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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. R E P R E S E N T A T I O N OP A D D I T I V E AND B I A D D I T I V E NONLINEAR F U N C T I O N A L S by P R I T A M SINGH A U L A K H B . A . , M „ A . , P a n j a b . U n i v e r s i t y , i960, 1962. A T H E S I S SUBMITTED I N P A R T I A L F U L F I L M E N T OF T H E REQUIREMENTS FOR T H E DEGREE O F MASTER OF ARTS i n t h e D e p a r t m e n t o f MATHEMATICS We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d T H E U N I V E R S I T Y O F B R I T I S H COLUMBIA A p r i l 1970 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced deg ree a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r ee t h a t t he L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r ag ree t ha p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y pu rpo se s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h Co l umb i a Vancouve r 8, Canada Date Mg^ \±J ; ? 7° S u p e r v i s o r : E . E . G r a n i r e r . A B S T R A C T I n t h i s t h e s i s we a r e c o n c e r n e d w i t h o b t a i n i n g a n i n t e g r a l r e p r e s e n t a t i o n o f a c l a s s o f n o n l i n e a r a d d i t i v e a n d b i a d d i t i v e f u n c t i o n a l s on f u n c t i o n s p a c e s o f m e a s u r a b l e f u n c t i o n s a n d o n L ^ - s p a c e s , p > 0 . T h e a s s o c i a t e d m e a s u r e s p a c e i s e s s e n t i a l l y a t o m - f r e e f i n i t e a n d a - f i n i t e . A l s o we a r e c o n c e r n e d t o t h e e x t e n d t h e p r e s e n c e o f a toms i n a m e a s u r e s p a c e c o m p l i c a t e s t h e r e p r e s e n t a t i o n t h e o r y f o r f u n c t i o n a l s o f t h e t y p e u n d e r c o n s i d e r a t i o n h e r e . A c l a s s ; o f n o n l i n e a r t r a n s f o r m a t i o n s o n L / ^ - s p a c e s , 1 < p < « , ; c a l l e d U r y s . o h n o p e r a t o r s . [11 ] t a k i n g m e a s u r a b l e f u n c t i o n s t o m e a s u r a b l e f u n c t i o n s i s s t u d i e d a n d we d e s c r i b e a n i n t e g r a l r e p r e s e n t a t i o n f o r t h i s c l a s s when t h e a s s o c i a t e d m e a s u r e s p a c e i s .an a r b i t r a r y 0 - f i n i t e m e a s u r e s p a c e a n d t h i s c h a r a c t e r i z a t i o n : e x t e n d s o u r p r e v i o u s r e s u l t s w h e r e t h e m e a s u r e s p a c e c o n s i d e r e d was a t o m - f r e e . T A B L E OF CONTENTS S E C T I O N 1: S E C T I O N 2: INTRODUCTION P R E L I M I N A R I E S S E C T I O N 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 THE VECTOR S P A C E OP R E A L V A L U E D M E A S U R A B L E FUNCTIONS ' S E C T I O N 4: R E P R E S E N T A T I O N OF A D D I T I V E F U N C T I O N A L S ON L P - S P A C E S S E C T I O N 5: EXAMPLES AND COUNTER EXAMPLES ON R E P R E S E N T A T I O N OF A D D I T I V E FUNCTIONALS S E C T I O N 6: R E P R E S E N T A T I O N O F B I A D D I T I V E F U N C T I O N A L S S E C T I O N 7: R E P R E S E N T A T I O N OP NONLINEAR TRANSFORMATIONS ON L P - S P A C E S B I B L I O G R A P H Y ACKNOWLEDGEMENTS I am thankful to my supervisor, Dr. E.E. .Granirer fo r introducing me to this topic and fo r his generous and valuable assistance during the preparation of t h i s 1 t h e s i s . I would also l i k e to thank Dr. Lee Erlebach f o r his nice suggestions. I am grateful to the University of B r i t i s h Columbia for the f i n a n c i a l support. Last, but not l e a s t , I. wish to thank Miss S a l l y Bate for typing this thesis. S E C T I O N 1 INTRODUCTION Some r e s u l t s o f A . D . M a r t i n and V . J . M i z e l [ l ] o n a n i n t e g r a 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 a d d i t i v e f u n c t i o n a l s d e f i n e d o n v e c t o r s p a c e s o f r e a l - v a l u e d m e a s u r a b l e f u n c t i o n s h a v e b e e n e x t e n d e d b y R. V . C h a c o n and F r i e d m a n [4]; F r i e d m a n a n d M . K a t z e '[J>]'3 a n d b y V . J . M i z e l [8], We s h a l l g i v e a u n i f i e d a c c o u n t o f t h e s e r e s u l t s a n d s h a l l make p r e c i s e some o f t h e r e s u l t s a n d s h a l l make some g e n e r a l i z a t i o n s . I n S e c t i o n 2 we g i v e t h e t e r m i n o l o g y a n d n o t a t i o n s o f m e a s u r e t h e o r y . A n a t o m i n a m e a s u r e s p a c e i s a l s o d e f i n e d a n d we p r o v e some t h e o r e m s t h a t we w i l l n e e d l a t e r o n . T h e e s s e n t i a l p a r t o f S e c t i o n J> o b t a i n s t h e r e s u l t s o f [ l ] i