  2  We c l a i m that  =N  = < | > ... '  2  " i . e . N^UN  s m a l l e s t i n t e g e r i n . J^UNg . not  s.t.  c  2  i  _ d  > 0 for  ^  ;  4 =< } >•  Thus  j_  dj_~Cj_ > 0 f o r  be the s e t o f i n t e g e r s s . t .  Suppose n o t .  1  1  be the s e t o f Integers  and N  ]  c. = d • , " i = 1 , 2 , . . .  only i f  Let I  e^  or i  be the  Q  Q  eN  2  but  t o both. Suppose  i  eN  n  .  Now i f  i e N,  and j € N  ,  ;  0  f  then .. c - d . Thus  E' (c.-d.)m. > m. jcN^ 1  i.e.  2" o r 1 .  and d .-c . .are e i t h e r  1  1  _  1  Q  E (c.-d.)m. >  > 2 E m. > 2 E (d.-c.)m. j=i +l - J N . J  J  o  €  J  J  2  E (d.-c/)m. . CO  But by hypothesis which i m p l i e s that  we have that .  1  1  E (c.-d.)m. = E (d.-c.)m. ieN^ JeN 1  1  1  J  2  c o n t r a d i c t i o n t o N^UN ={=<(>/.' 2  Hence.,  00  E c m . = E d.m. i=l i=l  =N  g  =,<j> .  J  J  1  1  which i s a  11.  2.39  Lemma.  s.t.  for  Let  tn^}'^'  e  sequence  a  o  p o s i t i v e numbers  f  09  n > n >_ 1 m. • ~ ° • •  > 2 E m. , i=n+l  3  .  Let  V  1  space of a l l r e a l sequences  s : s^,s ,...  be the v e c t o r |  such that  2  eo  E s.m. i=l 1  < « .  Let  S  be the subspace of  c o n s i s t i n g of 1  those sequences E m. iel  =  1  s  s.t.  E iel  s.m. 1  =  1  E s.m. jeJ J  whenever  J  E m. . jeJ J  Then the a l g e b r a i c dimension of Proof: the  V  1  Let  H = {(I,J),  S  is- i n f i n i t e .  IflJ = <j> , I and J are subsets of  integers} .  (a)  s. t.  E m. '= E m. . lei ' jeJ 1  For  J  a p o s i t i v e number •f  n  o  s.t.  l < n <n, — o —  (a) can ; I  be w r i t t e n u n i q u e l y as n• O  (b)  E i=l •or 1 , 1 =  . 8 0  c.m. 1  = E i=n +1 o 1,2,... .  1  1  c.m. 1  where  •  c.  has the v a l u e  -1,0  The r i g h t hand s i d e of (b) i s u n i q u e l y determined, i f it  i s determined, when t h e l e f t hand s i d e i s g i v e n . Thus there are a f i n i t e number o f r e l a t i o n s  are that (c)  (b) which  v a l i d as i s the number of r e l a t i o n s of (a) which proves H  is finfte. Consider the system of equations " •• .  E x.m. = E x .m. iel • jeJ 3  J  12.  ( I , J ) € II . where the unknown sequence The system  (c) i s f i n i t e  (c) has e x a c t l y  k  of (c)  k+1  H  K  i s a member of  is finite. k  can be zero a l s o .  i s a subset of the i n t e g e r s  whose support i s a subset of  i n t e g e r s each, then  con-  solution  K  i s a sequence of d i s j o i n t S^- ,S„. ,.... l *2  S .  So suppose that  members then there i s a nonzero  Now. i f ' £K.j^i>i k+1  since  equations where  In any case i f taining exactly  x : x^x^,...  subsets of  i s an i n f i n i t e  f a m i l y of  A  independent  s o l u t i o n s ; o f (c) . i '• •  Hence  •'  I'•  /  X  dim S = » . '  .  SECTION  3  REPRESENTATION OF ADDITIVE FUNCTIONALS ON THE VECTOR SPACES OF REAL-VALUED MEASURABLE FUNCTIONS  Let algebra  3.1  (XjEjii)  o f subsets of  Definition.  Let  be a measurable space w i t h X .  f , g : X - R* , then we say  a.e. i f [x : f ( x ) ={= g(x)} e E  3.2  Definition.  bounded  A function  Definition.  Is said  constant  f  to be e s s e n t i a l l y C s.t.  i s bounded  A sequence  f i n i t e a.e a r e s a i d  o  f  a.e .  f u n c  tions  t o converge a.e to a f u n c t i o n  i s f i n i t e a.e. i f E = [x : f ( x ) / f ( x ) } c E n  3.4  Definition.  A sequence  f (x). = g(x)'  and has u-measure zero. .  f  i fa a finite, positive  u{x : | f ( x ) | > C} = 0 ' i . e .  3-3  a a-  E  •(^ ^n-l  and  which a r e f ( x ) which u (E) = 0  o f measurable  n  functions  i s ^ s a i d "to converge i n measure to a measurable f u n c t i o n V  6 > 0  we w r i t e  Example;  we have f  u - f .  l i m u ( t x e X : | f ( x ) - f (x) | >_ 6})| = 0 and n-««' '  There exist x  n  ;  sequences o f f u n c t i o n s  measure and do not converge a.e. all  f |jLfj  Let  Lebesgue measurable subsets o f  that converge i n  X = [0,1] and  X .  E  be  F o r each i n t e g e r  n € I , define  = Xr ;>  f_  j+li  where ri = 2  Then  \{{x : | f ( x ) | > 6} ) <_  Thus  f  - 0  n  H 0 .  sequence  + j , 0 < J < 2  k  as  n -  k  V.- 6 > 0 '. 1  But on the other hand, i f x e [0,1] , the  (f (x)} n  converges nowhere on  [0,1] .  Throughout t h i s s e c t i o n , we denote by space of r e a l - v a l u e d measurable f u n c t i o n s on  M  the v e c t o r  (X,E,u)  where  two f u n c t i o n s a r e i d e n t i c a l when they a r e equal a.e.  3.6 M  Definition.  A real-valued function  F  on a subspace o f  i s c a l l e d an a d d i t i v e f u n c t i o n a l i f (i)  F(x+y) = F ( x ) + F ( y )  i i  i  f o r x,y e M s . t .  •  •.  •  •  u{supp x Tl supp y} = 0 l  ( i i ) F(.x) =-F(y) i.e.  i f x,y  i 1  a r e equimeasurable f u n c t i o n s  i f f o r every B o r e l s e t B  in  R ,  u(x~ (B)) « n(y- (B)) . 1  1  In t h i s s e c t i o n we w i l l c o n s t r u c t representation  of a nonlinear  on d i f f e r e n t subspaces o f d i t i o n s on atom f r e e .  F  M  an i n t e g r a l  additive functional under v a r i o u s  ••  -  defined  c o n t i n u i t y con-  when the u n d e r l y i n g measure"space •  F  (X,E,u) i s i j  15.  Let  B = B(X,E) = L (u) = { a l l e s s e n t i a l l y bounded 00  r e a l - v a l u e d measurable f u n c t i o n s } .  3.7  Definition.  A set function e > 0  continuous i f given u(E)  <  |cp(E) I  6  For given  x  Theorem.  for  €  -» x  n  satisfies  F(x) .  f  R  : R-  s.t. for E e E.,  6 > 0  x  n  -> x  boundedly  and there  be a f i n i t e atom-free measure space  (,X,E,|-t)  I f an a d d i t i v e f u n c t i o n a l (l) x  Then there  s . t . f(0) = 0  -• x  n  Let  boundedly  e B  a.e i m p l i e s function  ;  and V  x e B  * •  .  ,  • " -\  (*)  •/ be the constant f u n c t i o n ' a  for  !  CI  .  a  C  F : B - R  e x i s t s a unique. continuous  X (a)  instead  e x i s t s a p o s i t i v e constant  F(x) = J* (f.x)dn.  Proof:  a.e"  |x| <_ c .  the c o n d i t i o n :  F(x ) n  and  Let  n  a.e  |j(X) = ' {= 0 .  which  u-absolutely  .  [1] and-we w i l l use  s . t . |x^| <_ c  3.8  <  3  cp" i s s a i d to be  the f o l l o w i n g theorem, we w i l l ; f o l l o w the methods  of s a y i n g t h a t c  ;  e  (-eojco)  .  f ( a ) =11^1 . U(X)  Define ( r e a l s ) then t i n u i t y of  C„ - C n x  F ,  x  Thus f  ,  a € (-»,») .  boundedly, so  i s continuous.  - a n  F(C ) - F ( C ) ^ .  f ( a ) - f(a) i . e . n  If ' a  by con-  10  Now  supp C  supp x n supp C  If  Thus i f  F(C ) = 0  = F(C  there i s an  f  s i n c e f o r a =(= 0  then  that  + x) = F ( C ) + F ( x )  Q  Q  and thus  Q  x e B  v  = <j> which i m p l i e s  Q  F(x)  i.e.  = <p .  Q  f(0) = 0 .  s a t i s f y i n g t h i s theorem then i t i s unique F(C ) = J* . f « C  we have  du = f ( a ) - u ( X ) .  a  X  Now i t remains to show that  F(x)  If  S € Z ,  =. I" f ( x ( t ) ) d ( u t ) . X  f o r fixed  cp(S)  cp  For 9  (  F(ax )  = cp (S) =  s  a  S l  cp i s a d d i t i v e on i f S S 13  2  e E  US ) = F(a. 2  = F(a,x  s  1  S .  and  X S i U S 2  ) =  CP(S-L)  = <p  F(a.x  ) + F(a,y  q  )  Si  ^2 •  +  cp(S ) 2  .  S e E  and s a t i s f i e s :  ^  S^Sp  X =  x e B  a e (-»,») , d e f i n e  i s f i n i t e valued set f u n c t i o n V  (b)  V  '  +  (  then  a.^J supp a v  q  fl supp ayb  -  2  = (p)-  IT.  (c)  cp  If  is  S  e  • cp(Sn)  £  F  =  countably s.t.  (a,x  s  additive  (<d)  cp  Since |i(S)  0  n  .it  J(^T  =  cp(S)  r,s  In  fact  Since  U  =  ,  ,  is  f  =  0  to  X  then  show  is  =  0  ~wsr  since  that  boundedly that  so  cp  i is  count-  that  cp(S)  =0  whenever  cp(S)  aXg.  and  = F(axs)  C  =  (  F  C  are  o  D  )  =  equi0  .  ' i  defined  s  S  function  p  (0,u(X)]  <  s  S,R  e  F(axR)  .  ,  =  =  =-u^r = w  with  +  n ( S ) ± 0  and.  s.t.  rp(r')  =  ( s + r ).p( s + r )  C(s)  =  {S  e  £  :  u(S)  =  s}  .  =j= cf> •  then  a,Xg  and  e,x  are  R  equimeasurabl  Hence  a  S  £  (0,(i(X)]  s'p(s)  and  '< V  F  e  <  C(s)  C(s)  S  .  <_ u ( X )  atom-free,  V  on  and  e  i n d e p e n d e n t "of  ^  0  -  n implies  which  suffices  ( )  q>(S) =  it  follows  r+s  let"  F(aXg)  mr  a,x„"  continuous.  p(n(S))u(S)  If so  = 0  .  £  real-valued  for  )  P(ayR)  ,„s  a  F(C  u(S)  m  is  then  E  .  measurable,  ()  )  < •  on  •  on  If  e  I; cp  u-absolutely  |i(X) =  S .  ably  is  additive  e  C(s)  cp(R)  5w ,  'M > , \ ' w  and  h  l  c  h  i  m  P  ' cp(S) l  i  e  s  t  h  a  t  fEtfx. .  hence  = p-(s) '  i  S € C ( s )  w h i c h  18.  proves  that  ^js j  a  r  e  a  p  on  e q u a l i t y , we have that  X  l ~  v  a  l  u  e  f u n c t i o n of  d  j  (0,u(X)] . To prove the l a s t free,  there i s an  S e E  0 < s+r <_ |i(X) ,  with  u(S) = s .  Since  r <_ |j(X) - s = | i ( X - S)- .  space i s atom-free,  3  Re  Again s i n c e the u(R) = r  and  cp(SUR) = cp(S) + cp(R)  and \  S , RcrX-Ss.t.  since  SflR = <J> , i t f o l l o w s that  since  cp(S) = p(|i(S) )ji(S)  and  i s atom-  u(RUS) = |i(R) + i±(S) = r + s '< i  we have  p(u(SUR))u(SUR)  (r + s ) p ( r + s) = s p ( s ) p  (f)  numbers i n e S  {S l n  + rp(r)  p(|i(R) )u(R)  , and  .  vi(S ). = s 1  <_ s  2  with  s n  and  1  "*  s  •  Since  Sg  S , S  g  X c s  n  e E  with  3  i s atom-free 1  with  «  .. .' .  1  T h e r e f o r e there e x i s t s ' a monotonic sequence S  so  be a monotonic d e c r e a s i n g sequence of r e a l  (0,u(X)]  with = s  2  +  p(u(S))u(S)  i s continuous.  For t h i s l e t  n(S )  =  u(S ) n  = s  and  n  ... S  n  c S  ^ c ... c ^  S  n  ,  .  00  Le.t  S = " n S_ . n=l '  Since  u(S, ) < «  u(S) = l i m |a(S )r = l i m s n n additive,  X  so  l  l  m  p  (  s  n  )  i t f o l l o w s that  i  1  n  = s > 0  and  since  cp  i s countably! •! j  9 ( S ) - cp(S) n  -i*.f^  .  ..-  I i  p(.) .  19.  A s i m i l a r argument shows t h a t i n c r e a s e s to  s .  Thus  (g).  p  Let  s € (0,|i(X)] .  p  Since  X  and so -on, 3  X  d i s j o i n t and H  ±  l  c X _  i  i  with  1  S^Sj  u(S _) = ^ ( X _ j  ~ ^T s  s  i  = H >  = (J) i f  1  X  \i(X^)  c X  1  n  =  ±  =1  e  + .... n times )p(|- + £ +|'+  rS  s  p(^) = P(|[)  s,  since  s ^ 0  we have  (  H  s  )  (n ^ 0 )  = n  p  (n  }  where  m  and  n  are non-  then  +  n  p  (  H  }  +  ' ••  +  n  p  (  E  }  '  m  t  e  r  m  s  X  p(^ s) that  s  and  S  ... + 1)  , «/' \  s,  ^ s e (0,n(X)]  negative integers,  get  we have  /  I f now  .  =  and  +|  p  .  ±  i € I  ar;ei  S. € S  i  sp(s) = (£  sp(s) = n. . ^  .  then  ( r + s ) p ( r + s) = r p ( r ) + s p ( s )  s  — -s n  v  But s i n c e  n  n-1  u(Xv) = ^ 1'  i ± j .  P(s) = P(|) •  i.e.  n  c X , s.t.  -Pi  - n(X )  for  «/ v, „/' \,  .  s  s .  = nU^)  - X )  X  s.t.  o  S. = X. , - X. 1 i-1 i  ' *  i  s  i.e.  when  ;  i s atom-free 3  Furthermore t h e r e i s , •  =  ) "* P ( ) S  N  i s constant.  a(X ) = s . ^ o  1  S  i s continuous.  I t f o l l o w s that i f  I  P(  p  = p(^-) =  p(s)  is.constant.  ,  -and s i n c e  p  is. continuous,  we  20.  p = P  (h)  .  Since p = p (i)  a  V • a e (-»,«)  = f(a)  a  P(ax )  '  x  f(a) =  ^  = f(a)  Y  I f x e. B  )  • >  P  ( ^ (  a  ) )  x  =  P  a  ( u (  , s e E .  ) )  s  We have  a € (-«,«) .  and i f x  i s a simple f u n c t i o n then  P(X) = ; f(x(t))d ( t ) X u  Let  S-^aSg,. .., S  be a p a r t i t i o n  n  2. 2 '  ' ' >n  ^  these subsets.  Thus  x = E x^Xg  sets and  x  iX  x  s  e  ^  n e va  of  -"-  ues  X w  n  i  i n t o measurable subc  n  and s i n c e  mutually d i s j o i n t . s u p p o r t s , this implies  P(x)  =  E F(x x  i=l  s  )  1  =  E %  i=l  i  x  assumes on  x  (S ) V  have  that  P  .S  ntS^O  1  X-s.Xg  i  x  ^(S  ))iJ(S  1  )  1  = E f ( x ) n ( S ) = J* f ( x ( t ) ) d |i(t). . . i  i  Thus the theorem I s t r u e f o r P '  -  /  d e f i n e d on simple  '  functions. (j)  Now V  Since V  P(x) '= f f ( x ( t ) ) d n ( t ) . X  x e B , there e x i s t s a sequence  functions tinuous  x e B , we show that  F  s.t. x (  x n  )  F  theorem, we have  n  -» x  ( .) • x  ( 5 x  n  boundedly a.e. and s i n c e  o f simple F  l i s con- '  Thus by' Lebesgue dominated convergence  F(x)'"'= J f ( x ( t ) ) d n ( t ) = Jf ('x(t) )d|i(t) . n  I  Q.E.D.  21,  From now  onward we  will'.prove the n e c e s s a r y  s u f f i c i e n t c o n d i t i o n s that an a d d i t i v e f u n c t i o n a l on a p r e s c r i b e d subspace the form,  B c M  F(x) = f (f«x)d|ji  V  and  F  defined  permits a representation x e.X  where  f : R - R  of is  X uniquely  determined by  F .  