n t h i s m o r e g e n e r a l s e t t i n g a n d we a l s o p r o v e n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s f o r t h e i n t e g r a 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 a d d i t i v e f u n c t i o n a l s u n d e r v a r i o u s c o n t i n u i t y c o n -d i t i o n s . / I n S e c t i o n 4, we c o n s t r u c t t h e i n t e g r a l . r e p r e s e n t a t i o n o f an a d d i t i v e f u n c t i o n a l o n L ( ^ - s p a c e s f o r p > 0 a n d X T t h e n we e s t a b l i s h some n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n s f o r t h e i n t e g r a 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 a d d i t i v e f u n c t i o n a l s o n L - s p a c e s f o r 1 <_ p <_ <» . I n S e c t i o n 5> we g i v e some e x a m p l e s on i n t e g r a l representations and i n Section 6 we esta b l i s h analogous i n t e g r a l representations of nonlinear b i a d d i t i v e functionals. More p r e c i s e l y we e s t a b l i s h necessary and s u f f i c i e n t conditions f o r a. b i a d d i t i v e functional F defined on the Product. X-^ xXg of prescribed subspaces c , c M 2 to permit an i n t e g r a l representation where M^ and M^ denote vector spaces of real-valued e s s e n t i a l l y bounded measurable functions on (X,E,ia). In Section 7 we describe i n t e g r a l representations f o r a class of nonlinear functionals and nonlinear transformations p on the spaces L (X) y (1 <_ p £ «) associated with an arbitrary^ : • i a - f i n i t e measure space (X,S,|i) . The class of functionals considered here d i f f e r s from those considered i n [ l ] , [2], [3] and [4] and i t s study i s mainly motivated by i t s close connection with nonlinear i n t e g r a l equations i n [11]. Our characterization extends e a r l i e r results i n [l ] and [ 2 ] , / SECTION 2 PRELIMINARIES. We assume here some of the usual axioms of set theory and we use [ l ] , [ 2 ] , [3]> [4] and [8] as standard references except f o r some notations and d e f i n i t i o n s which we' give below. Throughout this thesis we w i l l use the following abbreviations " i f f " f or the phrase , r i f and only i f " "V" instead of "For every" " 3 " for "There e x i s t s " "e" f o r "belongs to" "w.l.o.g." f o r • "without loss of generality" "w.r.t." for I "with respect t o " a n d " s . t . " f o r "such that" . We w i l l use the following convections 2.1 (J>- denotes . the empty set. 2.2 R denotes the r e a l l i n e . 2 . 3 R* = R U-{»} U {-»} the extended re a l l i n e . 2 . 4 I = { 1 , 2 , 3 , } 1 2.5 A~B = {x e A and x £ B} • 2.6 ^ x i ^ i = l =' t xi> x2>--'» xn)-2.7 ^ x i ^ i e J = ^ x i 9 i € J} where J "denotes an index sc ' ^ \ ' s • . . . . . . Let X be any set. 2l P(X) = a l l subsets of X .' A denotes the closure of A , A <= X . 1 A~ = X~A . Let f be a function and A be a set. ' 1 2.11 f[A] = {y": y = f(x) f o r some x e A} 2.12 f _ 1 [ A ] = {x : y = f(x) f o r some y e A} Unless otherwise mentioned, i n this section, X w i l l be an ar b i t r a r y nonempty class with elements x,y,.... and E,P,A,B,.... . w i l l be subsets of X . 2.13 A. nonempty class S of subsets of X i s c a l l e d a r i n g i f EUP , E-F e T, whenever E,F e E and i s c a l l e d a CD a-ring i f U E € S whenever E e S V n e I . n=l n n 2.14 D e f i n i t i o n . A ring Y, of subsets of X i s c a l l e d an algebra i f X € -T, and a a- r i n g . T, of subsets of X i s c a l l e d a cr-algebra i f X e S . 7 2.15 Remark. Let £ be an a r b i t r a r y family of subsets of X . Let denote the i n t e r s e c t i o n of a l l a-algebras of subsets of X that contain 6 . . §(£) " i s also a a-algebra and i s the smallest cx-algebra of subsets of X containing £ . 2.8 2.9 2.10 2.16 D e f i n i t i o n . I f X i s a topological space, l e t B(X) b the smallest a-algebra of subsets of X that contains every open set. Then the members of (R(X) are c a l l e d the; Borel se of X . 2.17 D e f i n i t i o n . I f » c P(x) , a set function cp on H 'l±£ a function on M to R* written as cp : # -* R* i s a d d i t i | i f V A,B e ii , ADB = «J> , we have u(AUB) = u'(A) + \x(B) wheri W i s any family of subsets of X . 2.19 D e f i n i t i o n . An additive set function p : P(X) - [0,«] i s c a l l e d an outer measure i f the following hold: . ( i ) n (<|>)- = 0 and I ( i i ) u(E) < E n(E-t) whenever E c U E. c X , i e l 1 i e l 1 and |a i s c a l l e d a measure i f f ( i ) |i((j>) = 0 and ( i i ) n(E) S |i(E. ) whenever E = U E, .<=X and i e l . 1 / l e i 1 Ej_ n Ej = > , i 4= j . 2.20 D e f i n i t i o n . We w i l l use caratheodory outer measure \i f o r which we have: A set A c j i s immeasurable i f f V . E c X , |i(E) = |i(ETlA) + ^ ( E n A ~ ) . 2.21 D e f i n i t i o n . I f |i i s a measure on X , then we denote^ by M the set (A : A c X and A i s (^-measurable]. 2.22 D e f i n i t i o n . I f f o r every A £ M , |i(A) < « then u i s ca l l e d a f i n i t e measure. ' 2.23 D e f i n i t i o n . M i s c a l l e d a - f i n i t e i f V . A e M , 3 a, sequence { ^ ^ 1 o f s e f c s l n M , j s u c h t h a t A c ^ -^ n a n d \ i i ( A n ) .< co v n e I . J 2.24 D e f i n i t i o n . Let A c X then the measure H A(E) = . u ( A n E ^ on X i s c a l l e d the, r e s t r i c t i o n of |a by A . In future we w i l l take X = (X,S,|i) as a measure space where X Is a nonempty class of elements and E i s a nonempty class of measurable subsets of X which i s a a-algebra and u i s a measure on. ( X , E ) . 2.25 D e f i n i t i o n . Let (X,5Vu) / be a measure space. A e S i s c a l l e d an atom i f u(A) ={= 0 and i f B e E , B C A then| either u(B) = 0 or u(B") = u(A) . A space ( X , E , u ) w i l l be c a l l e d an atomic space i f every subset of X that belongs to S i s an atom. "N 2.26 Lemma. Let (X,S,|i) be a f i n i t e measure space and E e E 7. 0 < n(E) < » s.t. neither E nor any of i t s u-measurable sub-sets i s an atom, then E contains 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 does not contain any atoms, there exists F c E , ' F € S s.t. E = P U (E-F) and w(P') > 0 , n(E-F) > 0 Hence one of F and E-F , c a l l i t F.^  , s a t i s f i e s u(F^) _< •||i(E) . Now since F^ i s not an atom, continuing with the above method fo•decomposition, i t follows that E has subsets of a r b i t r a r i l y small p o s i t i v e measure. 2.27 Lemma. If (X,E,|_i) i s a f i n i t e measure space then there 00 i s a countable family {A.} of atoms'of X s.t. i f A = U A. then C = X~A i s atom free and X = AUC i s a decomposition of X into atomic and atom-free parts. Proof: Since X i s a f i n i t e measure space, any atom A c X i s of f i n i t e measure. By i d e n t i f y i n g the u-almost equal atoms we get that the.different atoms are d i s j o i n t . So the t o t a l . > number of d i f f e r e n t atoms contained i n . X i s at most countable 00 say An,A_,... . Let A = U A. and C = X~A then c l e a r l y C i s atom-free and the decomposition i s n-almost unique. 2.28 D e f i n i t i o n . x""= (X,S,u) has the strong interdediate value property "If V S e £ and V re a l number a , 0 <_ a <_ |a(S) 3 A e E , A cr S s.t. \x{k) = a . 2.29 Theorem. A measure space X has the strong Intermediate value property I f f I t i s atom free. [For Proof, see "Set Functions", by Hahn and Rosenthal Chapter 1, § 5 . 6 . ' ] •' 2.30 D e f i n i t i o n . (X,E , u ) . has the weak intermediate value | property i f f V r e a l number a , 0 <_ a <_ |i(X) ,3 A e E , A c X , s.t. u(A) = a . 2.31 Theorem. (Z. Nehari). Let (X,E,|-i) be a countable discrete measure space. Let — m 2 — * * * ^ e ^ e m e a s u r e s °? atoms of X . Then ( X , S , n ) has the weak intermediate value property CO ! i f f • m_ < E m, , n = 1 , 2 , 3 , . ' . . •. ! ^ ~k=n+l * . _ • ^ 2.32 D e f i n i t i o n . Let (X,E,n) be a measure space, then j f : X - R* i s measurable i f f V c e R* , {x : f (x). < c} e E / 2.33 D e f i n i t i o n , / x ^ = {Q X f E i s c a l l e d the character-i s t i c function. ' 2.34 Theorem. x E i s measurable i f f E e E . D e f i n i t i o n . A f u n c t i o n f : X R* w h i c h has o n l y a f i n i t e s e t c ^ , c 2 , . . . , c n o f v a l u e s and-''for w h i c h f ~ 1 ( c i ) = j. (s : s e X , f C s ) =,c^} e E i s c a l l e d a s i m p l e f u n c t i o n . 2.35 Lemma. Let f : X -* R* be measurable, then there exist a sequence {f } of simple functions s.t. f f . I f f > 2.36 D e f i n i t i o n . Two r e a l valued functions f,g are equi-measurable i f V Borel set S on (-«.,'») , f - 1 ( S ) and g - 1 ( S ) are measurable and have equal measure. •2.37 D e f i n i t i o n . Let (X,E,|j) be a f i n i t e measure space. j Let f be a bounded measurable function with i n f f(x) = m , n+1 \i V n [For proof see Taylor ] x sup f(x) = M . x n §(A) £ y S(A) mu(X) <_ §(A) </ S(A) <_ MQ|i(X) . Then sup §(A) = i n f S(A) = f f"d(j = f-f"du . ; A A X Before Nwe f i n i s h this section, we prove some lemmas which we s h a l l need l a t e r on i n Chapter 5 and fo r proofs we s h a l l follow [1]. 1 0 2.38 Lemma. Let m^ ,m2,m^ ,... be a sequence of p o s i t i v e 00 numbers s.t. n i > 2 E m , , n = l , 2 , . . . 1 1 . k=n+l K Let o'^,c'2,... and d^d^d^,... be two sequences of r e a l numbers having values - 1 , 0 or 1 . Then |. 