W e w i l l a s s o c i a t e these theorems  with f i n i t e or a - f i n i t e atom-free measure space we  (X,E,n)  and  s h a l l f o l l o w v e r y c l o s e l y [2] f o r p r o o f s .  3.9  Definition.  every  Let  (X,E,u)  be  the measure space.  For  denote the t o t a l v a r i a t i o n of n on E by n d e f i n e d as v..(E) = sup £ |n(E.)| where the supremum 1=1  E e £ , 'we  v (E) , u  1  •  •  »  is  taken over a l l f i n i t e  £  with  E^ 'c E .  [x  H  v (X) < Co  and  E e £  v (E) < c .  )i  if  sequences  (E^)  of d i s j o i n t  sets i n  i s s a i d to be of bounded v a r i a t i o n i f ; ;  i s s a i d to be of bounded v a r i a t i o n on a set ' •!•  ' Let  i; f , g : X .-» R , the r e l a t i o n  f u n c t i o n " i s an equivalence c l a s s of f u n c t i o n s from and l e t 3.10  P[X]  relation. /  X -• R  [f]  denote to  the f  = P(X,£,|j) . denote the set of a l l such sets  Definition.  c l o s u r e of simple TM(X)  Let  is a null  which are e q u i v a l e n t  Let  X  be  any  t o t a l l y measurable f u n c t i o n s on  by  "f-g  • •,.  functions i n  t o p o l o g i c a l space. X  F(X)  [f ] .  The  are the f u n c t i o n s i n the and  we  s h a l l denote them  22.  It that  i f  x  .tinuous  is  proved  and  functions  a  Theorem.  space  and  Then  in  III.2.11  x^  -» x  (2')  P(x)  =  an  function  that  the  a  V  of  finite  x  f  III.2.12  Is  a  con-  measurable measurable  P(X)  .'  atom-free on  B  measure  = L  (n)  .  equivalent:  i n measure  (f«x)dn  i f  totally  functional  are  i n Lemma  and  totally  subspace  additive  boundedly  Schwartz  Is  be  conditions  f  and  fox  linear  (X,E,^)  be  following  (2)  then  closed  P  Dunford  measurable  R  Let  let  the  on  also  form  in  totally  function  function  3.11  is  e  =^  F  (  (  x  n  ^  "*  F  (x)  B  a  (*)  n  d  .  X  with  the  that  f  representing is  function  continuous  and  Proof:  Let  -» x  a  measure  space,  finite  therefore •f(o)  =  by  a  a  =  f  •X  suppose  0 . Define  conditions  since  unique  so  F  continuous  (  x  n  (X,E,|i)  is  )  (x)  >  f  ,;  "*  F  function  /  Conversely, =  Then  -+ x • i n m e a s u r e  T h e p r e m 3.8,  the  0 .  a.e.  s.t.  0  satisfying  f(o)==  boundedly  F(x)  f(o.)  f  f.x  f  d^  :  V  R.'-» R  x  is  €  B  continuous  and  . ' F  on  B = L ^ u )  by  F(x)  = J  (f.x)d(i X  .  Since  f(o) = 0 ,  we have  c l a i m that:  F ( x ) -» F ( x ) n  Y  Let  x n  .'  follows f°x  n  x  -» x  n  n  r  t o t a l l y measurable and  e  that  -* f«x  B  and  f»x  i n measure and  s.t. x  n  -• x  in  f o r some p o s i t i v e constant  n  a  implies  x ,x e B .  |x | <_ c , |x| _< c  x  boundedly i n measure  be a sequence i n  measure and Since  s.t.. u{supp x n supp y} = 0  x,y e B  F ( x + y ) = F(x) + F ( y ) . • . Now  that  Y  f  c  i s continuous, i t  are t o t a l l y measurable and f°  > f°x  x n  are bounded.  Therefc re  by Lebesgue'dominated convergence theorem l i m F ( x ) = l i m J ( f o x ( t ) ) d u ( . t ) = J f . x ( t ) d u ( t ) = F(x) . n n n  n  Now we'take the case of a.e. convergence which i s ho :  n e c e s s a r i l y bounded and convergence i n measure.  3.12 Theorem. space and. F  Suppose  (X,E,ti)  i s a f i n i t e atom-free measure  i s >an a d d i t i v e f u n c t i o n a l on  M .  Then the  f o l l o w i n g conditions' are e q u i v a l e n t :  (3)  x  n  x  a.e. =25?  F(x)  V  x e B , where  s  f  F ^ ) ' -»F(x)  and  = f f o x d(i X  'satisfies  (a)  f  i s continuous and  (b)  Range ( f ) i s bounded.  (*)  the c o n d i t i o n s  f(o) = 0  and  24  Proof:  ;  Suppose  satisfies  F  and s i n c e f o r any sequence and  |x^| _< c  V  n  t } >]_ x  n  i  n  CO  L  n  and  which  F, = P/L ( u ) . , L ( u ) c. M  the given c o n d i t i o n and l e t  .1,  a.e.  M  i s an a d d i t i v e f u n c t i o n a l on  (> )  with • x  J  c o  |x| <_.c  CO  where  c  -• x  i s some  p o s i t i v e c o n s t a n t , we have by Theorem 3.8 that there e x i s t s a unique continuous f u n c t i o n P ( x ) = P(x) = J. (f.x)d|i  f : R -* R V  1  s.t. f ( o ) = 0  and  x e Lj\i) .  X  To prove ( b ) , suppose that {r 3  There e x i s t s a sequence  in  n  range ( f )  R  i s unbounded.  s . t . |r f -» »  and  !  1 _< | f ( r )| • » .  Since the measure space i s atom-free, there  e x i s t s a sequence \ u(x)  {B } ,  Let  n  x^^ = r^Xg  B  , *  B„ n 1 & . n, x. ' -» 0  and s i n c e  e E  n  and  B n  +  1  F(x )  = J(f ox )d  F(x )  / F(o) = 0 • which i s a c o n t r a d i c t i o n .  n  n  n  U  = f(r ) , n  U  f  i s bounded.  J <f> « t . s  n  ^  } }  xn € L o o  x  x /  n  eM  , - we ' have 7  ,, = + n(X) + 0 Hence  range ( f )  /  Now l e t sequence  (  B  i s measurable and hence  n  a.e. and s i n c e  v  £  (s } n  x e M be any f u n c t i o n .  o f simple f u n c t i o n s  There e x i s t s a  s . t . s^ -»,x  a.e. and  i s continuous, f°s^ -• f»x a.e.'where Ifos 1 .. ' n n |fox[ _< c f o r some p o s i t i v e constant c . Thus by Lebesgue  since  f  1  convergence theorem,  °* d|jt = l i m J*(fos )dn n  Conversely, l e t f  = l i m F(s ) = F(x) n  s a t i s f y - ( a ) and ( b ) .  n  Define  25.  F : M - R F and  F(x) = J*(f.x)d|j . .  by  e x i s t s and i s r e a l - v a l u e d  range ( f )  F  i s bounded;  because  and s i n c e  f  i s continuous  f ( o ) = 0 , we have  that  i s additive. Let  x e M .  be a sequence s . t . x  n  Since  f o}^ -» f»x that  {x } e M f  i s continuous,  ^  o X n  n  -• x  > f°x e M  a.e., and  a.e., by Dominated Convergence Theorem i t . f o l l o w s  F ( x ) - F(x) .  Q.E.D.  n  Since convergence a.e. i n a f i n i t e measure space i m p l i convergence i n measure, we have  3-13 C o r o l l a r y . (X,£,|i)  Let  F  be an a d d i t i v e f u n c t i o n a l on  a f i n i t e atom-free measure space.  the c o n t i n u i t y c o n d i t i o n F(x ) n  -» F(x)  (4):  n  ~*  i f f ('*) holds w i t h  f  x  x  Then  i n measure satisfying  F  M  satisfies  implies' (a) and (b) i n  Theorem 3.12. (The  with  • Proof i s s i m i l a r /to the above theorem. )  I f the space  (X,E,|i)  i s a - f i n i t e and  |i(X) = »  then the above theorems become:  3.14 Theorem.  x  The a d d i t i v e f u n c t i o n a l  F on . •  s a t i s f i e s c o n d i t i o n . (3) i n 3.12 or c o n d i t i o n respectively i f f  F s 0 .  L (a) .  or  09  (4) i n 3.13  M  "  26.  Proof:  Case ( i ) . (X,E,u)  Since A  1  U A  2  =  A  Let  be  i s atom-free 3 ( i)  a n d  r  A  = ^( 2^ = A  u A  If  A e S  00  \ i ( A ) = \i(X - A ) = « .  s.t.  i* 2 A  €  * ^DAg  s  = (j) s . t .  •  i s any constant  then  rx  fl  >  r  Xn l  and '  A  are equimeasurable.  Since  rx'  = rx» , ,  fl  A  |i{supp rX  H supp rx«  fl  l  R  that  2  P(r.x'  A  A  )+  and which i m p l i e s that  A-^Ag  , A  1  A-^UAg  we have, by  F(rX  A  n  + rX  A  A  2  A-^  and i  the a d d i t i v i t y of  F ,  ) = 2 F ( r X ) ' (by e q u i m e a s u r a b i l i t y ) A  F(rX ) = 0 . A  Assume t h a t f o r any 3 A  A e E , p.(A) < »  )  there e x i s t s ,  which t o g e t h e r w i t h t h e i r complements are of  2  i n f i n i t e measure and of  = rx  A  A  F(rX ) =  Case ( i i ) .  } = 0  A  r ' . 2  F , we have  A = A^~A  F(rX ) = 0 A  a simple f u n c t i o n , then  2  , then again by the a d d i t i v i t y  when  |i(A) < <» /  F(x) = 0 .  Now  v a l u e d f u n c t i o n f o r which each v a l u e has of i n f i n i t e measure, then as above we  if  Hence i f X  is a  support and  get that  x  v  is  countable complement  F(x) = 0 .  N  I'  SECTION 4 REPRESENTATION  4.1  OF ADDITIVE FUNCTIONALS  Definition.  For  measurable f u n c t i o n s  ON L  1 <_ p < co ,  L  SPACES  0 < p < » , L (X) = { a l l real-valued P  f  on  X  = ( f |f| du X P  s . t . || f || P  For  P  i s a Banach space but t h i s i s no  p  < »] . longer  true f o r 0 < p < 1 , f o r the t r i a n g l e i n e q u a l i t y does not j hold f o r 0 < p < 1 .  4.2  Proposition.  (a) (b)  But i n s t e a d we have:  Let. 0 < p < 1  ||f g|| .< +  p  2  (|| f | |  P  i s a metric  :  where  F i r s t l y we show  (1)  " •2 " (1+x )-_> ( l + x ) 1  p  Let and i.e. and  f'(x) = 2 ~ .p x P  1  P r l  p  (l+x)  - p(l +x) ~ p  1  P  =0  r=5> 2x = 1 + x  x = 1 f"(x) = 2 " . p ( p - l ) x " ? p  1  - p(p-l)(l+x) -  p  f"(l) = p.(p-l)[2 p  Hence  1  a.e.  for 0 < p < 1  p  1  p  2  --2 - ] < 0 p  e  n  p  d ( f , g ) = 0 =^ f = g  f(x) = • • 2 " ( l + x ' ) = p  h  + ||g|| ) . p  Proof:  P  t  f , g e X , d(f,g) = J | f - g | d i i  I f f o r any two f u n c t i o n s then d  p  and l e t ..X = L [0,1]  2  x =: 1 . 'is a maximum p o i n t f o r f ( x ) and  f(l) = 0 .  »  28.  2  Thus we have  P _ 1  (2)  We show that  Let  g(x) = 2  and  'g'(X) = 2  ( l + x ) >_ ( l + x ) p  .  p  2(—-l) 2 (l+x) > (l+x ) ;  p  for.0 < p <  1 / / p  1..  (l+x) - (l+x )" p  p  1 -1  -i.e.  2  (  p  '  4 = (l+x )x" p  l  -  )  i (l+x ) p  i.e. 4 = x  p  . -  1  Also' g'(x) = _ [ ( I . i ) ( i  and Hence  g'^I) ^)  +  x  + 1  _ p  = 0 x = (i)  .xP" •'•+ ( p - l ) x - ( l + x ) j p  2  p  ;  p  1  p  < |f(x)|  p  (2)  and we have  |f(x).| ^ 0, , then  Assume that < |f(x)|  |]  p  .  Now f o r the p r o o f o f ,(a). + g(x)|  .  1 / p  1-1 1  i s a minimum p o i n t f o r g(x)  1 / / p  |f(x)  or  1  2  p  > o .  1  x = (i)  )  p  .p x ? "  p  . (1  +,-[f[f) )  . 2 "  1  p  (1 +  P  >g(x)lP) |f(x.)| .  (1)  by  P  I'sP^dfCx)^  v  f+g||  P  p  = J | f ( x ) + g(x)| dn < 2 - [ J | f ( x ) | d u ; | f ( x ) | d ] >  p/  p  1  p  p  +  = 2 - ( | | f || | | g | | ) p  1  p  p  +  Suppose that  + |g(x)| )  || f || ^ 0 , then  .  U  29. i .  f  p-1  ..  ..  |L < 2 (|| f ||P ;n g H ) p y y p  + g  p  p  < 2  1-1  = 2  f|l  II  (iii) (Iv)  •s  II S II  • a +11^' y <>  p  fe  •• •  ,  J|'f-g| dn  p  and  d(f,g) = 0  p  d(f,f) = 0 . f = g  i;  a.e.  d(f,g) = d(g,f) . Let  I  f,g,h € X , then  d(f,g) = J | f-g | d|i = J* | f-h+hp  2 - (|f-h| p  1  +  p  |h-g| )dn • (by P  < J|f-h| dn + J|h-g| du p  d  •  2  p  • <J  Hence  '  (||f p|| + ||g|| ) . , then  P  ( i ) d(f,g) >_ 0 (ii)  p  p  .l|f + g||p<2 I f d(f,g) =  (  '• - 2 ( i - l ) '  +  (II f || + ||g||)  P  i-1  (b)  _ 1  (  +  1- —  ' ..." I' ' = 2 > || f || i J U J l £ ) i f p y f . . : x  1/p  i s a metric on  p  r=  dn  p  (1)'  d(f,h) +. d(h,g[)v  X .  So, f o r 0 < p < 1 , we have somewhat'.weaker c o n d i t i o n I' l f  f-1 +  f  linear  2  Hp -  2  [  "  f  l Hp  t o p o l o g i c a l space.  D. H. Hyers ological  +  'I 2 "p^ f  a  n  d  W  e  h  a  V  e  t  h  a  t  X  i  s  a  I t f o l l o w s from the theorems of  [9] and J . V. Wehausen [10] t h a t such a l i n e a r  space \ L  P  i n which the neighbourhoods  are spheres o f r a d i u s F r e c h e t metric."-'  of a point  6 > 0 , can be g i v e n an e q u i v a l e n t  T h i s suggests t h a t - w h i l e many theorems on  topf  !°  3.0.  Banach spaces which can be p >_ 1  may  still  fail  a p p l i e d to the space  to h o l d i n spaces where  remain many theorems on Frechet  •spaces which may  be  [7]  0 < p < 1  spaces and  may  pseudo-normed  has  j  of the c l a s s of a d d i t i v e f u n c t i o n a l s .  shown t h a t i f the u n d e r l y i n g  i s atom-free then any l i n e a r f u n c t i o n a l on  lP  i d e n t i c a l l y zero, which proves that almost no on the use  , there  i t appears t h a t the c l a s s of l i n e a r  f u n c t i o n a l s i s a subclass Day  with  P  applicable.  In general  B.ut M.M.  L (X)  measure space  , 0 < p < 1  is  r e s u l t s dependin  of l i n e a r f u n c t i o n a l s can be u s e f u l l y a p p l i e d  to  these spaces. In t h i s s e c t i o n we theorem of N. representation Lp  f i r s t l y give the p r o o f of Katz [3]  Friedman and M.  which i s the g e n e r a l j!  theorem f o r an a d d i t i v e f u n c t i o n a l d e f i n e d  , p > 0 , which reduces to the standard r e p r e s e n t a t i o n p >_ 1  f o r l i n e a r f u n c t i o n a l s when  .  [4].  on .. X  || f || > 0 consider  4.3  7  theotrem  w i l l be  the  same as  /  Let defined  on  '  The methods used i n the p r o o f s in  the  M  be and  the v e c t o r f o r each  which may  f e M  there  i s defined  be regarded as a g e n e r a l i z e d  a c o r r e s p o n d i n g space  Definition.  space of a l l r e a l - v a l u e d  F..: M - M  1  M^"  and  say  functions  a number  norm.  We  that:  i s an ADDITIVE TRANSFORMATION i f  31.  (1)  Continuity:  V  € > 0  II f II <'k , II g:-ll 1 k for  f,g e M .  (2)  Boundedness:  II  P(f)  II <  and  and  ^Y  k > 0  3  || f - g || < 6  k > 0  a  6 = 6(k,€) s . t . || F ( f ) ' -  H = H(k)  P(g)'|| < €  s . t . || f || <_ k  H .  . i i  (3)  Additivity:  for  s € X .  = L (X,E,|Ji) p  4.4  (X,E,ii) ,  p  Theorem. F ..  >  ( i ) K(0,s) - ( i i - ) " K(x,'s,) (iii)  0  .  K(x,s)  = J K(f(s.),s)a(s)du  X  Y  ,  L i f f P  f e LI  P  =0 i s a measurable f u n c t i o n o f  s 'Y  i s a continuous f u n c t i o n o f  x  ad|a - a . a . s . (iv)  =0  be a f i n i t e atom-free measure space and  i s an a d d i t i v e f u n c t i o n a l on  F(f)  where  I f f(s).g(s)  \  Let M  F ( f +g.j = F ( f ) + F(g)  k > 0  a  x . '•  for  11  i  1  / H = H(k)  ] k ( x , s ) | <_ H  s . t . fx| <_ b  f o r .ad|i.- a.a.s .  • ,(v) ) i f F ( f ( s ) ) = K ( f ( s ) , s ) a ( s ) f o r m a t i o n from  implies  Lp  then  F  i s a trans-  t o L^ .  P r o o f of the Lemma i s g i v e n I n Lemma 4.12 next and c o n d i t i o n (v) f o l l o w s by u t i l i z i n g  4. 3 . f o r  F .  the c o n d i t i o n s  ( l ) and (2) o f D e f i n i t i o n ] •  4.5  Definition.  Let  extended r e a l - v a l u e d  (X,E)  be a measurable space. \i  set function  d e f i n e d on  Z  An i s called  a signed measure i f i t s a t i s f i e s the f o l l o w i n g : (i) . u (ii)  assumes a t most one o f the v a l u e s =0  n(<p) 00  (iii)  +», '-«  CO  n(U E. ) =  .1  I |i(E. ) i=l  f o r any sequence  1  E.  of d i s j o i n t  1  • measurable sets where the e q u a l i t y taken means that i • the s e r i e s on the r i g h t converges a b s o l u t e l y i f n(U  4.6  E^) < oo  Definition. '  u (E)  + U~(E)  +  and t h a t i t p r o p e r l y  If  u  diverges  i s a signed measure then  i s c a l l e d the t o t a l v a r i a t i o n o f  otherwise;  |u|E =  \i where  p.  +  •r and  u  4.7  Definition.  tinuous  a r e c a l l e d the p o s i t i v e and n e g a t i v e  with respect \i(A)  which  A measure  v  u .  i s s a i d to be a b s o l u t e l y con-  to a measure  u  i f v(A) =0  V  set  A  for  .= 0 .. /  4.8  v a r i a t i o n s of  Lemma.  Y  h e  (-00,00)  there  which i s a measurable f u n c t i o n o f  .  exists a function  K^(s) j  s  defined  and i s u n i q u e l y  . i  up to t-i-null sets s . t . (a)  K (s) = 0 ,  (b)  P(hy >=. J K (s)dp , • B  Proof:  s e X  Q  B  Define,  h  n ( B ) . =' P ( h ^ ) . n  B e S Clearly  u (<p) = 0 . h  33.  (1.), (2) and (3) i n 4.3 imply  Conditions  measure and s i n c e E e E  | i  h  ||i |.(E).= ^ ( E )  |j  + ^(E)  h  i s of f i n i t e v a r i a t i o n on  Finally i f ,  u(B) = 0 ,  B e E  4.9  (1)  Remarks.  o  r  e  a  c  n  s  e  t  B  K  h  s  h  i  K  h  that  satisfies  If F  i s l i n e a r then we have  = F(hx ) = hP(x ) = h.fi (B) , B € E B  h  hence  f  .  . u (B)  and  08  ^ ( ) = J* ( ) d u = 0 B \x . Hence by Radon- •  Nikodym Theorem, there e x i s t s a f u n c t i o n (a) and (b)  <  i s a signed  E .  i s a b s o l u t e l y continuous w.r.t.  h  that  K (s) h  B  = hK-^s) ,  1  s e X . 1  (2)  L e t K (-s) = K*(h,s) , then we have h  F(hx ) = J K ( h x B  #  s ) B  ,s)d  U  ,  B € E .  , i  •  !  4.10 ing  Lemma. •  There e x i s t s a k e r n e l  K(x,s)  /  and  :'•  d ' s'atisfy-  (1) - ( i v ) o f Theorem 3-4.  Proof: 4.9  v  By Lemma 11 i n [ 4 ] , i t . can be shown t h a t  (2) i s continuous i n x  f o r |i - a.a.s.  of Lemma 12 i n [ 4 ] , we can o b t a i n requirements.  4.11 Remark.  K  -  If F  i slinear,  then  and  a  K^(x,s) i n  Next by the proof t o f u l f i l the  ;}4  F(hX-g)'•'=' h F ( X g )  K(x,s)  Now  the  following  For  = x  and  lemma y i e l d s  each  f  and  e L  •  a(s)  the  we h a v e  =  that  K-^s)  proof  of  the  Theorem.  define  P  ' p  (f)  = f K(f(s),s)asdn  .  i 4.12  Lemma.  Proof:  F^(f)  Since  a  = F(f)  simple  ,  f  e L  function  .  f  is  a  finite  •  ation  of  we h a v e  characteristic that,  i f  Now exists  a  s.t.  f„  f  is  F(f)  = J  suppose  f  a  that n  a.e.  and  n  continuity is'a lim  of  F  continuous  ,  l I 1 f  K  vergence  lim n  f  s  s  n  theorem,  F  L  (f  n  )  of  >  0  simple  lim n  II f lim n  of  x  F  (fn)  for  we g e t  H'B  H(b)  =  P  F  for .  (f)  = 0 (f)  <_ b  .[fn(s)| .  .a.a.s  There  <_ b  ,  since •' ,  |i:'-a.a.s.  the i K(x,s) ;  we h a v e and  Hence by bounded  ( s ) , s )a( s ) d u . = J  .  Hence by  and  that  = l i m -J K ( f n X  ,^ \  |f(s)|  >,adu -  F  then  = F  II  ' •  a d d i t i v i t y of  s.t.  f  combin-  '  functions,  = K( f (s) ,s )a( s)  ! ( ( )> )I 1  b  n  the  linear  '  function,  b  3  we h a v e  function  K(fn(s),s)a(s)  simple  by  K(f(s),s)a(s)dp  Cf l  sequence -  functions,  .  that  since) coni  X  K ( f ( s ) , s )a( s )du L :  4(f)  ii, 35and  since  bounded  ^ ( f ^ ) = F ( f ) , i t follows  <x(s) > 0}  ,  and  f n  i f |f(s)|<_n  f (o)  0  i f |f(s)! > n  l i m || f n  - f ||  (1) i n 3.3, we have  and s i n c e  n  f  (f ) n  =  We have  <_ n} ,  E  *>  V  f  n  ||  <  p  i s bounded,  n  = || f  f  F(f ) .  = {s : : [ f ( s ) |  n,l  and  = 0 . .  lim F ( f ) = P(f) ,  n  E = {s : K ( f ( s ) , s )  = f(s)  Hence by c o n d i t i o n  A  and l e t  bounded, we d e f i n e  n  F l  p  f (s) n  hence  f e L  G = {s : K ( f ( s ) , s ) a ( s ) < 0} .  To make  Let  = F(f) for  f . F i n a l l y consider  and  F-Jf)  that  2  =  n  || f ||p , hence  = EHA^  n  F  = GHA  n  n  " l l f ^ ' l f p  <  If  f  ||  p  , i = 1,2  By c o n d i t i o n (.2) i n 4.3, we have  X < n,i' F  Therefore  f  =.r  F ( f ,) = f K ( f  •uniformly bounded i n n  ( n,l)lB(Hf  ll )  , (s),s)a(s)dp  i .= 1,2 a r e  K x  f  and we can w r i t e  p  1=1,2  36  F(f  ) = J K(f • X  n  *  ( s ) , s ) a ( s ) d p = J*  E  K(f(s),s)a(s)du  and by Lebesgue monotone convergence theorem, we have  lim n Similarly  F(f n  )  lim F(f n  Therefore  .) = X  r..K(f(.s),s)a(s)du E  ) = I* K(f(s)','s)a(s)dp G  11  P(f) = l i m F ( f ) = lim{F(f  -,)+F(f  -)} = P. ( f ) Q.E.D.  Now i n the f o l l o w i n g theorems we again prove the i n t e g r a l representation  o f an a d d i t i v e , f u n c t i o n a l on 'L  1 <_ p < co , under d i f f e r e n t c o n t i n u i t y c o n d i t i o n s on  spaces,  F , when the u n d e r l y i n g measure space i s atom-free and f i n i t e or c r - f i n i t e .  For t h i s purpose we s h a l l f o l l o w  4.13 Theorem. space.  F  Let  (X,E,u)  [2].  be a f i n i t e atom-free measure  i s an a d d i t i v e f u n c t i o n a l on  L (u) , 1 _< p < »  t  'ST  (3): x  then the f o l l o w i n g a r e e q u i v a l e n t : F(x)  (*)  and  .  where  F(x) = / (f.x)du X • f  satisfies  V -  x  €  L (u)  the c o n d i t i o n s :  p  n  - x  a.e. =29* F ( x ) n  37.  (a)  f  i s continuous and  (b)  range ( f )  Proof:  Let  f(o) = 0  .  i s bounded.  f  satisfy  the c o n t i n u i t y c o n d i t i o n  Then-as-in S e c t i o n 2,  condition^ (b). —  f u n c t i o n a l defined by  i f . F .: L (u) -* R  F(x) = [ (f«x)du  (3):  x  x  n  •  (  F  ^ ~* ( ) •  x  F  n  an a d d i t i v e f u n c t i o n a l on (3)  u i t y condition x  x a.e.,  n  x € L (u)  1  <_ | f ( r ) | /  y  n  co  3 a decreasing U ( B  n  }  =  n  *  L  n  e  t  n = n*B  that  as u s u a l ,  F(  for  o f simple f u n c t i o n s  J(f«x)dp  n  x  - f°x  n  .  uniqueness of  f  follows  ?! •  ,  v  value  •  e  n  x  n  e  M  n  d  x  n "*  ^(x^) = J*(f  have  ~*  property  0  ] a  ' e  )<^u = _+ U-(X)  Thus range ( f ) i s bounded.  x e M , there  s.t.  a  ^  n  "*  x  a  « e  boundedly a.e.  e x i s t s a sequence 1 a n  d by the conThus  = l i m J(f.S )du = l i m F(S ) = F(x) . n • . :\  Then  and  00  intermediate  r  w  .  of measurable sets s . t .  n  x  (3)  a l s o ..satisfies n  £B )  s'. t-.  1  f o r some constant  s.t., 'fz* |.-»  n  e ^(u) >  t i n u i t y of, f , f « S and  ^^  :  which c o n t r a d i c t s  {S }  L  By the strong  However, s i n c e ' x  Now  x  n  {r }  sequence  Iffr-TT  x  is  the c o n t i n -  ^(l- )  c  l L l l •.<. b  F^ = F /  e x i s t s a sequence  which s a t i s f i e s  a sequence  and  co  condition  Conversely, i f , F  x  L (u)  then f o r  b > 0 , we have that there  ••I  P  a d d i t i v e f u n c t i o n a l and i t s a t i s f i e s a.e.  is a  x e L ( u ) j , i t is  V  X a well-defined  (a)' and a l s o  f«x e L ( p ' ^ ^ 1  The  •  from Theorem 3-8 by a p p l y i n g  i t to Q.E.D.  38.  Since  (X,E,u)  i s a f i n i t e measure space, convergence  a.e. i m p l i e s convergence i n measure, we have  i 4.14 C o r o l l a r y . by a . c o n d i t i o n  I f Condition (4)  "*  x n  x  i  measure  n  the above theorem I s s t i l l  ( 3 ) i n Theorem 4.13 i s r e p l a c e d F ( x ) - ' F ( x ) , then n  true.  I f the u n d e r l y i n g measure space f r e e a - f i n i t e and  4.15 Theorem.  u ( X ) = co  If  then we prove that  (X,E,n)  an a d d i t i v e f u n c t i o n a l on  (X,E,u)  is a-finite,  i s atom-  F = 0..  u(X) = »  L_(u) , 1 <. P < °> , then  and  F .is  F  s a t i s f i e s c o n d i t i o n (3) i n Theorem 4.13 i f f F = 0 . Proof:  Let  = x x 1  , if.  A  e E  and  0 < u(A) < » . Ii:  x  l  e  V  u )  "•' »  Since  (X,E,u)  I s atom-free, we can f i n d  a sequence  £ } >;i_ A  n  n  A C\Aj = <J> , -i =}= j and A e E ' V i >_ 1 , s.t. ±  ±  • • . . u{A ) = u ( A ) n  Thus for  x  n  this  =  x n  X  A  sequence  V  1  for  n >_ 2  (^l  However s i n c e  w  e  F  F(x ) n  and  n >" 1  x^  have that  are equimeasurable and x n  "* 0  a.e.  I s a d d i t i v e , we have that  = F(  x 1  )  Is constant.  39  F('x) = 0 By  f o r functions  (3) of 3.12,  The  4.16 on  P  i t follows  M. M.  Day  , 0 < p < 1  and  4.15  we  space  case, we  4.17  Y  x e L (p)  (X,E,u)  F  j  !  Q.E.D.  on  zero, where  i s atom-free^ but  'L (p,)  u  from Theorems  1 X p _<  any  that s a t i s f i e s  i s atom-free -ex-finite and  p(X)  f o r analogue of C o r o l l a r y 4.14  .is  nonlinear  (3)  of  |.  measure!  = » .  i n the c f - f i n i t e  have  Theorem.  space and on  =0  i s i d e n t i c a l l y zero when the u n d e r l y i n g  (X,E,p)  Now  F(x)  |j  has proved that any l i n e a r f u n c t i o n a l !  have seen that f o r  additive-functional Theorem 3.12  that  is identically  Lebesgue measure and 3.14  u(supp x) < »  where  converse i s v a c u o u s l y t r u e .  Remark. L (X)  x  Let  suppose  (X,E,p) p(X)  be  = » .  L (p) , 1 <_ p < » .  a c r - f i n i t e atom-free measure Let  F  be a n / a d d i t i v e  Then the f o l l o w i n g c o n d i t i o n s  functional are  equivalent; (4)  x  (*.)  F(x) = f ( f ' x ) d u X  n  -* x  i n measure =??  F  (x ) n  r* F(x)  and i  (a*)  f  Y  i s continuous and  x € L  (p)  w i t h an  f  satisfying:  1  p  ' i f[-h,h]  =0  f o r some  h > 0 .  40.  Proof:  Let  •restriction Define  P  B  B e S , 0 < u(B) of  u  to  on  L (u ) p  B .  d e f i n e d f u n c t i o n a l and we  ^  where  < p < » . — — (4).  the p  B  satisfies  be  > Y* e L ( u )  B  F- (y) = F(y')  L (u) •", 1 • p  a d d i t i v e f u n c t i o n a l on  and l e t  y ' = yx  Let  by  B  < »'  = J f«y X  B  f(o) Now  =0 we  and  P  i s an . i i s a well-  p B  Hence by Corollary-  dp  where  f : R - R  i s continuous,  For i f  f  C c E  determined and  by  F  i s independent' of  B  0 < u(C) '< u(B)  <" » \ we have "by  the. s t r o n g i n t e r m e d i a t e v a l u e . p r o p e r t y t h a t , there e  '  S  B  number f,g  4.14  range ( f ) i s bounded.  