00 CO ' ] E c m = E d.m. only i f c. = d • , "i = 1 ,2 , . . . 1 i = l 1 1 i = l 1 1 1 1 Proof: Let be the set of Integers s.t. c j _ _ d ^ > 0 for i e N-j_ and N 2 be the set of integers s.t. dj_~Cj_ > 0 for i e N 2 • We claim that = N 2 = <|> ... '; Suppose not. "i.e. N^UN2 4= <}> • Let I Q be the smallest integer i n . J^UNg . Thus i e ^  or i Q e N 2 but not to both. Suppose i e Nn . Now i f i e N, and j € N 0 , ;f then .  c - d . and d .-c . .are either 2" or 1 . Thus E' (c.-d.)m. > m. > 2 E m. > 2 E (d.-c.)m. j c N ^ 1 1 1 _ 1 Q j = i o + l J - J €N 2. J J J i . e . E (c.-d.)m. > E (d.-c/)m. . CO 00 But by hypothesis we have that E cm. = E d.m. . i = l 1 1 i = l 1 1 which implies that E (c.-d.)m. = E (d.-c.)m. which i s a ieN^ 1 1 1 JeN 2 J J J contradiction to N^UN2 ={=<(>/.' Hence., = N g =,<j> . 11. 2.39 Lemma. Let t n ^ } ' ^ ' e a sequence o f p o s i t i v e numbers 09 s.t. f o r n > n >_ 1 3 m. > 2 E m. . Let V be the vector • ~ ° • • , i=n+l 1 | space of a l l r e a l sequences s : s^,s 2,... such that eo E s.m. < « . Let S be the subspace of V consisting of i = l 1 1 1 those sequences s s.t. E s.m. = E s.m. whenever i e l 1 1 j e J J J E m. = E m. . i e l 1 j e J J Then the algebraic dimension of S is- i n f i n i t e . Proof: Let H = {(I,J), IflJ = <j> , I and J are subsets of the integers} . (a) s. t. E m. '= E m. . l e i 1 ' j e J J For a p o s i t i v e number n s.t. l < n < n , (a) can •f o — o — ; I be written uniquely as n • O . 8 0 (b) E c.m. = E c.m. where c. has the value - 1 , 0 i = l 1 1 i=n +1 1 1 • o •or 1 , 1 = 1,2,... . The r i g h t hand side of (b) i s uniquely determined, i f i t i s determined, when the l e f t hand side i s given. Thus there are a f i n i t e number of relations (b) which are v a l i d as i s the number of re l a t i o n s of (a) which proves that H i s f i n f t e . (c) Consider the system of equations E x.m. = E x .m. " •• . i e l • j e J 3 J 12. (I,J) € II . where the unknown sequence x : x^x^,... i s a member of S . The system (c) i s f i n i t e since H i s f i n i t e . So suppose that (c) has exactly k equations where k can be zero also. In any case i f K i s a subset of the integers con-taining exactly k+1 members then there i s a nonzero solution of (c) whose support i s a subset of K Now. i f ' £K.j^i>i i s a sequence of d i s j o i n t subsets of k+1 integers each, then S^- ,S„. ,.... i s an i n f i n i t e family of A l *2 independent solutions;of (c) . Hence dim S = » . i '• • •' ' . I ' • / X SECTION 3 REPRESENTATION OF ADDITIVE FUNCTIONALS ON THE VECTOR SPACES OF REAL-VALUED MEASURABLE FUNCTIONS Let (XjEjii) be a measurable space with E a a-algebra of subsets of X . 3.1 D e f i n i t i o n . Let f,g : X - R* , then we say f (x). = g(x)' a.e. i f [x : f(x) ={= g(x)} e E and has u-measure zero. . 3.2 D e f i n i t i o n . A function f Is said to be e s s e n t i a l l y bounded i f a a f i n i t e , p o s i t i v e constant C s.t. u{x : |f(x ) | > C} =0' i . e . f i s bounded a.e . 3-3 D e f i n i t i o n . A sequence o f f u n c t i o n s which are f i n i t e a.e are said to converge a.e to a function f(x) which i s f i n i t e a.e. i f E = [x : f n ( x ) / f(x)} c E and u (E) = 0 3.4 D e f i n i t i o n . A sequence • ( ^ n ^ n - l of measurable functions i s ^ s a i d "to converge i n measure to a measurable function f |jLfj V 6 > 0 we have l i m u(tx e X : | f n ( x ) - f (x) | >_ 6})| = 0 ;and n-««' u ' we write f - f . Example; There xexist sequences of functions that converge i n measure and do not converge a.e. Let X = [0,1] and E be a l l Lebesgue measurable subsets of X . For each integer n € I , define f_ = Xr ;> j + l i where ri = 2 k + j , 0 < J < 2 k Then \{{x : | f n ( x ) | > 6} ) <_ - 0 as n - V.- 16 > 0 '. Thus f H 0 . But on the other hand, i f x e [0,1] , the sequence ( f n ( x ) } converges nowhere on [0,1] . Throughout this section, we denote by M the vector space of real-valued measurable functions on (X,E,u) where two functions are i d e n t i c a l when they are equal a.e. 3.6 D e f i n i t i o n . A real-valued function F on a subspace of M i s c a l l e d an additive functional i f i i i ( i ) F(x+y) = F(x) + F(y) f o r x,y e M s.t. • • •. • i u{supp x Tl supp y} = 0 1 l ( i i ) F(.x) =-F(y) i f x,y are equimeasurable functions i . e . i f for every Borel set B i n R , u(x~ 1(B)) « n(y- 1(B)) . In this section we w i l l construct an i n t e g r a l representation of a nonlinear additive functional F defined on d i f f e r e n t subspaces of M under various continuity con-ditions on F when the underlying measure"space (X,E,u) i s i atom free. • • • - 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 real-valued measurable functions}. ; 3.7 D e f i n i t i o n . A set function cp" i s said to be u-absolutely continuous i f given e > 0 3 6 > 0 s.t. for E e E . , u(E) < 6 | c p ( E ) I < € . For the following theorem, we w i l l ; follow the methods given [1] and-we w i l l use n x n -> x boundedly a.e" instead of sayingthat x n -» x a.e and there exists a p o s i t i v e constant c s.t. |x^| <_ c and |x| <_ c . 