claim that t h i s  B e E .  l  |  have P (y)  B  .  l  C  B  s  r , rx  " "  = u(C)  t  , rx^  B  represent  P  F(rx  B  Since f o r any  are equimeasurable,  and  B  .  P^  we  exists real  have t h a t i f  then  .) = P ( r x ) = ^ F ( r x c  B  B  ) =  P (rx ) c  c  /  /  •  ""  f ( r ) a ( B ) = g(r)|i(C) f  1  f ( r ) = g(r)  Hence i f above we  x e L have  , \  i.e.  p(supp x) < »  f = g s i n c e u(C)  then w i t h  f  = u(B ) x  ^ 0  determined .  I  41.  'P(x)  =  J  (f-x)du  = J(f.x)dn  V  x  e Lp(u)  ,  supp(x) |i(supp  Now  i f  f  does  a  null  a  sequence  not  £an5  sequence (an5  Aspairwise  satisfy o  condition  reals  f  i-s called  in  Theorem  disjoint  sets  s.t.  null  A^ e  E  the  Let  x  addivity  of  A  a  n  )  then  ^ 0  v  converges  i ^ > i  be  there  exists  '  where  n ' ,  to  a  zero.  sequence  o f !•  s.t.  ±  by  it  £  [i(A ) = p ( A 1 ) < co Therefore  (  f  i f  let  4.15,  (a)*,  X) < oo .  F  |J  T i >_ i .  9  ,  for  any  integer  m ,  we  have  I = a  v m U A.  •1 (i)  ..|F(xn;)|.  (ii)  ess.  sup  IxJ  =  ess.sup  fxn(t)f  since  x  i n measure  f(xn) Let  / E  0  1  — 0 =  F(o)  .  e  :  =•. { t  x  n  = xx  E  .  I ; in  m  s.t.  1  =  |an[  inf  {M :  we h a v e  Hence (s(t)f  /  f >_  n(t  : ; fx  (t)f  > M) =  contradiction,  satisfies and  for  (a)*  0}  and  for  '  .  arbitrary  x  e  Lp(^)  X  n let  X  we c h o o s e  >li  where  n  where  measure  Thus  |j[supp  ( x n ) ] ' = \i(E1)  "  <  - •; ' ^ '  »  and  x  n  -  x  42.  by Lebesgue's  •  limit  theorem,  we  have  F ( x ) = l i m P ( x n ) = J f . ( x x )d^i E  n  and i  since  <• h  for  by  h  chosen i n  (a)*  that  f»xx  (a)*  we h a v e ,.i t  = f»x  E  for  all  h- . w i t h •  follows  that  n. F(x)  =  <  f(fox)du  .  Conversely, x  e Lp(u)-  we h a v e  let  as  f  above  satisfy that  V  (a)*  and  integer  (b). m ,  For U^E^) < »\.  •• and  (a)*  gives  that  m ,  . V  < h  ,  ' ' I  f « x = fo(xx  1 ?  )  which  1  m together  with  function  with  (b)  implies  supp.  that  c'E^  f « x  i s : d o m i n a t e d by  a bounded  .  m f(x) x  e  e L^u)  L-nd- ) .  we h a v e  (  i}  1  a  n  d  V  m >  I  s  and  0  n  m  Now s u p p o s e  is x  defined V -  x  x  \ . in  H—»  xE  = J'(f«x)dp  '  ^  n\  x  x  F(x)  additive..  m  (ii)  so  xx E  where m  KJ! = X -. m  in  in  measure,  i43.  A l s o we see that c o n d i t i o n s and  condition  ( a ) * and.(b) imply c o n d i t i o n s (a)  (b) that range ( f ) i s bounded.  We have by C o r o l l a r y 4.14 and w i t h the c h o i c e that  ( i ) implies ( i i i )  J*(f•V E (  )du 1  - K  f e x  X  E i  m and  of m  i f we take  C^  )dU =  J(f»x)dn - P(x) ,  m  h  = (t e ^  : | x ( t ) f <> h}c{t : n  |x (t)-x(t)|>|} n  m we have by ( a ) * that  supp (f°x \„ ) <= C  ,  and by ( i i ) we have  m that  u(C ) - 0 . nh  Hence the boundedness o f range ( f ) i m p l i e s n > 1  }  f . x x„ n  the f u n c t i o n s  that f o r  a r e dominated u n i f o r m l y by m  bounded f u n c t i o n s w i t h support  ( = c n  ^  which i m p l i e s  r . f . ( x x )du - 0 • • 1 / m  (iv)  v  Thus by ( i i i ) and ( i v ) we have that  lim F(x ) = J[f-(x x ) + fo(x x )]du n  n  Ei  n  m •  • • = lim x  = F(x)  m  J[f.(x x  n  p  ^1 m .  .  Ei  )]du + l i m n  f f . ( x v . )d|i •  1' . "  m  Q.E.D.  44  4.18 Theorem. space  and  Then  F  (X,E,n)  Let  F  an  additive  satisfies  be  a  finite  functional  (5):  x  -  n  iff  x  in  .(*)  f  (c)  |f(r)|  Proof:  Since  measure on  is.continuous  in  a  Lp(^)  a  < k(l  [r|)  convergence  finite n  +  and  F  d  in  measure  ]_ = / L F  V  L  ( )  w  u  with  =  -=4>  an  f  F  (  x  n  >;  < »  )  .!;  "*  ,  satisfying  0  -norm  and  some  implies  for  have  e  1 <_ p  norm  p  r e R  space,  (u)  measure  .  f(o)  p  L  L  holds  conditions  (a)  on  atom-free  as  K >  0  .  convergence  F  an  additive  in  T h e o r e m 3-8  in  functional that  CO  F(x) Now  = F^-(x) suppose  sequence {Bn)  =  '  be  Since  f(f-x)dp  f  does  {r_} a  Jfr  c  ! du n . P  B  not  R  c Loo(u).  |f (r  of  = •  x  satisfy  s.t.  sequence  x  V  n  (c).  X,  +  ) =/|r  n  |  B  p  satisfies  there  | r l )  s.t.  n  f  •' T h e n  ) f > n(l  sets. In  |rn[Pp(B  where  e  |f(r  n  .  p  S  1  )[  W  exists  (a). a  Let V  n  and  (X)  rpu(X) <  =rn ( l  we h a v e  But  F(r  that  v  contradicts  n  r  )  n  X  B  -  | r  in  0  =J!(f'y  +  B  n  n  [ )  p  < ^ H('X) " n  0  L^-norm.  )dp  the ...continuity'of  =  f(r F  )uB  )  Thus  =  + f  u(X) >  0  satisfies  which (c).  j  45..  Now  i f B = ( t :' | x ( t ) | < c^} . and  c o n d i t i o n (c) i m p l i e s s.t. on  |r[ > c  for B  and  that there e x i s t s constants |f(r)|  ±  <_ k | r |  |f«(xxg)| <. K|xxg|  s e l e c t a sequence L^u)  c  L  ( ) , a  e o  i s a dense subset o f  P  p  .  Thus  , and s i n c e s.t. x  c-^  and  | f - x f i s bounded x e L (|a)  we can  -• x a.e.' because i j  Lp(u) . . .  Thus by the c o n t i n u i t y of ' P we have that  B = X~B  and by p r e v i o u s  theorems  F(x) = l i m F ( x ) "= l i m f ( f . x )du i f n n n  f •x . - f .x e L ^ u ) n  Now by the c o n t i n u i t y of a.e.  we have . f o x ^ - * f°x  and  M-L  s.t. for  a'.e.  f  for  |t|> K , |f(t)|  is valid K  l  =  |sup 11  u(B) < 6 ' ( f ) for  6'(e)  ( f (Kt ) |  '•  <  we get  < M |t|  p  1  V .€ > 0  2K~3  '•  3  = f (fox)dii X  K  6'(€) > 0  V  n > 1 • which ! -  :  I  w n e r e we l e t  "  p = 1 , by V i t a l i ' s Theorem, (X,E,|a)  that  f«x  Hence we have the r e q u i r e d r e p r e s e n t a t i o n  P(x)  - x  n  /  have, f o r f i n i t e measure space  f^: R R  x  .  f |f«x |du < € B  = min {6(-^-)>  Thus f o r the case  Conversely i f  that  A l s o by (c) 3 constants  Thus from ( i ) we have ( i ) ' : s.t.  and the f a c t  satisfies  i s well-defined  n  we  -• f»x e L^('u) .(*")'.  !  (a) and (c) then the f u n c t i o n a l  and has the a d d i t i v e  property.  he £  Now i f a sequence  II  3^  of  t  n  3 > i ^ -^(u)  i  s  s  u  c  that  n  n  - x || •-• 0 , then as above, f o r every subsequence x  n  3  i xm^>  which converges p o i n t w i s e as w e l l as i n norm, we have  that in  x  ~*  f  L (u)  o  X  €  L  i and s i n c e every norm convergent sequence  converges I n measure,  i t c o n t a i n s an a.e. convergent  subsequence. I t f o l l o w s that every subsequence o f c o n t a i n s a subsequence which converges i n Hence  {  f i x  n}ri>i  i ^  3  8  1  ^  converges to  ^  o X n  ^n>l  norm to  f»x  in  f«x .  ' norm and  hence  l i m F ( x ) = l i m f f«x du = f f i x dp = F ( x ) n n X X n  V  n  An analogue t o the above .theorem, when the space i s a-finite, i s  4.19 Theorem. . L e t space w i t h Lp(u) . 5: (*)  u(X) = »  (X,£,p)  be a d - f i n i t e atom-free measure  and l e t  F / be an a d d i t i v e f u n c t i o n a l on  Then the f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t : x  n  -* x  in L  p  norm  F(x) = I* "(f «x)du X  -•—^ Ftx^) x e. L(\x)  V  F ( x ) and with  f  satisfying  p  c o n d i t i o n s '• .'. (a)  f  i s continuous and  (d)  )f ( r ) | <_ k | r |  p  'V  f(o) = 0  r e R  and some  K > 0 .  Proof:  If  satisfies FB  F  an  condition  obtained  there  is  from  exists  F  (a)  and  satisfies  (d).  of  (X,E,n)  FB(y)  4.18.  Theorem  Then  there  s.t.  a  \|f(a  atom-free,  I  |  a n X B  jPa  for  f  exists  in  :  a n  X  we h a v e  |  )  u  n  F(a x  However  n  establishes  e LpCt- )  ^  = x x ^  1  this  a  n  By  for d  f  then  sequence  Jf.xJ  that  f  •'-}.<= L p ( u )  sequence  .  p  B  e  =  E  and  1  If(0  0 =  F(o)  .  I  [J a x  which  contradicts  |! -. 0  B  ,J  n1  1  .and  n  <  .  p  .  (5).  This  (d).  Now x  B  ) /  B  which  = ,:ajP (Bjy,aJ ^_<J^!  U  {a x  that.'  < p(B)  R  null  s.t.  p(B  functional  0  R -  n we h a v e  which  all  We c l a i m  n  .nee  (|i)  B  ) > n | a  sequence  L  F(yx ) ,  =  by  be  is  by  on  4.17  function  reals  fB^}  Theorem  continuous  not.  non-zero  Let since  defined  (c)  Suppose  functional  then by  (5)  a'.unique  satisfies  {anl  additive  o  the r  E  defined  c  || x n {xnl  representation  ,  -  x ||  p  f"*^ "*  f  of  F  in  Theorem  0  and  o  X  |xn(  a.e.  ,  we h a v e  4.17  that  <_ | x |  and b y  (d)  for i f  .  Thus  for  ,  < K[x|p e L1(u) . LebesgueNdominated  convergence  theorem,  F ( x ) = l i m F ( x n ) «= l i m J f - ( x x n  n  E  n  we h a v e  that  = J(f.x)dn .  »  48.  For Property define  (5)  the of  F(x)  converse F  to  ,  be  an  Appealing in  Lp norm,  || x ^ . -  x  Hp  sequences, a.e.  -* 0 by  there as  the  contains  the  to  f « x  Hence  subsequences  as  xm  Theorem  of  in,  -• x f  ensures  subsequence the  n  )  on  ^  that  sequence  = lim J(f«x n  x  and  we h a v e  { f « x }  continuity (d)  L  on-. F  (u)  Theorem,  a.e.  norm.  x  and  functional  exist  well  lim P( n  (a)  the V i t a l i ' s  i n . ' L-^ n o r m a n d h e n c e  converges  show t h e  to  continuity  a  only  conditions  additive  again  Thus V i t a l i ' s  Cf°xn] fox  then  for  we n e e d  m  ^  i f  x  -» x  n  s.t.  for  such  that  every  .  f  o  X  m  sub"*  f  °  x  subsequence  of  w h i c h ' c o n v e r g e s ' to ^  o X  n^n>l  itself '  )du = J ( f « x ) d u  = F(x)  .  SECTION EXAMPLES AND  5  COUNTER EXAMPLES ON  REPRESENTATION OF ADDITIVE FUNCTIONALS  Let  B = L (n)  be the s e t of a l l  CO  r e a l - v a l u e d measurable f u n c t i o n s on continuous f o r which  'f(o) = 0 .  F(x) = J (f X F  5-1  (a)  If  X .  essentially Let  For every  bounded  F : R -* R  be  x e B , consider  x(t)d (t)  —  U  —  (*)  satisfies  x,y  have d i s j o i n t  n(supp x D supp y) = 0  and  support, then s i n c e  f ( o ) = 0 , we have  F(x+y) =  F(x) + F ( y ) . 5-1  (b)  If  1^1  , |x| <_ M^  'F(x-). - F(x)  C }  i  x  n  sequence i n  sa  B  s.t.  ' f o r some, p o s i t i v e constant-  since  •  f ( x ) - f ( x ) a.e. and  •  x  n  M-^.  x j a . e . ahq then'  | f ( x )[" <_ sup  f (y|)  - • - Ty|<c  by Lebesgue'd dominated convergence theorem  that  l i m F ( x ) = l i m J*f . ( x ^ t ) )d|i(t) = Jf(x(t))dn(t) =F(x) n  •  n  ,  .  ...  F- i s continuous. 5.1  (c)  If  x,y e B  s.t.  x',y  a r e equimeasurable, t h en  :  50.  'F(x') = F(y). .  For i f  f ( x ) and  and hence  'f(y)  x,y are equimeasurable  then so are | j  P(x) = J f'(x(t))du(t) = J f ( y ( t ) ) d , i ( t ) = F(y) . ; X  X  X  In t h i s s e c t i o n we w i l l be concerned to the to which P r o p e r t i e s  5-1  { ( a ) , ( b ) , (c)} c h a r a c t e r i z e  extentj functionals  of the type (*)'.'. example shows that Theorem ~5.8 i s f a l s e  Our f i r s t if  the u n d e r l y i n g  measure space  (X,i:,|i)  i s atomic and we. s h a l l  f o l l o w m a i n l y V... J . M i z e l and A. D. M a r t i n [1] i n t h i s  5.2. X}  Example.  .  Let  u(2) = m For  Let  u  , S = { a l l measurable  be the measure o n . (X,E)  ,  2  X - {1,2}  =(= m  d e f i n e d by  section.  subsets of ja(l) = m  .  2  x e-B(XyS) , l e t  x ( i ) = x.. , i = 1,2,  i '  For each  . x e B(X,E)  F(x) = f- (x^)m L  and  define  •+ f ( x ) m  1  2  f^'o) = 0  2  for  2  F(x(  i  x  n )  n  ^  x  i°^  ) = f (x[  n )  1  where  P(x^ ) o )  .  \  For a d d i t i v i t y o f  F  f^ : R - R  by are continuous  !i :  i = 1,2.  and )m  j'  the f u n c t i o n a l  Thus by. the c o n t i n u i t y c o n d i t i o n of then  ,  1  1  x^ ^ n  -» x^°^ . i m p l i e s  + f (x^)m 2  R , i f  2  - f ^ x j  •• F :  . -  x^ ^  -• x^°^ boundedly  n  that 0  ^  + f (x^° ' ) m > )  2  ?  '  51.  < (!)• i f x-^x^ e B  s . t . supp x.^ n supp x^ = <p  x-^ ^ 0 , and x  2  =  X  2  (2) x  (2)  • implies (ii)  Since  x^ ^ 0 w  h  that  e  r  then  ^(1) =  e  x^ =  )  then i f a  n  d  1  i > ^( )=  m  2  '  m 2  F ( x + x ). = F(x )'+ F(x ) . 1  2  1  2  m^ ^ m^ , two p r o p e r subsets o f  X  have equal  measure i f f they a r e e q u a l , which i m p l i e s ' that x,y e B And  since  are-equimeasurable i f f x = y .  m^' = |a{V : x(w') "= x^}. = |i{w .: y(w) = x^} i . e .  {w .:, oc(w) = x- } = {w' : 'y(w) = x, } It follows  that  x,y  tinuous f u n c t i o n _ f : R -R  F(x) = f ( x ) m 1  1  1  But i f x^ = 0  F(x). = f ( x ) m 2  Similarly l  f  2  a l w a  implies that  x = y .  a r e equimeasurable i f f x =y=^> F ( x ) = F ( y )  Now i f the theorem i s , to h o l d  f  .  2  , f(o) = 0  + f (x )m 2  s  2  f o r some  2  =.f(x )m  f^ = f .  y -  t r u e we would have f o r some con-  2  = f(x )m  2  1  x  2  that  1  + f(x )m 2  2  .'  i:  then  .i.e. 'f  g  = f  .  .But i t i s not n e c e s s a r y that Q.E.D.  So we have Seen i n the above example t h a t , presence of atoms i n the..measure false.  space makes the' R e p r e s e n t a t i o n (*)  !52  5.3  Theorem.  X = AUC parts  I f X =T (X,E,u)  i s a f i n i t e measure space and  Is the decomposition of  i n t o atomic and atom f r e e  then the s u f f i c i e n t c o n d i t i o n f o r Theorem 3-8 to h o l d  true i s t h a t , f o r every atom • A Proof:  If  ,  hence  of  i  x e B(X,E) , l e t x  —————  x = x  + x  0  £L  X ,  u ( A ) <_ ia(C) . i  = x x ' and  x  fl  a*  J\  = xx C  which by the a d d i t i v i t y o f  o  and  n  L/  P  implies  that  C  F(x) = P ( x )  +  &  C = (C,E,n)  Now  X  supp x ' n  [  a  supp x  = cp]  c  becomes an atom-free measure space when \x  i s r e s t r i c t e d to  C" and" i t s • measurable subsets, and i n the same  way we may take' x  as a member o f ' B  Q  as i n Theorem 3-8, 3 >f• : R  R  = B(C,E,ia) . .  Hence  continuous, s . t . .  P ( X ) = j f(x (t»du(t) = ; f(x(t))dp(t) . c c C  c  Now  A-^,Ag,...'  a r e atoms of  f u n c t i o n i s constant on each If  = m  \s(A ). ±  A^  and any measurable  with value  , i = .l' 2,...  x^ , I = 1,2,...  , then V  3  1  X  x.e  B(X,E) ,  CO  x  e B(A,E,|_i) a  A = U A.  •  1  P(x ) a  and  and s i n c e  since  W  •1  x x  n E x, x . 1 i  n l i m F(£ x  A  ±  t  CO  OJ  = F(E  1  1  )  ±  A  )  = P(lim E x  i•  -• E x. x 1 i  n-»» 1  A  x  A  ) = l i m F(E x  i  n-*o>.  boundedly we have  1A  n = l i m E F C x ^ ) "= "E f ) m .  1  x  A  )  i•  53 ?(yx where  )  A  P(o«x ) At  f, (y) = — — — — .  and  f . (o) = — — — —  = 0  i = 1,2,..'.  , then'  as i n  Theorem 3-8 . Now i f we can show that  f^= f  V  F ( x J = S f ( x ) m, = J* f(x(t))dn(t) 1  a  A . .  i.e.  '  •  i f x e B = B(X,S,u)  F ( x ) = F ( x ) + F ( x ) = J f(x(t))du(t) + J f ( x ( t ) ) d p ( t )  then  a  Q  = j f(x(t))d (t) X  ' ;  U  and  hence the theorem f o l l o w s  5 0 we now show that Since 5  1  .  1  f = f^ , i = 1,2,...  .  ^(A^) < u(C) , by n o n a t o m i c i t y o f  <= C  s . t . n(S ) = i  r e a l number  a , axg_  \ ± { A  ±  )  and  C , S  , i = 1,2,... ax  A>  S  ±  e Y, ,  , and hence f o r every  a r e equimeasurable and i t  f o l l o w s that  F(ax  So and  s>  ) = F(ax ^) = f ( a ) m A  i  J'f(ax )du = J f(a)x d|i' = C  since  g>  l  C  s>  l -.  i  .'•.1=1,2,....  f(a)u(S ) = f ( a ) p ( i  A i  '.  ) = f(a)m . ;  m^ > 0 , X  f ^ a j n ^ - = f(a")m . ±  f ( a ) = f(a) V real ±  a . Q.E..D.  54.  We g i v e an example i n Theorem 5.3  to show.that the c o n d i t i o n  given  i s not n e c e s s a r y f o r the theorem to h o l d  true.  I t may be p o s s i b l e that the atom-free p a r t of  X  i s empty  while i t s atomic p a r t i s nonempty.  5.4  Example.  sets o f  li(n) =  X  2  X = {1,2,3,...  Let  and l e t the measure  } , E = a l l measurable sub-  |i be d e f i n e d  on  E  by  n = 1,2,3,... •  ,  So here . C = the atom-free p a r t i s empty but we show that the theorem s t i l l holds, t r u e . For  x e B , take  which s a t i s f i e s 5-1  x(n) = x  (a) and 5.1  n  .  The f u n c t i o n a l  F '  (b) can be d e f i n e d by f n (xn') v  F(x) =  Now  T  n  '= {n+1,  real  since  eo  F(a .  •  X  f  —S-v  2  f  n  —rrn  2  n _ 1  -  n  a  ' Xm  d  k £ '-TT  -,  =  „  H  S„ = {n}  and  2  f =  r  f -£ 2 n  e  equimeasurable and Y f = S ~4 k=n+l 2  n  oo  •  f  k=n  f •.. n-1  a  a  T  00  2  n  ) = F(ax ) . V  1  1  ='•• E ' —^ , the setsk=n+l 2  have the same measure and hence f o r any  aXc  ai  2  00  —. 2  n+2,.".. }  a e (- > )  A l s o then  1 •  E n=l  • —  2  k  n  which i m p l i e s by s u b t r a c t i o n  . ^ x.e.  . ~ f = f  ,  Y  „  ^ _ n > 2 v  ~  that  55.  and  hence  5.5  the  Theorem 5-3'Is  Lemma.  whe r e  m^.  Let is  a  s t i l l  true.  {1,2,3,....  X =  positive  number  }  and  define  s.t.  for  every  u(n.)  '  E k=n+l  7 1  m,  ,  subsets  are  equal.  of  Proof:  Assume  Suppose  IUJ  e  J  then  E i e l  .  two  nQ  1,2,...  n =  CO  B I >  = m. n  m.  X =  (X,E,p)  that  IflJ  4= <P > l  and not- to  e  t  n  =  E jeJ  1  have  =  tp =  Q  both.  claim a  n  =  J  and  same m e a s u r e  n  Let  I  hence  3  the  and ^  m  m. — ^  e  I  that n  ,  IUJ  thus  d  only  =  nQ  i f  they  cp . e  I  .or  Then  CO  ' E iel  m.  > m  1  n  > o  E k=n  m,  >  +1  E jeJ  m.  and  o which  are  is  a  ;.  Hence  I  So  subsets  .  two  =  J  =  It'-follows'  equimeasurable  F  Em. > E m. 3 I e l 1 .. j e J  contradiction.  equal.'  Condition  therefore  3  only (c)  5.1  i f  is  • ' . • , „ „ :  cp . of  X  that they  have  two are  vacuously  real equal  equal  measurable and  satisfied  s.t.  measure  as and  in  only  i f  functions  =  s  fi(xi)m.  hence  every  functional  '  = J>f(x(t))dp(t)  ....  satisfies  T h e o r e m 3.8  x  T  t  e  and  I (1)  N  V  are  Example.5.2,  '  F(x)  they  provided  f(o,t)  =  0  f^('x). = and  f(x,t)  the- s e r i e s  (l)  is  continuous converges  In  '  56.  uniformly  and  absolutely,' for  |x J  n  , V  n  ,  to  be  uniformly  bounded.  Hence  for  The n e x t for  Theorem  measure  discrete of  gives  5-6,  hold'true  Theorem  for  a  a  is  3-8  not  necessary  countably  true.  condition.,  infinite  discrete  space.  Theorem.  5.6-  (X,E,p),  theorem,  to  3.8  this  the  Let  measure  atoms  of  X =  (X,E,p)  be  space  and  m^ >_ m 2  X  Then  a  .  the  countably  >_ m-^ >..  necessary  .  be  infinite, the  condition  '  measures' j  for  Theorem! 00  to  3.8  be  true  is  that  for .infinitely  many  n  ,  in ^  < ~  2  E m, k=n+l k  (2) ' Proof:  Suppose  number  of  n  condition  and.take  is  (2).  n  >  false  (where  1  o  for  all  but  n  .chosen  a is  finite the  same  o  v  CO  as  in  Lemma  Let  ??'=  5.1  (b),  s.t.  2.39)  {all  V . n i>' n  functionals  5-1 ( c ) } . • •  F  ,  °  m  >  n  on  /  2  £ m, k=n+l  B(X,E,p)  (3)  which  satisfy  5.1  (  '  !: i.  Then  for  every  real-valued  F  satisfies  5.1  (a)  =  £ i=l  i f f  f,  (b)  iff  •  £ i=l  f.  :  R -• R  ;  m  1  (4 ) ;  1  f±(o)  =0  V  I  =1,2,...  -  CO  5-1  function  f.(a.)m. x . • 1  and  satisfies  j converges  uniformly  and  absolutely'  1  57.  for  a^  i n a n y compact s e t o f  5.1 ( c ) i f f ' V  a e  (-co,00)  R .  And l a s t l y  , t h e sequence  ^ ± ^  a  F  satisfies satisfies  ^ ± y ±  i  —  that  ——-(5)  £ f.m. = S f.m. f o r (I,«T) e H iel JeJ • 1  1  3  ' -' I  3  I  as i n Lemma 2.39. Let f  Q <= 3  •. f 2 =  i  be t h e s e t o f those f u n c t i o n a l s  F  1  \  f o r which  ••• •  =  We c l a i m t h a t  5 = Q  f o r t h e a b o v e theorem' t o b e  true. Let containing in'2.39  k+1  ('c) .  support i s  K  K  be a f i n i t e  subset o f the p o s i t i v e  i n t e g e r s where L e t £ j_}j_>i S  b  integers  k, i s t h e number o f e q u a t i o n s  e  t  n  a n d l e t g : R -» R  e  °l  s  u  t  i  o  n  o  f  2.39  be a continuous  ( c ) whos function  which vanishes only a t zero.  Let  •  so  Then 5.1  F =  ( b ) a n d 5-1  S  f  _ f.m.  1  i = j.S  f  ±  f  o  r  = 1,2,...  1  f o r 1 ={= j  ± f j  ______(6j  ( 5 ) ' a b o v e a n d a l s o 5.1 ( a ) , -  satisfies  F e 3?  ( c ) and hence  and b y (6), F | Q  c o n t r a d i c t s our assumption.  which Q.E.D.  The f o l l o w i n g e x a m p l e shows t h a t t h e c o n d i t i o n i n Theorem 5.6 i s n o t s u f f i c i e n t .  Even t h e s t r o n g  Theorem 2.31 w h i c h i s e q u i v a l e n t  t o t h e weak i n t e r m e d i a t e  property,  i ssufficient  i f f  r > 1  where  condition of  0 < r < 1  and  value  58  = r  5.-7  1  for  Example.  between  \  Let p > 1  and  be an Integer..  (_) = r p  lies  1 .  Since it  1,2,...  i =  r  also s a t i s f i e s  2x  i s an a l g e b r a i c number o f degree Since'  we have f o r any  E r k=o  p  +  k  '  for r  p  = \  —  — ( 8 )  p  i = 1,2,...  1  —(7)  -1 = 0  p .  = •= 1 l - r  p  p  that  _. rP+kP-hi.  =  (9)  k=o Let  X = {1,2,.-.. }  that  m-^  m  2 _L •••  m< ^ So that  X  and a  n  d  = r  also  1  = (i)  1 / / p  = m  i  , we have  that  E m, • k=n+l k  satisfiesthe  weak i n t e r m e d i a t e  p _> 1 .  value property  provided  /  We c l a i m that Theorem J5..8 does not hold f o r t h i s X = (X,E,n) .  For i f F =  i E f.r i s a f u n c t i o n a l on i=l .  B  which  s a t i s f i e s c o n d i t i o n s 5-1 ( a ) , 5-1 (b) and 5.1 ( c ) then by (9) above, i f g  ±  = -,£ r^= ±  f  ^ ,  i = 1,2,...  , " then  59  J  S i  =  g  and  i  g  f .  Thus  fp ]_*^p+2*""" 1 n  0  2  Now V  a  n  '  = ±  hence  d  ? i+p ~ 2 g  .  F  can be. completely  R  f^,f ,. ..,f 2  f^(o) = 0  and  are. a r b i t r a r y con-  p  for, i = l,2,...,p .  as i n Lemma 2.39,*  (I,J) € H  E r iel  1  = £ r ' jeJ .  -E f . r iel  _____(12)  J  1  =  E f .r ' jeJ  .  J  (13)  J  D i v i d i n g by the lowest power (13)  determined  a r e known.  p  functions'on  implies  ( 1 0 )  W  i = 1,2,...  for  ±  We c l a i m now that tinuous  —"  g k p + i  i  S  +  f ,f ,...,f  P +  " i+p ~ i+p  i.e.  if  = f  k  g 2  %i  + i  (11) from ( 1 0 ) , we get  g  p  f  i+P = k  subtracting  +  =o p^p g  k  r  h  of  r , (12) and  can be w r i t t e n as  1 =  E c r k=l  (14)  k  K  and  •  / f  h  = ^ f  h  +  k  r  k  " ,-  ^  — - ( 1 5 )  60.  ? Furthermore where  fact  d^ =  that  icity  E n=o k  k  " ^ 2"  (14)  becomes  r'K  p  = •'a^-p  + b^.  for  k  -and  in  of. f u n c t i o n "sequence  Since of  degree  dn 1  =  d_ = 2  p  t  r by  . . . '=  comparing = .  0  a  .__(i6)  where  we u s e  t h e .same way b y . u s i n g ,  unique  (2)  k  = E d, r . k=l •  =• 1 , . • . . , p  f ^ f ^ . . . .  satisfii.es  d, •_ k=l  1  and  (16)  (15)  can  be  irreducible we g e t  the  the  period-  written  as.-  polynomial  that  and  .dp - 2 .  and r  h  since =  2r  h + P  2rr p p  =1  and  i.e. (17)  f  h' ~  f  d rrp p = 1 P reduces to  h+p  '  h  ~  1  ,  ,  relation  ^ • • *»  2  /  •  (12.)  reduces  to  Q.E.D.  6  SECTION  REPRESENTATION OF B I A D D I T I V E FUNCTIONALS  6.1  Definition.  If  X,Y  a r e two s e t s  XxY = { ( x , y ) : x e X , y e Y] of  X  6.2 any  and  i s called  then the c a r t e s i a n product  Y .  Definition.  I f . A c X'  s e t of t h e form Let  where  AxB  B c Y  i s called  -(X,S^ji_^)  and S  and  and  then  AxB c X x Y .and  a rectangle.  (Y^S^,^)  be measure  are a-algebras of subsets of • X  2  spaces and  Y  respectively.  6.3  Definition.•  B e Sg  6.4  A s e t o f t h e f o r m - AxB  i s called  a measurable  Definition.''  If  where  A e S^  /  rectangle.  E a XxY  for /  x e X  and  y e Y , .  define  E  x  E  E  v  • and  i v e l y of  E'y E  y  -  :  x  = {x : ( x , y ) € E} .  are c a l l e d and  ( ^ y ) e E} .  E  Y  the x-sections  c Y , E  y  c X .  and y - s e c t i o n s  respect-  62  6.5  Theorem.  for  x e X  6.6  Definition.  each  I f E e S-^Sg , then  and  x e X  by  6.7  'With each f u n c t i o n  we a s s o c i a t e a f u n c t i o n  XxY .  Similarly ' f  Let f  be an  on  and  E  y  e• S  1  y  f  XxY  defined  and w i t h on  Y by  i s the f u n c t i o n d e f i n e d on  S^xS^-measurable f u n c t i o n on  Then .1 x e X , f  (b)  V  y e Y ,' f  i s an S -measurable f u n c t i o n . 0  y  i s an S-^-measurable f u n c t i o n .  Theorem.. ' L e t ( X ^ S - ^ i ^ )  measure spaces.. E )  Suppose V  y  (Y.,S ,n )  and  E e S xS n  2  0  .  x e X , y e Y , then  \Jr . i s S -measurable and *  and  f  2  y  (a)  I); (y) =  € S  f ( x ) = f(x,y) :  Theorem.  '6.8  x  y e Y .  