3.8 Theorem. Let ( , X , E , | - t ) be a f i n i t e atom-free measure space fo r which | j ( X ) '={= 0 . I f an additive functional F : B - R * • s a t i s f i e s the condition: ( l ) x n -• x boundedly a.e implies F ( x n ) F(x) . Then there e x i s t s ; a unique. continuous function f : R - R s.t. f(0) = 0 and V x e B . , • " -\ F(x) = J* (f.x)dn. (*) X • / Proof: (a) Let C e B be the constant function' a for ! . CI a e (-eojco) . Define f(a) =11^1 , a € (-»,») . I f ' a - a . U(X) n (reals) then C „ x - C boundedly, so F(C ) -F(C ) by con-n ^ . t i n u i t y of F , x Thus f ( a n ) - f(a) i . e . f i s continuous. 1 0 Now supp C Q = <p . Thus v i f x e B then supp x n supp C Q = <j> which implies that F(x) = F ( C Q + x) = F(C Q) + F(x) i . e . F(C Q) = 0 and thus f(0) = 0 . I f there i s an f s a t i s f y i n g this theorem then i t i s unique since f o r a =(= 0 we have F(C ) = J* . f«C a du = f(a)-u(X) . X Now i t remains to show that F(x) =. I" f ( x ( t ) ) d ( u t ) V x e B . X If S € Z , f o r fix e d a e (-»,») , define cp(S) = cp a(S) = F(axs) cp i s f i n i t e valued set function V S e E and s a t i s f i e s : (b) cp i s additive on S . ^ For i f S13S2 e E and S^Sp = <p then 9 ( S l U S 2 ) = F ( a . X S i U S 2 ) = F(a.xSi + a.^J = F(a ,x s ) + F(a,yq ) ( supp av q fl supp ay- = (p)-1 ^2 • b 2 X -= CP(S-L) + cp(S 2) . ' IT. ( c ) cp i s c o u n t a b l y a d d i t i v e o n E . I f S e £ s . t . S I; cp t h e n a,x„" - 0 b o u n d e d l y so i • . • n cp(S n ) = F(a,xs ) F ( C ) = 0 w h i c h i m p l i e s t h a t cp i s c o u n t -n a b l y a d d i t i v e o n £ (<d) cp i s u - a b s o l u t e l y c o n t i n u o u s . S i n c e | i ( X ) < • , i t s u f f i c e s t o show t h a t cp(S) = 0 w h e n e v e r | i ( S ) = 0 . I f u ( S ) = 0 t h e n s i n c e aXg. and C o a r e e q u i -m e a s u r a b l e , . i t f o l l o w s t h a t cp(S) = F ( a x s ) = F ( C D ) = 0 . m,„s P ( a y R ) ' (e) J(^ T = U( S) i s d e f i n e d V S e £ w i t h n ( S ) ± 0 a n d . i s a r e a l - v a l u e d f u n c t i o n p o n ( 0 , ( i ( X ) ] s . t . cp(S) = p ( n ( S ) ) u ( S ) a n d s 'p(s) + r p ( r ' ) = ( s + r ).p( s + r ) f o r r , s , r+s e ( 0 , u ( X ) ] . < I n f a c t l e t " 0 < s <_ u ( X ) a n d C ( s ) = {S e £ : u ( S ) = s} . S i n c e X i s a t o m - f r e e , C ( s ) =j= cf> • I f S , R e C ( s ) , t h e n a,Xg a n d e,xR a r e e q u i m e a s u r a b l so F ( a X g ) = F ( a x R ) . H e n c e q>(S) F'<aV cp(R) ' M > , \ ' ' cp(S) mr= ~wsr = = 5w w h l c h i m P l i e s t h a t fEtfx. . i s i n d e p e n d e n t "of S e C ( s ) , a n d h e n c e i f ^ = -u^r = w = p - ( s ) ' S € C ( s ) w h i c h 18. proves that ^ j s j a r e a l ~ v a l u e d function of p on j (0,u(X)] . To prove the l a s t equality, we have that X i s atom-free, there i s an S e E with u(S) = s . Since 0 < s+r <_ |i(X) , r <_ |j(X) - s = |i(X- S)- . Again since the space i s atom-free, 3 R e S , R c r X - S s . t . u(R) = r and since SflR = <J> , i t follows that cp(SUR) = cp(S) + cp(R) and \ since cp(S) = p(|i(S) )ji(S) and u(RUS) = |i(R) + i±(S) = r + s '< i we have p ( u ( S U R ) ) u ( S U R ) = p ( u ( S ) ) u ( S ) + p ( | i ( R ) ) u ( R ) , and so (r + s ) p ( r + s) = s p(s) + r p ( r ) . (f) p i s continuous. For this l e t { S n l be a monotonic decreasing sequence of re a l numbers i n (0,u(X)] with s n "* s • Since X i s atom-free 3 e S with vi(S 1). = s 1 and Sg S , S g c s 1 with « n(S 2) = s 2 <_ s 1 .. .' . Therefore there exists'a monotonic sequence S n , S n e E with u ( S n ) = s n and ... S n c S ^ c ... c ^ . 00 Le.t S = " n S_ . Since u(S, ) < « i t follows that n=l n ' 1 i I i u(S) = l i m |a(S )r = l i m s = s > 0 and since cp i s countably! n n •! additive, j X 9(S n) - cp(S) so l l m p ( s n ) - i * . f ^ . ..- p(.) . 