f (y) = f (x,y.) . X  E  J cpd|_ X  p  I f cp(x) '=. |_ (E )' , cp - i s S-^-measurable  = P d|_i Y  _ U ( E ) = ..J X ( x , y ) d u ( y ) . V 2  X  E  be a - f i n i t e  2  and s i n c e . x  p  e X  we have that .  J ^ ( x J J * X ( x , y ) d u ( y ) '= J d|i_(y)J X Y. Y E  2  X  5.9  Definition.  If - ( X ^ S ^ ^ )  measure spaces and i f  E e S^xS  2  l  X (x,y)'d^(x) • "  f o r i =1,2, then  define  E  are a - f i n i t e  63,  (  U l  xp2)(E)  = J* p 2 ( E x ) d  U l  (x)  =  J  X  The p r o d u c t (For  Proof  U^xUg see  In of  the  R.  this  biadditive  finite  P.  is  to•the  spaces If  B^  x  also  that  associated  for  this  purpose  results  hold  when  also  u-^X!-^  when t h e  cr-finite  finite  and  non-atomic  non-atomic  is  Let  integral  we  a vector  sub s p a c e  on are  the  B^xB2  1  S  a '  measure.  'CT~f i ^ ^ e .  representation  measure  shall  space  follow  additive  .  respectively.  for  underlying  shall  space  space  is  the  we  .(X^,S^,^)  be  N  but  measure  let  N ( x , •')  e B-,  and  is  functionals and  .  is  closely  i n [2].  functional and  Halmos)  and  the  Definition.  5.10  measures  we p r o v e  Similar space  (Ey)du2(y)  section  non-atomic  proofs  of  U l  Y  ,  of  i  only.  •  = 1,2  for  said  restrict  to  every  be  i  ourselves  '  be  measurable  of  measure  two  functions  = 1,2,  then  biadditive  function  measure  y  e  i f B  p  on  X^  a.' N(*,.y)  and  I  . /  The to  the  shall  ones  results  proved  establish  for  the  functional  prescribed  subspaces  the  form  biadditive  additive  necessary  biadditive  of  for  N B^ c  functionals  functionals  and'sufficient  defined M^- ,  on cz  in  are  analogous  Section  conditions  the'product' permit  a  B-^xB^  We  2. that  a  of  representation  64.  (**)  N(x ,x ) = 1  J*  2  (x ,x )d(vju xu )  tp  1  2  L  2  for a l l x i  where  6.11  cp  e B  ±  = 1,2  .I  |[  i s a unique continuous r e a l - v a l u e d f u n c t i o n on  Definition.  A function  continuous i f  cp(x, • )  6.12  If  Remark.  and  ep(x,y)  s e p a r a t e l y then i t may  cp  i s s a i d to be  cp(«,y)  ,  ±  R  p  separately-  are continuous  V  x,y e R  i s continuous i n b o t h v a r i a b l e s  not be continuous i n both v a r i a b l e s  I  jointly. r Consider . cp(x,y) = -I ^2 x  Example.  v  2  '  0 Now  cp(o,y) = 0 , y 4=  0  •  ,  Hence it  9  But  For l e t  lim x-»o Y-o  y  2 i *  n  u  I  y^ = 0 '  .  cp(o,o) = 0 •  .  '; i  \ 9(0,0)  .  | ;  i s continuous f o r a l l y  i s continuous f o r a l l x  +  x^,+  and s i n c e  •  o. . lim cp(o,y) = 0 = y-o  2  Therefore  l i m . cp(o,y) = 0 = cp(o,y ) y-»y  Y  and by symmetry;,  also.  cp(x,y) ^ 9 ( 0 , 0 ) .. ...  S  =  ( x , y ) — • ( 0 , 0 ) along the l i n e  0  y = mx  .  . Then  65.  lim x-»o y-o  Hence  atomic on  L  (m 00  )  (**)  (a,a):  0  and  (u0).  i  let  Then  is  .  N N  discontinuous,  be  =1,2, 'be  a  -  functional  |  boundedly  a.e.  N(x^,y)  -  N(x,y)  V y  e  B  g  y  boundedly  a.e..  N(x,yn)  -  N(x,y)  V x  e  B  1  representing  function  cp  is  V  c,d  cp  Proof:  Let  with a  separately € R  is  continuous  and  cp  x/  and  on  e L<a(ii1)  continuous  =  every  .  satisfies  N(x1,o)  cp(c,o)  and  0  denote-.  x  ,  Y  A  I  i  i  cp(o,d)  =  0  (l):  R - » R , f  additive is  Theorem 3.8,  l  x  2  an  N(x^,-)  f  :  '  .  is  by  2  1,2  R  hence  (fx^ox )dn  =  of  N(x1,«)  condition  function  =  =  '  subset  Then  x  N(x1,x2)  satisfying';  ,  bounded  functional.which  v-  non-  x  holds  (b,b):  a unique  ( 0 , 0 ) .  condition:  •  tinuous  at  .. f i n i t e  biadditive  satisfies  and  Let  '  00  yn.-  iff  +  function  ( X ^ S ^ , ^ )  spaces  x L  J_ ^.xn  (1,1)  given  Let  measure  ^  = —  1+m  the  Theorem.  5.13  cp(x,y)  V  x2  (o)  con-  there =  0 ,  exists s.t.  l...  e L (vi ) tt  2  (l).  66.  Define  cp : R  - R  2  by  , ' \  r.  1  ^>^=^VqjS c  Since  N  ^(cx^dXp). p  * l  (1,1)  i s biadditive,  l ( X l  T — - ( g )  )^(x ) 2  I m p l i e s that  cp  i s separately  continuous. Let  E^  fixed  be the d i s j o i n t measurable sets i n . For a . . k Xg e L^dig) and f o r each simple f u n c t i o n x^ = S c . x • 1=1  00  we have by the b i a d d i t i v i t y o f k N(  N  1  that  •k  S  c  .1=1-  . ,  v 1  *i  =  S  ^  c  I f each  )  x  .  (  . i=l  = c  ±  N  and  1  c  ,  x  _ _  )  x  .  1  .  u (E ) = ^(E^) 1  (3)  then we  i  get  N  (  c  l  x  ft E . ^ 2 ^ 1=1  ^ ( ^ E  =  ^ 2 1 X  1  and hence- i f  ^ i^i_]_  i  E  s  •  )  a . p a r t i t i o n of  1  X^  then  Un (E-, ) (5)  If  ^(F)  we have  i s an i n t e g r a l m u l t i p l e of  N(cx ,^.) = J  f  p  •. ;. By (5).and ( l )  x  Q  X F  2  «(-)du ••  2  u (X ) 1  1  then from ( l )  E  I.  67. • ' N  (  (?)  a  ^ - . ) = ^ ^ )  c  •N(cx ,-  i.e.  F  )  N  =  '  (  c  n(F)  •'  ='t^\)  X i , 0  ia1(F)p2(X2)cp(c,.  P (P)|i (X )«p(c,.) = J 1  2  .x i.e.  for As  .  f  F e  c  S-^. ,  in. 3.8(g),  x  (•)  ;  2  2  = !a (F)cp(c,.)  (6)  1  s.t. by  )  .(v)dn '  f  2  U1(X1)!a2(X2)cp(cJ.).  ia-]_(F)  i  applying  s  a  the  integral  n  multiple  additivity  again,  of  (5)  p^(X^). implies  MF) • that and  is  (6) hence  true- whenever  the  ^  continuity of  is N  a  rational  implies  that  number  (6)  is  true  general.  (3)  can be  k  ^  in  I  Now  N  also  C i  written  as  k ^ (S )  \  , X 2 )  ~f i ^ "  N  k U (E. )  "J  < i*i' 2> c  x  = E p ( E . ) J/ p(c ,x )du 1  C  i=l  X  2  i  ' .  P^TT ..  X  ( c "2J^  2  f  A  !  (7)  2  ..  g  k ' =  X xX 1  which proves simple  the  function.  cp( E c x  J  ±  ,x )d(u xp )  E  2  1  when  x^  2  1 2  representation  (**)  e L^p^)  is  a  68.  To p r o v e that  cp  that  satisfies  rectangle  is K  unbounded *  is  =  max  assumption are  .'.  •  on  Q, .  and  as  "  <_ i  There  =  K *  {Kc  c*,d*,  :  ' max  ,  s.t.  exists  ,cp(c,d)  l e t |cp(c,d*)|"  .  Since  are well-defined.  [ c | <_ K^}  and  A  g  cp  Butb y  =• {i^ :  | d | <_ K g }  b e a sequence  of positive  numbers  s.t.  <_ n - 1  then  c  Or .  n > 1- .  of points  s.t.  > ~  Let sets  .K  c  . { ( c ^ , d^) }  > 40"" 1 "  In general,  i, 6 . a i 1  choose  measurable  " for  n  a sequence  choose  > 26"inE1 ~ ' n i=l  n  0  i  n  Q by  follows:  = K  ,d_  and  firstly  '  Choose  |cp(c1,d1)[  n  and  =  1  (b,b) .  | c | <_ K ^ , ' [ d | <_ K g }  E 6. < \ j>n+l 3  Choose  induction  c  we p r o v e  3  :r  n  :  continuous,  both . A  not satisfy  For fixed  |cp(c*,d)|  l  K  does  {(c,d)  L e t {0.1  ]' E 6 . = 1 J>1 J  1  i n general,  unbounded.  '  M  cp  Q=  separately  holds  (b,b) .  Suppose .a  (**)  c + 2  so  n  n + 1  having  n  0~1a n  i n  X-^  a  l P( c  n  (c^d^)  for  where  n  s.t.  ^j[^i>l  chosen  s.t.  that  -, n-1 ' _ . (f) E K 2 0. 1 . • 1=1 . C i  d  and . d^  d  and  +  c  n  n-1 _ . i - 1 E 2 1 E I, a 1=1 J=l j  , d  n^  ' ^j^j>i X^  =  ^  0,. + n J  ^c  e  s60!1161106  respectively  ° f  s.t..  disjoint  69,  p1(Ei)  = 2  1  | _ 1 ( X 1 ) _ and  2  = j ^  X  Also,  d  H-(Fj)  J  X  F j  X  •'  the sequence  6 .^(X.)  =  2  £  x,  l  =  S  c  i>_l  Thus  i ^ E  a  N(x^,x2) Now  which  r  e  i  """a, (l-1^)  n  i  we h a v e  -  N(x1,x2)  consider  N(x£,x2)  =  =  J<P(  d  l "*  x  .n -  when  -  x  l  a  n  "[  the  d  function • x  boundedly a.e."  oo .  the I n t e g r a l  n .S^x-^ ,  representation  x^  or  x  is  £  a  for  . N(x-^,x2)  simple  1  ( E . ) p  d  1  p  L  2  • -|  cp(c , d )vi ( E ) n ( F ) , J . . 1 • -. J -  E :>i  we h a v e  .^ jX .)d(uu xn : •  -  S i=i  ^ % ( c i , d . ) p  n  '  }  that  n  Also  as  established  f u n c t i o n , - we h a v e  a  Let.  = E c.x-c• 1=1 1 ^ i  x  x  > 2  L  n  n  of functions  . •  that  2  f o r each  1 < i  ^ . 2 -  ( F . ) | ,  1 2 -  1  ( K  6, c  1  - -  < n  e ^  1  If (8)  ,  ( X  1  ) p  2  ( X  2  )  ,  ^ - 21d )u1(X )n2(X2) i  1  (9)  and  |  E cp(c y d W ( E ) ^ J J L j>i+l . 1 . . - .  J  (F -  )[  < S K 2~ie.H1(X ) p ? ( X ) c J - j>i+l i . • .. •. ±  _, e. < 2~ (-±) X  K  c  .p1(X1)p2(X2)  1  —(10)  70.  Hence  f r o m . (8),  (9).and  (10)  we h a v e  that  cp(c ,d )e .|i1(E )p2(X2)|  | N ( x ^ x 2 ) | > | _S  n  J  j  n  n-1"  Z 1 _S 9 ( c 11 , d .3 ) e ,3 y j u L ( E ) n - ( X 2 : i = l jXL i  .  n-1 ^  [  K  c „  n  6  "  s  X^J'i  X — J-  >  which  contradicts  f o r .(**)  exists  a  -• x ^  K ^]2K c  J —-_L  -L  n  l  i1(X )u2(X2) 1  J  _L  n  Hence  Now  e. "-  e.  the  9  to  has  hold  sequence  1  2  N(x^,x2)  property  c  .  (b,b).  let  L ^ f ^ )  -• N ( x 1 , x 2 )  of  x^  e L^n^)  simple  •.  There'  functions,  s.t.  a.e.  we h a v e  N(x ,x )  that  i n general,  {x^}  boundedly  Hence  fact  that  = lim N(x£,x2)  = lim  J  cp(x^,x2)d(ti1x|i2)  .  1 2 - — ( 1 1 ) V  x  2  e -LJM2)  Now (b,b)  we h a v e  converge  .  by  the' separate  that  the  continuity  functions  boundedly pointwise  to  = ' h =  of  cp  cp(x^,.x2)  cp(x-^,x2)  and :  the  property  X^xXg  outside  -> R • a  set  '  7  1  l i  of  the  form  (Ti^xX^U^xNg)  Hence, b y we h a v e  the  from. (11)  where.  Lebesgue  N  are  ±  dominated  null  sets  in  convergence  Xi:  theorem  that i  N(x1,x2)  =  J  cp(x1,x2)d(>xLxii2)  x  V  e L ^ ^ )  ±  and  XxxX2 x  and  this  proves  Conversely,  From N  . cp  the  let  N(x^,x  ) =  above  is  the  5.14  Remark."  that  it  proofs  s t i l l  (2,2)  x  in  It  J cp(x1,x2)d(u1xia2)  •,  x  n  V  e R  cp  is .  is  to  and  (1,1)  x  we h a v e  that  jointly  (a,a)  i;  it  that  a  satisfies  finite (1,1)  valued^ follows  2.  from is  the  is  proof  of  Replaced b y  boundedly•a.e., and  ,  2  X  i  (b,b).  (11')'  clear i f  l  and  Section  holds  -»' N ( x , y )  (a,a)^ c,d  leading  well-defined  from  N(xn,yn)  ( Si y St )  satisfies  steps  €  (**).  X  where  2  y  n  -» y  replaced  continuous  and.  the  above  condition  boundedly by  theorem  a.e.  condition:  cp(c,o)  = cp(o,d)  =  0  :  72.  Since boundedly  a.e.  in  a  totally  implies  finite  convergence  measure  space,  boundedly  convergence  i n measure,  we  •  have  ' 1 j  i ii  5.15  Corollary.  let  N  -be  Let  (X. ,S  ) i . functional  a biadditive  be  as  in  on  L  (u, )xL CO  x  N  satisfies,  (3.3)  x  T h e o r e m 5-13, (u0)  CO  v  " j _  .  I  ^ J J  condition: "*  n  I  boundedly  x  i n measure  N(x  ,y) V  y  and j I Then  •* y  n  boundedly  i n measure  ^>  N(x,y) y  N(x,y  e  ) -  Y  x  B  2  N(x,y) e  B  1  or (4.4)  x  measure  n'"*  ^>  boundedly  x  N(xn,yn)  -  i n measure,  N(x,y)  ,  i f f  y  (**)  -  n  y  holds  "boundedly with  in.  a  2 cp :  R  •-• R  Proof: then  We n e e d N  measure xm--» such by a  then  there  boundedly  this  implies  L-L(p-^xi-ig) .N(xn,xr)  show  a.e.  _ N{.  X I  and  hence  ;X ) .  or  x^  -  to  cp(x^,x2) itself  by Lebesgue  x^  {x^}  -• c p ( x ^ , x 2 )  (a,a)  satisfies  T h e o r e m 5.13.  £cp(x^,x2)}.  2  cp  every'subsequence  converging  norm and  i f  Suppose  cp(x^,x2)  that  that  (a,a)  e x i s t .subsequences  same a r g u m e n t  subsequence  properties  (.3,3)-  subsequences  the  the  only  satisfies  i  x  having  and  1  ( b , b ) . .•'  (a,a)  and  boundedly of  Implies  {x^}  (b,-,b  in s.t.  that  for  all  i n . L^(IJ^XU2)-norm and of  (cp(x^,x2)}  in  L-^IJU^XIJ  converges limit  to  theorem  contains  ) norm  and  cp(x^,x2)  in  N^x^xlJ) -» N(x ,x-)  Similarly  Theorem.  5.16 a  biadditive  condition:  (5,5)  Let  ( X ^ S ^ , ^ )  functional  •  .  1  N  be  on  L  as  in  Q.E.D.  T h e o r e m 5.13.  ( ) x L ^ ( )  a  Then  satisfies  '  x  n  -  x  a.e.  ^  N(xn,y)  -» N ( x , y )  y  n  -  y  a.e.  =£p N(x,y ) n  N(x,y)  for  all  for  all  y  e  X-  v  x e I .  O iff  (**)  (b,b)^ types  holds t  cp  for  Proof:  with a  is  cp :  bounded  all  h >_ 0 ,  S^=  {(c,d)  Let  cp  R  R  satisfying  on f i n i t e  strips  |d|  S2  :  satisfy  N(x1,x2)  i  < h},  (a,a)  =  and  J  =  of  (a,a) the  {(c,d)  (b,b)^  and  ;'  following  :  [c|  and  <  h}  let  cp(x1,x2)d(^1xu2)  X^xX2 It  is  just  theorem  a  routine  implies  that  Conversely, L  £  a  (u1)xL  o  o  (n ) 2  verification, N  that  satisfying.  N  be  (5,5).  bounded  convergence  (5,5).  satisfies  let  the  1  a biadditive Then  N  functional  also  on  satisfies  2  (1,1).  Hence  satisfying (x1,x2)  there  condition  e Loo(u)xLeo(u2)  exists (a,a) .  a unique and  (b,b)  function  cp :  s.t.  holds  (**)  R  -» R for  all  '.  74.  Now we c l a i m Suppose  not.  =  Then  {(c,d)  :  that  there  |d|  cp  exists  <_ h}  or  1 s.t.  cp  is  unbounded of  unbounded on  on  positive  S^  . s.t.  We c h o o s e as  Let  a  K  =  Since  cp  by  continuity  the is  strip  !  S^,  =  {(c,d)  :  | c | < h|'.}  2 or  E 9. iXL .  .  strip  •  S^  (b,b)^  S^/  .  let  =1  sequence  ^i^>]_  and  of  Suppose be  £ 9. i>n+l  points  ^ (  c  that a  cp  ,  n > 1 ~ i  n  i ^  d  is,  sequence^  < | 9  i ^ i > i  follows:.  ^  separately  ,  continuous,*  condition  of  (3)  =  I,  a  K  _  sup  .  |cp(c,d)|  .  r o < c < c 0  is  c  .  ^  n  "  max | c p ( c , d ) [ d <h .  c  is  a  a  As b e f o r e ,  reals  inductively,  satisfies  Theorem  welldefined on  4.13  and  N(-,dx ) > 2  finite.  Since  cp ^  is  unbounded  functions  of  s.t.  >_4  Kc  :  fcp(c-, , d - , ) |  S^ 1  = K  and  unbounded on c  and  then  d  S?" 3, h  , ' | d |  take  d^  e  K c  and  < h  .  Having  chosen  {(c.,d.)}  K.  > ~  S ' 9. 1 . • + n •2 1 a -i=l i  are  Choose  [-h,h]  .  £ , d  c-,  s.t. 1  < i  < n-1  ,;  1 select  c  s.t.  c  choose  d  sequence xp  =  s.t^.  n  of  £ d x  l M  0  /  x2  9„ n  ^ ^ )  measurable p  2  e L  I" =  K  sets  such  (ijp)  .  c  •  F  that •-  o  r  n +  and  then-  n  ^Fi^i>i  up(F.) •  9~1  a  =9.Hp(Xp)  dis:j.oim; ,  let  75.  Let in  {E^}  be  a nested  sequence  of  measurable  sets j  s.t.  ^(E..)  As  =  2~iu1(X1)  •  Theorem 5.13, i f  in  x^  '  c x  =  n  then  E  ! i  . N(x^,x2)  > n p  ( X  1  1  ) u  2  ( X  2  )  .  ••  i i  And  since  N(x^,x2) the  x^ -» 0  property  -• 0 a . e . , .  convergence theorem  (6,6)  is  (.b,b)-^  a.e.  n  -  Since • are  true  for  for  Since  true  x  continuity of  H e n c e we h a v e  Corollary.  5.17  the  x  in  a  finite  convergence  a.e.,  y  above  n-additive  that  c o n t r a d i c t i o n which  totally  condition  the  implies  j,  establish,  cp .  implies i f  a  N  n  (5,5)  -  y  i n measure,  is  replaced  a.e.  theorems  functionals  measure  N  are for  true n  (  x  n  space,  the  1  above  by. c o n d i t i o n :  : , y  for  n^  "*'  n =  finite.  2  N  (x»y)  ,  .•  these |  -  SECTION  REPRESENTATION  OP N O N L I N E A R . . . T R A N S F O R M A T I O N S ON  Let s.t.  |i(T) <  T eo  Definition.  said  to  if  it  be  a  LP-SPACES  subset  • where  }  7.1  be  7  of  is  \i  A real  the  valued  of. C a r a t h e o d o r y  the  type  function for  T ,  R -* R  is  continuous  (b)  cp(--,c) : T -* R  is  measurar.able  :  Remark. i f  M(T)  If  since  function  cp°x  M(T)  and  since  {Xj^}  of  using  7.1(a)  for  simple  of  then defined  cp :  T x R -> R .  denoted  that  real  each  cp«x  .  for  by  Au(s)  on  t  for' a l l  €  simple  T  c  e  is  cp e  Car(T)  ,  .  R .  measure  function  x  ,  belongs  there  exists  to  x  ,  the  operator  it  space  measurable  = cp(t,x(t))  converging  We d e n o t e  functions  each  e M(T)  € M(T)  -  real-valued  (cp<>x)(t) x  a.a  a a-finite  class, of  by  functions  Definition. set  }  X  for  is  (T,Y, \i) the  on  the  T =  denotes  functions  7-3  Rn  satisfies.  cp(t,-)  and  space  Lebesgue "measure.  (a)  7.2  n-dimensional  a  the to  sequence  follows  by  .  by  A ,  ' T  by  = cp[s,u(s)]  where  ,  cp e  defined '- :  CAR  (T)  on  77.  In interested when for  it  the  in  acts  this  beginning  the  by V.  entation  of  shall  the  J.  nonlinear  section,  of  [8]  we  shall  the.operator  A  to  p-^,p2  L^  this  for  2  f o l l o w M. A.  end  Mizel  of  iF^-  space  we  Towards theorems  this  properties  from a  purpose  of  for  which prove  we  the  shall  mainly  the  case,  >_ 1  Krasnoselskii  section  transformations  be  and-  [11].  state  integral  on. L p - s p a c e s  two  repres-  and  also  i  extend  our  linear  functionals  or  earlier  results  of  defined  essentially  a-finite.measure  Lemma.  J A  (V.  Let operator which also  converges  measure  =  Suppose, to  V. Nemytskii  be  a  set  u  (s e G s . t .  of  s  e  G  of  implies  non-  atom-free •finite  measure.  a  Then  (un(s)}  sequence  of  ,  the  s  e  G  functions  --—(l) which  measure.  that  (s)  the. s e q u e n c e  for  for  the  of  ;  sequence  into  s  given  e  G .  6 > 0  | c p [ s o u ( s o ) ] - cp(s,u) | " < € _ i continuity  for  [12]).  finite  every  i n measure in  representation  spaces.  transforms  converges  Proof:  GK  A  G  integral  /  ,  Let  converges  ,  cp(s,u)  G  2  c w.r.t.  in  .  |u^(s)-u(s)|  Gx c  function  {un(s)}  <  .. .  ^  ,  u  j  and  for  the  a. a.  -  that 00  'V( U G K ) . K=l  =  U  (G)  ;;-=_*> l i m | i ( G K ) K-*co  =  u(G)  (2) .  *  '78.  and  hence/  given  n >  P  n  |i(F n )__>  (s  6  (i(G)  -  =  functions  /  3  G :  fuo(s) o for  -  a l l  CAun(s)}  s.t.  k-j^  ) > u(G)  p(Gk  Let  0  u  -  ri/2  v, (s) n  .'  < TT 1 "} • K. o  n > N .'  Choose'  Consider  the  N  s.t.  sequence  of  where  i Au^s) and  = cp[s>un(s)]  •  .  f  let D n . = .{s  Then  we h a v e  €  that  G :  G,  |cp[s,uQ(a)]  flF^ c D „  and  since  and  -  cp[s3un( s ) ] |  it  follows  < f .  that  o l-i(Dn)  >  u(G)  completes  the  r\  and  sequence  lim  Let  (x^"  d(xn,x)  =  i  kP  n  || x n - x  d(x,y)  ||  0  P  Theorem.  L  into  a  t)  If  function  A  are  arbitrary,  this  || x - y ||  for  x,y  strongly  to  e L x  ,  p  e L  then i f  p  .  l  : «L  in  =  converges  =  r' 7.Q  g  proof.  Definition.  7.5 a  -  L  P  2  -» L  transforms  (p-^Pg  >_ l )  every  then  A  function is  in  contin-  uous .  Proof: Let A8  =  Case  @  be  6  and  (i).  the .zero we  show  Suppose  |i(G)  function  in  that  operator  the  the  < <» . space A  p  l  L ^ is  .  Assume  continuous  at.  that the  79  zero • 0  .  Suppose  it  is  not P  exists  a  sequence  converges  cpn(s)  strongly  to  continuous l  0 .  Then  there  •  e L  (n = 1 , 2 , . . .  )  and  <Pn(s)  s.t.  0  |Acp ( s ) | P 2 d s  J  at  > a  (n = l , 2 , . . .  (i)  )'  G for  some  positive  number  a  .  co  Assume  We  that  construct  functions  by  cp k  n  following  (b)  n  induction,  a  and  .  k  < «  sequence  sets-  +  l  are  G,  c=. G  (4)  of (k  numbers  £^  = 1,2,...  )  , s.t.  the  satisfied:  < *  ti(Gk)  \  < e  k  ...  J - G  |A cp .  k  n  4  • 2 2 ( s ) | .- > a P  (c) -  '  ds  .  conditions  £  |cp ( s ) |  f G  (s) '  ;  p  _ n=l  v  n  k  5  .  pi  (d)  For  any  D  If  ,  6, *  ^k+l which  5  w  is  '  suppose  cp^ ( s ) and n k 'e  s e l e c  '-  t  D c  a  P 2  k+1  d*<  k  G, k  % *  that  number  possible.by  |i(D) <_ 2 e.  G ,  |A cp n ( s ) |  J  We  set  t-^ =  have  s.t.  virtue  |i(G)  been  ,  cp R  the  = cp1(s)  constructed,  condition of  (s)  (d)  absolute  is  then  ,  G^ = G  for  satisfied,  convergence  of  the  8o. • 2 .d s P  integral  P IA cp ( s ) | G k. .  satisfied,  since  By and  a  set  the  k  +  %  , A  G  k  +  i  =  Also  (6).  ~. F i c + i  G  by  fA cp  I r  M  (3)  is  (s) k •  satisfies  possible  < WI  ( 3 ) I  [  + 1  w i t h . U(G) - u ( P  Let  i t  cp  k + 1  '  ) < e  ] 1 / P 2  f  to  find  °  and  By  (b)  (s)| ds = J|Ac p n ( s ) | n r  u  is  satisfied  k+l  P 2  (c).  .Consider  the  D,  ±) 1  <  -i_  s  by  ds.-  "\  sets  e  i=k+l  ± 1  >- - W T ^  8  a  .  condition  i=k+l  k +  ....  satisfies  G  n  ——(5)  •  ( a ) _and_-_(b )__ we h a v e - t h a t -  n(  a number  (c).  (5).  3  .  also  -(6)  Then . c o n d i t i o n  •  this  is  condition  V  R 8 £  J • U^ ( ^ l ^  and  (a)  k + 1  P 2  k+1  condition  s.t.  G  - ^ c  F  and hence  function  7.4,  Lemma  ',  =  - --  K  • |  G,  -  CO . U  i-k+1  G. 1  (k  +  1  = 1,2,. . . ) j  .  . . ( k = 1,2,...)  < 2e  :  (7)  ( G )  811 Define the function cpn t(s)  by  i|r(s)  (s)  if  s. e D^  =  (k=l,2,. ....). ... — — ( 8 )  „  0  i f s | U D. . 1=1 1  From (c), (d) and (7) we have for  J  |A|(s)|%s  = J ]A tp ( s ) |  k =1,2,...  > J |A cp ( s ) | d s -  P 2  P 2  n  n  | A tp (s)| as > §  J  shows that that  Po "  i|f e L  (4),  and by hypothesis  A \|r f L  J |A C-  p  IJJ(S).  .  Since.  A f e L  CO  | ds >_ ' E f | A . k=l D k 2  .  But (9)  D. (ID. = cp , i 4 ' = j , i t follows I|J(S) |  contradiction proves that the operator continuous at  (9)  P2  n  Pn  By  that  P 2  .  ds = » , and this A s.t.  '  A0 = 9 , i s  6 .  Now we prove.in general 7 that  A i s continuous at  p uQ  e L  .  f[s,uQ(s)]  Consider the function for The  where  %  £  L  s e G and u e  operator  Pi  .  A-^ . determined by the function  A-Lu(s) = g[s,u(s)]  A-[_6 = 9  g(s,u) = f[s,u Q (s)+u] -  g(s,u)  s a t i s f i e s the condition that  and we have proved that i t i s continuous at the point  '  -  ...  -  82.  Case  (ii).  Suppose  that  Assume, w . l . o . g . at  8  and  A3 =  9  n ( G ) . . = <» .  that  the  operator  A  is  discontinuous  .  !»  • •  •1  Pi As is  a  sequence  of  assume  functions  some p o s i t i v e ' n u m b e r  a  .  also  we c o n s t r u c t  (s)  and  (a) (b)  u  sets  .  D.  D  | A cp n k  .  We l e t  G.  , (s)|  •cp„(s)  • '  e L  (n  • i  S T |cp' ( s ) | n n=l G .  a  •  p  ds  sequence  1,2,...)  (k=  l  = 1,2,.  11  ——-(Id)  < »  of  (ll)  I  functions  s.t.  '  D n D . = cp I + j . i  P 2  J  ds  f  >  (k  =  1,2,...)  I  ' '  k  I  cp n  (s)  = cp1(s)./  and  construct  by  1 virtue  of  i  (10). Having  constructed  cp n  CO \s( U D . )•<  that '  '  .1=1  •  '  continuity s.t.  j. . )  'f  '  ( n = l , 2 , . . . )  induction, c  (Dk)< »  . J  that  by  that  s.t.  a  Again k  (i),"  >  G  Assume  cp  Case  ,P |Acpn(-s)| dds  J  for  in  •  of  A-  co  and.by  for  u(G)  k.  Case  •  '  <  co  (s)  and  • D,  ,  we  see  by  K  (i)  . .  ,  (a) ...  i n w h i c h we r a v e p r o v e d  we c a n  |  I  t h e \ :  find  •  an  '  integer  •  n -J  |  k+  I  83.  J  | A cp  J  U  n D  I  (s)|  2  ds  (12)  < |  k+l.  2  1=1 • and  t h e n we c a n f i n d  J G  Let fied  a set.  | A cp  •(s)|  2  ds  <  °° ' '^" s  > a  (13)  -  k+1 k U  LV - = G. _ -  k+1  k+1  D.  ±-j_  , condition  (a) i s s a t i s -  (s)|  J  and s i n e e. .  J D  |Acp  (s)|  n  P  2  ds=  k+1  j * |Acp G  i t f o l l o w s from t h i s  n  P 2  ds  k+1  and  > ^^k+l^  a  -  n  (s)i  D  1=1  2 ~ 2  that the function  ( b ) . ..  cp  k+l  (s)  satisfies  / /  Again define a function \jf(s) . a s i n ( 8 ) , t h e n b y /' . -------p (11) we s e e t h a t i|r e L and b y h y p o t h e s i s A\|r € L J-j . Biit p  x  by c o n d i t i o n  ( b ) , we. h a v e t h a t  This  Aty $, L  .  c o n t r a d i c t i o n p r o v e s the theorem.  An o p e r a t o r  i s s a i d t o be bounded i f i t t r a n s f o r m s  any s e t w h i c h i s bounded  p 2  U_ i  n  condition  |Acp  ( i n t h e s e n s e o f norm) i n t o  another  ds  84-.  bounded s e t . iff  We know, t h a t a l i n e a r o p e r a t o r  i t i s bounded.  Example.  C o n s i d e r the space  cp = [Q^ C ,. . .} }  i s continuous  But. f o r a n o n l i n e a r o p e r a t o r , t h e n o t i o n s  o f c o n t i n u i t y and b o u n d e d n e s s a r e - i n d e p e n d e n t  7.7  A  1  p  o f one  of numerical •  another.  sequences 1  w i t h norm d e f i n e d b y  2  |  II cp ii = { E *Jy* . i=l  ;  1 •  1  ,  2 Let  F  be a f u n c t i o n a l i n  1  F(cp) = E C l ^ f  where t h e sum e x t e n d s d e p e n d i n g on  over  d e f i n e d by  - D-i .  those values of the index  cp , f o r w h i c h  1^ | >_ 1  .  i ,'  For each element  2 cp e 1  t h e r e i s a f i n i t e number o f s u c h v a l u e s  The. f u n c t i o n equal  F(cp)  i s continuous  t o z e r o on t h e s p h e r e  of the index.  a n d i t i s bounded- and i n . f a c  || cp || '<_ 1  on a n y s p h e r e w i t h r a d i u s l a r g e r t h a n  and i t i s n o t bounded one.  The a b o v e example.'shows t h a t ' t h e b o u n d e d n e s s o f a n o n l i n e a r o p e r a t o r does n o t i n g e n e r a l f o l l o w f r o m i t s c o n tinuity.  7.8  Theorem.  