1 9 . A si m i l a r argument shows that P ( S N ) "* P( S) when s n increases to s . Thus p i s continuous. ; (g). p i s constant. Let s € (0,|i(X)] . Since X i s atom-free 3 X c X , s.t. - P i n-1 -s a(X ) = s . Furthermore there i s Xn c X s.t. u(Xv) = — ^ o , • 1 o ^ v 1' n and so -on, 3 X i c X i_ 1 with \i(X^) = s . I t follows that i f S. = X. , - X. then S. € S ar;e-1 i - 1 i i . i d i s j o i n t and u(Sj_) = ^ ( X i _ 1 - X ±) = nU^) - n(X±) = I 1 = H ± l s ~ ^T s = H > i ' e * =1 f o r i € I and S ^ S j = (J) i f i ± j . But since (r + s)p(r + s) = rp(r) + sp(s) we have sp(s) = (£ + | + .... n times )p(|- + £ +|'+ ... + 1) s«/rSv,s„/'s,\, ,s«/'s,\ i . e . sp(s) = n. . ^  p(^) = SP(|[) and since s ^ 0 we have P(s) = P(|) • / I f now ^ s e (0,n(X)] where m and n are non-negative integers, (n ^ 0) then n s p ( H s ) = n p ( n } + n p ( H } + ' • • + n p ( E } ' m t e r m s . . . X i . e . p(^ s) = p(^-) = p(s) , - a n d s i n c e p i s . c o n t i n u o u s , we g e t t h a t p i s . c o n s t a n t . 2 0 . (h) p = P a = f(a) V • a e (-»,«) . P ( a x x ) ' Since f(a) = ^ ) • > P a ( ^ ( x ) ) = P a ( u ( s ) ) , s e E . We have p = p a = f(a) Y a € (-«,«) . (i ) I f x e. B and i f x i s a simple function then P ( X ) = ; f ( x ( t ) ) d u ( t ) X Let S-^aSg,. .., S n be a p a r t i t i o n of X into measurable sub-sets and x2.iX2s' ' ' > xn ^ e ^ n e v a-"- u e s w n i c n x assumes on these subsets. Thus x = E x^Xg and since X-s.Xg have mutually disjoint.supports, this implies that P(x) = E F(x x s ) = E % (S ) V . S P ^ ( S )) iJ(S ) i = l 1 i = l x i 1 n t S ^ O x i 1 1 = E f ( x i ) n ( S i ) = J* f ( x ( t ) ) d |i(t). . . Thus the theorem Is true f o r P defined on simple ' - / ' functions. (j) Now V x e B , we show that P(x) '= f f ( x ( t ) ) d n(t) . X Since V x e B , there exists a sequence ( x n5 of simple functions s.t. x n -» x boundedly a.e. and since F l i s con- ' tinuous F ( x n ) F( x.) • Thus by' Lebesgue dominated convergence theorem, we have F(x)'"'= J f ( x n ( t ) ) d n(t) = Jf ('x(t) )d|i(t) . I Q.E.D. 21, From now onward we will'.prove the necessary and s u f f i c i e n t conditions that an additive functional F defined on a prescribed subspace B c M permits a representation of the form, F(x) = f (f«x)d|ji V x e.X where f : R - R i s X uniquely determined by F . W e w i l l associate these theorems with f i n i t e or a - f i n i t e atom-free measure space (X,E,n) and we s h a l l follow very c l o s e l y [2] for proofs. 3.9 D e f i n i t i o n . Let (X,E,u) be the measure space. For every E e £ , 'we denote the t o t a l v a r i a t i o n of n on E by n v (E) , defined as v..(E) = sup £ | n ( E . ) | where the supremum u • • » 1=1 1 i s taken over a l l f i n i t e sequences (E^) of d i s j o i n t sets i n £ with E^ 'c E . H i s said to be of bounded v a r i a t i o n if;; v ) i(X) < Co and [x i s said to be of bounded v a r i a t i o n on a set E e £ i f v (E) < c . ' •!• ' i; Let f,g : X .-» R , the r e l a t i o n " f - g i s a n u l l function" i s an equivalence r e l a t i o n . Let [f] denote the / class of functions from X -• R which are equivalent to f and l e t P[X] = P(X,£,|j) . denote the set of a l l such sets [f ] . 3.10 D e f i n i t i o n . Let X be any topological space. The t o t a l l y measurable functions on X are the functions i n the closure of simple functions i n F(X) and we s h a l l denote them by TM(X) • •,. 2 2 . I t i s p r o v e d i n D u n f o r d a n d S c h w a r t z i n Lemma I I I . 2 . 1 2 t h a t i f x i s t o t a l l y m e a s u r a b l e f u n c t i o n a n d i f f I s a c o n -. t i n u o u s f u n c t i o n o n R t h e n f o x I s t o t a l l y m e a s u r a b l e f u n c t i o n and a l s o i n I I I . 2 . 1 1 t h a t t h e t o t a l l y m e a s u r a b l e f u n c t i o n s f o r m a c l o s e d l i n e a r s u b s p a c e o f P ( X ) . ' 3 . 1 1 T h e o r e m . L e t ( X , E , ^ ) b e a f i n i t e a t o m - f r e e m e a s u r e s p a c e and l e t P b e a n a d d i t i v e f u n c t i o n a l on B = L (n) . T h e n t h e 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 : ( ( 2 ) x ^ -» x b o u n d e d l y i n m e a s u r e =^ F ( x n ^ "* F ( x ) a n d ( 2 ' ) P ( x ) = f ( f « x ) d n V x e B ( * ) . X w i t h t h e r e p r e s e n t i n g f u n c t i o n f s a t i s f y i n g t h e c o n d i t i o n s t h a t f i s c o n t i n u o u s and f ( o ) = = 0 . P r o o f : L e t -» x b o u n d e d l y a . e . T h e n s i n c e ( X , E , | i ) i s a f i n i t e m e a s u r e s p a c e , -+ x • i n m e a s u r e so F ( x n ) "* F ( x ) > t h e r e f o r e b y T h e p r e m 3.8, a a u n i q u e c o n t i n u o u s f u n c t i o n f , ; • f ( o ) = 0 s . t . / F ( x ) = f f . x d ^ V x € B •X C o n v e r s e l y , s u p p o s e f : R.'-» R i s c o n t i n u o u s a n d f (o . ) = 0 . . ' D e f i n e F o n B = L ^ u ) b y F ( x ) = J ( f . x ) d ( i . X Since f(o) = 0 , Y x,y e B s.t.. u{supp x n supp y} = 0 we have F(x+y) = F(x) + F(y) . • . Now claim that: x n -» x boundedly i n measure implies that F(x n) -» F(x) Y x n,x e B . Let be a sequence i n B s.t. x n -• x i n measure and |x n| <_ c , |x| _< c f o r some p o s i t i v e constant c Since x n.' x a r e t o t a l l y measurable and f i s continuous, i t follows that and f»x are t o t a l l y measurable and f°x n -* f«x i n measure and f ° x n > f°x are bounded. Therefc by Lebesgue'dominated convergence theorem lim F (x n) = l i m J(fox n(t))du(.t) = Jf.x(t)du(t) = F(x) . n n re Now we'take the case of a.e. convergence which i s :ho necessarily bounded and convergence i n measure. 3.12 Theorem. Suppose (X,E,ti) i s a f i n i t e atom-free measure space and. F i s >an additive functional on M . Then the following conditions' are equivalent: (3) x n x a.e. =25? F ^ ) ' -»F(x) and F(x) = f fox d(i (*) X V x e B , wheres f 'satisfies the conditions (a) f i s continuous and f(o) = 0 and (b) Range (f) i s bounded. 24 Proof: ;Suppose F i s an additive functional on M which s a t i s f i e s the given condition and l e t F, = P/L ( u ) . , L ( u ) c. M .1, CO CO and since f o r any sequence t x n} n>]_ i n L c o ( > J ) with • x -• x a.e. and |x^| _< c V n and |x| <_.c where c i s some p o s i t i v e constant, we have by Theorem 3.8 that there exists a unique continuous function f : R -* R s.t. f ( o ) = 0 and P 1(x) = P(x) = J. (f.x)d|i V x e Lj\i) . X To prove (b), suppose that range (f) i s unbounded. There exists a sequence {r n3 i n R s.t. |r f -» » and ! 1 _< | f ( r )| • » . Since the measure space i s atom-free, there exists a sequence {Bn} , B n e E and B n + 1 £ B n J <f> s«t. \ u(x) Let x^ ^ = r^Xg , * n i s measurable and hence x n e M and since B„ 1 & ., x. ' -» 0 a.e. and since x € L , we have n v n n o o x / - 7 ' F(x n) = J(f ox n)d U = f ( r n ) , f U ( ^ } } ,  = + n(X) + 0 F(x n) / F(o) = 0 • which i s a contradiction. Hence range (f) i s bounded. / Now l e t x e M be any function. There exists a sequence (s n} of simple functions s.t. s^ -»,x a.e. and since f i s continuous, f°s^ -• f»x a.e.'where Ifos 1 .. ' n 1 n |fox[ _< c f o r some p o s i t i v e constant c . Thus by Lebesgue convergence theorem, °* d|jt = l i m J*(fos )dn = l i m F(s ) = F(x) n n n Conversely, l e t f sa t i s f y - ( a ) and (b). Define 25. F : M - R by F(x) = J*(f.x)d|j . . F exists and i s real-valued because f i s continuous and range (f) i s bounded; and since f(o) = 0 , we have that F i s additive. Let {xn} e M be a sequence s.t. x n -• x a.e., x e M . Since f i s continuous, ^ o X n > f°x e M and f o}^ -» f»x a.e., by Dominated Convergence Theorem i t . follows that F(x n) - F(x) . Q.E.D. Since convergence a.e. i n a f i n i t e measure space impli convergence i n measure, we have 3-13 Corollary. Let F be an additive functional on M with (X,£,|i) a f i n i t e atom-free measure space. Then F s a t i s f i e s the continuity condition (4): x n ~* x i n measure implies' F(x n) -» F(x) i f f ('*) holds with f s a t i s f y i n g (a) and (b) i n Theorem 3.12. • " (The Proof i s s i m i l a r /to the above theorem. ) If 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. The additive functional F on L (a) or M x . • . 0 9 s a t i s f i e s condition. (3) i n 3.12 or condition (4) i n 3.13 respectively i f f F s 0 . 26. Proof: Case ( i ) . Let A e S be s.t. \ i ( A ) = \i(X - A ) = « . Since ( X , E , u ) i s atom-free 3 A i * A 2 € s * ^DAg = (j) s.t. A 1 U A 2 = A a n d u(Ai) = ^ ( A 2 ^ = 0 0 • I f r i s any constant then rx f l > r X n and r ' . A l ' A2 are equimeasurable. Since rx'fl = rx» , ,A = r x n + rX A and A A-^UAg A-^  A 2 i |i{supp rX f l H supp rx« } = 0 we have, by the a d d i t i v i t y of F , Rl  A2 that F(rX A) = P(r.x'A ) + F ( r X A ) = 2F(rX A) ' (by equimeasurability) and which implies that F(rX A) = 0 . Case ( i i ) . Assume that f o r any A e E , p.