function i n the operator  s  L  Suppose t h a t t h e o p e r a t o r into a function i n  A'. , i s b o u n d e d .  L  A  transforms  ( p - ^ P p 2l  every Then  85.  Proof:  We can assume w . l . o . g .  continuous at  0  by Theorem 7.6,  P-i |cp(s)|  f  ±  that  A6 = 0 .  Since- A  is  there exists an  r > 0  s.t.  PT  ds < r  x  PO  implies  | A c p ( s ) | ^ds < 1  f  G  G Suppose that  p  u(s)  e L  l  (l4)  ~ p  and  nr  l  P-,  <_ [| u ||  <_  PT  (n+l)r  x  ,  n  p a r t i t i o n of  i s an integer. G  s.t.  So when  [u(s)f  J  G.  .  Let ?-,  G^,  , . . .,&n+1  P-, <_ r  ds  = 1,2,.. . ,n+l)  x  ,Po l / P o | | A u ( s ) | | = { J j A u ( a ) | 2 ds} < G 2  we have by .(14)  (i  be a-  II „  I!  [(-H-^U-)  1/Po  Pn  r  that Po  n  f |Au(s)|,^ds <  +  P-  1  _ 1=1  f |Au(s)| ^ds < n+1 G. Q.E.D.  1  7.9  Definition..  space.  Let  A function  p-class for. satisfies  T  T = (T,Y,,[i)  cp e Car (T)  denoted by  cp°x e L 1 ( T )  for  defined on  T  If by  A : L  i s said to be i n Caratheodory  cp e Car p (T)  for  1 < p <_ »  }  if• i t  x e LP(T) .  Pi 7.10 Theorem.  be a a - f i n i t e measure  Po -> L  i s an operator  Au(s) = f [ s , u ( s ) ] .  where  + b|u|  x  1  f e Car. (T) , then  PT/PO  f (s,u) |. < a(s)  P^Pp h  (15)  d  Pp  where-•-b---is-a - p"ositive constant and -a(s)-e L  86  Proof:  By T heorem 7 . 8 , we can find a p o s i t i v e number Pp  f |f(s,u(s)| T  ^ds <_ b  whenever  d  I* | u ( s ) | T  • f|f(s,u) - b|u| cp( s ,u) = \ 0  Def ine  PT/PO x  P-i 1  •  '  |f(s,u)|>b|u|.  |cp(s,u)| P 2 <_ | f ( s , u ) | P 2 - b P 2 | u | P l p  Consider an arbitrary function  u(s)  'P-i/p  . i f | f (s,.u) |<bfu|  L  We have that  s  d s <_ 1 .  if  d  b  if  1  P-, /P 1 .  cp(s,u) ± 0  l  e L  and l e t  T + = {s e T : cp[s,u(s)] > 0} P1 Let  f |u(s)|  0 _< € <_ 1 .  T  1  3  T  2  , .  Then  ds = n+f The set  ..,T  s.t.  n + 1  where T+  n  i s an integer and  can be partitioned into P-,  J^|u(s)|  x  ds < 1  i = 1 , 2 , . . . ,n+l  M f [ s , u ( s ) ] j P 2 d s < (* J f [s,u( s ) ] | P 2 d s Po  .b  '  'PT  |u(s:)i ds  J  2  1  n+1 . sets  PO  <_ (n+l)b  2  -b  PO 2  (n+e)  P < b 2  Let T  l  (T^} C  w.r.t  T  2  u  (u-^ts)}^^ u-k (s)  =0  (16)  be a sequence of sets of f i n i t e measure " " '  C  a  n  d  T  =  U  T  CO  k ' i=l  at almost a l l  Since  s e T ,  cp(s,u)  is  cp[s,u (s)] v  and .=  max -k<u<k  s.t.  continuous  we can define a sequence,  of functions defined on almost a l l when, s | T,  ..  cp("s,u]  T  s.t.  87.  p  So  "^(s)  €  i  L  a  a(s)  n  =  d  w e  s  sup  e  t  cp(s,u)  = l i m cp[s,uk(s)] k-»oo  -oo<U<co  and F a t o n ' s lemma i m p l i e s  16)  Po f I a ( s ) I ^ds < sup "T ~ k  a(s)  =  sup  P^f |cp(s,u, ( s ) |. ^ds K T  Po < b d  Po e L  a(s)  Since  that  c p ( s , u ) _>  sup  { | f ( s , u ) | - b|u[  1  2  ]  ,  -£»<U<oa  -03<U<00  P-1/P0  we h a v e t h a t u  e  (-»,oo)-  7.11  |f(s,u)|  *  for  s e T ,  .  Remark.  space.  <_ a ( s ) + b [ u |  It  Let follows  1 < p < 00 i f f  T = ( T , _.,(_)  be a f i n i t e  atom-free  from t h e above theorem t h a t  |cp(s,u)|  <_ a(s') + b | u |  7.12 D e f i n i t i o n .  For  nonlinear integral  operator  s e S , A  t e T  cp € C a r P ( T )  f o r some  P  and  u,e  measure  a e L1(T)  (-00,00)  }  the  d e f i n e d by  Acp(s) = f K [ s , t , c p ( t ) ] d t T is  c a l l e d P.  functions  S.  .,  (17)  U r y s o n ' . s o p e r a t o r and i t  t o m e a s u r a b l e f u n c t i o n s where.  takes S,T  measurable are  Lebesgue  n measurable  subsets of  R  and  f u n c t i o n which i s measurable  on  K : SxRxT -* R SxT  is  a real valued  f o r each f i x e d v a l u e of  88  its  second  argument  ments.in  7•13  Remark.  S  denote  S ,  is  compact. cp . i s A  the c l a s s  all  argu-: 1  of'continuous,  t h e n an i m p o r t a n t  of U r y s o n ' s  the o p e r a t o r defined  C(S)  d e f i n e d on  the k e r n e l  s u b c l a s s of  o p e r a t o r s whose r a n g e i s  in  (17)  ,C(S)'  This subclass included the.case' i n which  independent  of i t s  first  reduces to a r e a l - v a l u e d  argument  so  functional  that F  by  F(x)  ' In this measure space  = J cp(x(t),t)dt T '  section,  T = (T,E,|j)  w h i c h have  t h e f o r m (17)  functionals  on  LP(T)  characterization and V . J .  Mizel  In previous  essentially Let  A : LP(T)  Hausdorff  — C(S)  and a l s o i n p a r t i c u l a r ,  of  we  the form g i v e n i n ( l 8 ) . results  concerning functionals  spaces,  , 1 <_ p < co , characterize, This  o f A . D.  and V . J M i z e l a n d K . S u n d a r e s a n  sections,  F(x)  for a l l cr-finite  and a l l compact  extends our e a r l i e r [l]  _____(18). ' •  we c h a r a c t e r i z e  the n o n l i n e a r t r a n s f o r m a t i o n s  defined  f o r almost  r  Let  the c l a s s  where  on. R  SxT .  functions is  and c o n t i n u o u s  of  the  Martin [2] g i v e n  form  = | cp(x(t))dn(t) T on n o n a t o m i c  T = (T,E,p)  (19) ,  a - f i n i t e measure  be a f i n i t e measure  later  spaces.  space.  •  7.14 1  Lemma.  <_ p  <  Let  co ,  (i)  F  which  F(x+y) x,y  (ii)  F  each (iii)  F  -  subset  is  Proof: same  €  = co ,  By  type  S  is  F  then  a  where  xy  0  =  F^  j  L P ( T ) <,  whenever  by  L°°  norm on  . L  norm,  P  bounded  every  to  a.e.  i f  real'number v^(E)  =  . p  <  »  convergence h  F(hx ) E  ,  ,  the  set  for  measure.  +  C^,  which  C_,  =  0  condition  ,  is  a  functional (i)  reduces  of  the  to  the  1  i.e.  F(x+y)  =  F(x)  + F'(y)  whenever  v ((p) = F(hxi,) = 0 .  Let  h  for  ±  4= J  E .s?  then  n  E  n  E  implies  n  = U E. , hx  hence.  ' W Hence  on  a.e.  E n E . = cp and  to  w.r.t.  for  relative  = F  with  0-^ = 0  Now  ±  constant,  LC°(T)  yi-continuous  '  case  of.  defined  taking-  as  C_, =  relative  continuous  function E  =  continuous  continuous  p  functional  a . e.  bounded  and  real-valued  F(y)  uniformly  is  if  F(x)  0  is  a  satisfies .  -  =  be  is  =  F  (  h  X  a c o n t i n u o u s  E n  )  -  P(hx ) =  measure  E  on  T  vh(E)  .  £  and  - hx  E  90.  Now we s t a t e t h e f o l l o w i n g  theorems by V . J .  that prove the i n t e g r a l r e p r e s e n t a t i o n formations  on L ^ - s p a c e s  7.15  Theorem.  let  F  that  Let  of n o n l i n e a r  Mizel  [8]  trans-  , 1 <_ p <_ co .  T = _(T,E_,|_i)  be a f i n i t e m e a s u r e s p a c e and  be a r e a l - v a l u e d f u n c t i o n a l cn  LP(T)  , 1 <_. p <_ <» >  satisfies (i)  F(x+y) - F(x) xy = 0  (ii)  F  is  - F(y)  = constant  uniformly continuous r e l a t i v e  F  is  is  continuous w . r . t .  p =  where (a)  F(x)  cp  RxN  exists  a.e.  = 0  with.  Cp e R ,  and__(iii)—  Lp-norm, i f  p < co and  convergence  a function  N  — •  (*)  for  cp e C a r p ( T )  if  a null V  • r  satisfy  set  in  "  the form  T .  cp c C a r p ( T )  defines "  s.t.  x e LP(T)  and i s u n i q u e up t o s e t s o f  Conversely, every  to  bounded a . e .  = - C „ + J cpox dp T  c a n be^ t a k e n t o  cp(0,*)  L°° n o r m on  .  Then t h e r e  (*)  to  L°°(T) .  continuous r e l a t i v e  CO  whenever  a.e,  each bounded subset of (iii)  = C^,.  • s a t i s f y i n g - (a)  a functional satisfying ' -  .  —  ' and (i),  _ | for (ii),  91.  The above results.extend and the proof for  p = »  to a - f i n i t e measure spaces  i s as i t i s and for  v a l i d i f the phrase "bounded subset of bounded subset of  L°°(T)  p < <» i t i s '•'  l/°(T)"  i s replaced by-  which i s supported by a set of f i n i t e  measure.  7.17 Theorem. and i e t S  Let  A ' LP(T)  T = (T,T,,|a) C(S)  i s a compact•Hausdorff  be a f i n i t e measure space  1 <_'p~ <_ » space.  be a transformation where  Suppose  A  satisfies  the  conditions (ia)  (ila)  A(x-t-y) = A(x)  -i- A(y)  whenever  A  is. continuous r e l a t i v e to  p = cs . cp : S -  (**)  Car p (T)  L p norm i f  p < »  and •  convergence i f  s.t.  A(x)(s) = f cpox. d|_' T  cp(-s)oO = 0 a.e.  for  each  s  in  T .  Moreover,  V  L°° norm on',  Then there exists a transformation  The transformation  (b)  a.e.  L C °('T). .  i s continuous w . r . t . bounded a.e.  (a)  = 0 .  A' i s uniformly continuous r e l a t i v e to . each bounded subset of  (ilia)  x.y  V  cp  can be taken to  s e S , i n which case  up to- sets- of the form  RxN . with  cp i s unique N  a null  cp "has the following additional  The mapping -s - c p ( s ) ° x e L 1 ( T ) "Is x e LP(T) .  satisfy  set  properties:  weakly continuous  92. (c) to  The m a p p i n g  x — "cp(s)<>x  i s uniformly continuous  L ° norm'on e a c h b o u n d e d s u b s e t o f c  (d)  The m a p p i n g  uniformly i n  lim P (cp(s)ox !-i(E)-0 E ' if  p  =  L °(.T) , u n i f o r m l y i n ^s. C  x -• cp ( s ) »x . i s w e a k l y  s , i f p < <» '/'If  )dp -» 0  x  n  continuous'-on  -• x  }  satisfying  means o f (**)  a  uniformly in. s  P  satisfying  n  spaces.  p  determines, by  C(S)  ( i a ) , ( i i a ) and  For  cp : s -»'Car (T)  ( a ) , ( b ) , '(c) a n d '(d)  The ab.pve r e s u l t  E.  and  then  transformation  A : L (T) -  (e)  p  o .  <_ p <_ OD  is  L '(T)'  " b o u n d e d l y . a. e.  Conversely every transformation 1  relative  (iiia)..  a l s o extends  to  cr-finite  measure  p = <» , i t i s v a l i d i f t h e - f o l l o w i n g c o n d i t i o n  added. If ^ <J> ,  x  -* x J  i  boundedly  a . e . , t h e n f o r any s e q u e n c e  (cp(s)ox ) d u - 0  uniformly i n  s  and  'n .  E.  i  For o f . L°°(T) "  p < co , I t i s v a l i d i f t h e p h r a s e  "bounded  i s , , r e p l a c e d b y "a s e t o f f i n i t e m e a s u r e " .  subset I  The p r o o f i n Theorem 7 . 1 5 . u t i l i z e s t h e Lemma 7 . 1 4 and t h e r e p r e s e n t a t i o n •(*) i s t h e n e s t a b l i s h e d b y u s e o f t h e Vitali  convergence. The c o n v e r s e u t i l i z e s  [12]  t h e Lemma 7 - 4 b y N e m y t s k i i ' i n  a n d B a n a c h - S a k s Theorem. The l a s t  theorem, 7 - 1 7 , u t i l i z e s  Theorem 7 - 1 5 a n d  V i t a l i - H a h n - S a k s Theorem o n c o n v e r g e n c e . o f m e a s u r e s .  9^ i• BIBLIOGRAPHY  [1]  A. D. M a r t i n and  [2]  and V. J . M i z e l , A r c h i v e f o r R a t i o n a l Mechanics  15, No.  Analysis, Vol.  5,  V. J . M i z e l and K. Sundaresan, A r c h i v e f o r R a t i o n a l  30, No. 2. pp. 102-126  Mechanics and A n a l y s i s , V o l . [3]  r  N. Friedman and M. K a t z , : " A d d i t i v e f u n c t i o n a l s on LP •  [4]  pp. 353-367-  s p a c e s " , Canad. J . Math.  18 (1966), pp. 1264-1271.  :  R. V. Chacon and N. Friedman, " A d d i t i v e f u n c t i o n a l s " , ; A r c h i v e f o r R a t i o n a l Mechanics and A n a l y s i s ,  Vol.^18,  No. 3, pp. 230-240. [5]  N. Dunfo'rd and. J . T. S c h w a r t z , L i n e a r O p e r a t o r s , P a r t I , New Y o r k , I n t e r s c i e n c e .  [6]  E. Hewit and K. Stromberg, R e a l and A b s t r a c t S p r i n g e r V e r l a g , New Y o r k ,  [7]  M.. M. Day, The spaces L  p  Analysis,  1965-  w i t h 0 < p < l y B u l l . Amer.  • "  Math. Soc. 46. (1940), pp.- 816-823. [8]  V. J . M i z e l , R e p r e s e n t a t i o n o f n o n l i n e a r on L  p  spaces,  transformations  Bull... Amer. Math. S o c , V o l . 75, No. 1,  (1969), PP- 164-168. [9]  D. H.. Heyers, A n o t e on l i n e a r t o p o l o g i c a l spaces, B u l l . Amer. Math. S o c , V o l . 44,  [lO]  (1938), pp. 76-80.  J . V. Wehausen, T r a n s f o r m a t i o n I n l i n e a r t o p o l o g i c a l ' ' spaces, Duke Math. J.' V o l . 4  (1938), pp. 157-169-  [11]  M. A. • K r a s n o s e l ' . s k i i , T o p o l o g i c a l methods i n t h e t h e o r y of n o n l i n e a r • i n t e g r a l e q u a t i o n s , . t r a n s l a t e d by J . B u r l a k , M a c m i l l a n , New York, 1962. pp. 20-32.  [12]  V. V . . N e m y t s k i i , E x i s t e n c e nonlinear  and u n i q u e n e s s theorems f o r  i n t e g r a l e q u a t i o n s , Mat. Sb. 4l, No. 3,  (1934). [13]  A. C. Zaanen, " I n t e g r a t i o n " , N o r t h H o l l a n d Pub. Coy. Amsterdam,  1967.  

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