(A) < » ) there exists , A-^Ag , A 1 3 A 2 which together with t h e i r complements are of i n f i n i t e measure and A = A^~A2 , then again by the a d d i t i v i t y v of F , we have F(rX A) = 0 when |i(A) < <» / Hence i f x i s a simple function, then F(x) = 0 . Now i f X i s a countable valued function for which each value has support and complement of i n f i n i t e measure, then as above we get that F(x) = 0 . N I' SECTION 4 REPRESENTATION OF ADDITIVE FUNCTIONALS ON L P SPACES 4.1 D e f i n i t i o n . For 0 < p < » , L P(X) = { a l l real-valued measurable functions f on X s.t. || f || = (f |f|Pdu < »] . P X For 1 <_ p < co , L p i s a Banach space but this i s no longer true f o r 0 < p < 1 , for the tr 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 . But instead we have: 4.2 Proposition. Let. 0 < p < 1 and l e t ..X = L [0,1] t h e n (a) | | f + g | | p . < 2 P (|| f | | p + ||g||p) . (b) I f f o r any two functions f,g e X , d(f,g) = J | f - g | p d i i » then : d i s a metric where d(f,g) = 0 =^ f = g a.e. Proof: F i r s t l y we show (1) " •2 P" 1(1+x p)-_> ( l + x ) p f o r 0 < p < 1 Let f(x) = ••2 p" 1(l+x p') = ( l + x ) P and f'(x) = 2 P~ 1.p x P r l - p ( l + x ) p ~ 1 =0 r=5> 2x = 1 + x i . e . x = 1 and f"(x) = 2 p " 1 . p ( p - l ) x p " ? - p ( p - l ) ( l + x ) p - 2 f " ( l ) = p . ( p - l ) [ 2 p - 1 --2 p- 2] < 0 Hence x =: 1 . 'is a maximum point f o r f(x) and f ( l ) = 0 . 28. Thus we have 2 P _ 1 ( l + x p ) >_ ( l + x ) p . 2 ( — - l ) (2) We show that 2 ; ( l + x ) > ( l + x p ) 1 / / p f o r . 0 < p < 1 . . Let g(x) = 2 p ( l + x ) - ( l + x p ) " 1 -1 and 'g'(X) = 2 2 ( p ' l ) - i ( l + x p ) p .p x?" 1 = 0 - i . e . 4 = ( l + x p ) x " p i . e . 4 = x _ p + 1 or x = ( i ) 1 / p . . - 1 2 1 -1 Also' g'(x) = _ [ ( I . i ) ( i + x p ) p .xP" 1 •'•+ ( p - l ) x p - 2 ( l + x p ) p |] j and g ' ^ I ) 1 ^ ) > o . . 1 Hence x = ( i ) 1 / / p i s a minimum point f o r g(x) and we have Now f o r the proof of ;,(a). Assume that |f(x).| ^ 0, , then (2) |f(x) + g ( x ) | p < | f ( x ) | p . (1 +,-[f[f) ) P < | f ( x ) | p . 2 p " 1 (1 + >g ( x ) l P ) |f(x.)| P. by (1) v I ' s P ^ d f C x ) ^ + |g(x)| p) f + g | | P =J >|f(x) + g(x)| p /dn < 2 p - 1 [ J | f ( x ) | p d u + ; | f ( x ) | p d U ] = 2 p - 1 ( | | f ||p +||g|| p) . Suppose that || f || ^ 0 , then 29. i . p-1 .. .. x_ 1 ' .." I' ' f + g |L < 2 p (|| f ||P + ;n g H p) 1 / p = 2 > || f || ( i + JUJ l£) i fp p y y y f .: 1- — '• - 2 ( i - l ) ' II S II ' < 2 p II f|lp • s p • a +11^' fey <2> • 1-1 •• • = 2 P (II f ||p + ||g||p) i - 1 , . l | f + g | | p < 2 P (||f ||p + ||g||p) . (b) If d(f,g) = J|'f-g|pdn , then ( ( i ) d(f,g) >_ 0 and d(f,f) = 0 . ( i i ) d(f,g) = 0 f = g a.e. i; ( i i i ) d(f,g) = d(g,f) . I (Iv) Let f,g,h € X , then d(f,g) = J | f-g | pd|i = J* | f-h+h-• < J 2 p - 1 ( | f - h | p + |h-g|P)dn • (by (1)' < J|f-h|pdn + J|h-g|pdu r= d(f,h) +. d(h,g[)v Hence d i s a metric on X . pdn So, f o r 0 < p < 1 , we have somewhat'.weaker condition - f - 1 I' f l + f 2 Hp - 2 [ " f l Hp + 'I f 2 "p^ a n d W e h a V e t h a t X i s a l i n e a r topological space. I t follows from the theorems of D. H. Hyers [9] and J. V. Wehausen [10] that such a l i n e a r top-o l o g i c a l space \ L P i n which the neighbourhoods of a point f ! ° are spheres of radius 6 > 0 , can be given an equivalent Frechet metric."-' This suggests that-while many theorems on 3.0. Banach spaces which can be applied to the space L P(X) with p >_ 1 may f a i l to hold i n spaces where 0 < p < 1 , there may s t i l l remain many theorems on Frechet spaces and pseudo-normed •spaces which may be applicable. In general i t appears that the class of l i n e a r j functionals i s a subclass of the class of additive functionals. B.ut M.M. Day [7] has shown that i f the underlying measure space i s atom-free then any l i n e a r functional on lP , 0 < p < 1 i s i d e n t i c a l l y zero, which proves that almost no results dependin on the use of l i n e a r functionals can be u s e f u l l y applied to these spaces. In this section we f i r s t l y give the proof of the theorem of N. Friedman and M. Katz [3] which i s the general j! representation theorem f o r an additive functional defined on Lp , p > 0 , which reduces to the standard representation 7 theotrem for l i n e a r functionals when p >_ 1 . ' The methods used i n the proofs w i l l be the same as i n [ 4 ] . / Let M be the vector space of a l l real-valued functions defined on .. X and f o r each f e M there i s defined a number || f || > 0 which may be regarded as a generalized norm. We consider a corresponding space M^" and say that: 4 . 3 D e f i n i t i o n . F..: M - M 1 i s an ADDITIVE TRANSFORMATION i f 31. (1) Continuity: V € > 0 and k > 0 3 6 = 6(k,€) s.t. II f II <'k , II g:-ll 1 k and || f - g || < 6 || F ( f ) ' - P(g)'|| < € for f,g e M . (2) Boundedness: ^ Y k > 0 a H = H(k) s.t. || f || <_ k II P(f) II < H . . i i (3) A d d i t i v i t y : F ( f +g.j = F(f) + F(g) I f f( s ) . g ( s ) = 0 for s € X . \ Let (X,E,ii) be a f i n i t e atom-free measure space and M = L p(X,E,|Ji) , p > 0 . 4.4 Theorem. F i s an additive functional on L i f f . . P F(f) = J K ( f ( s . ) , s ) a ( s ) d u , f e LI X P where ( i ) K(0,s) = 0 -( i i - ) " K(x,'s,) i s a measurable function of s ' Y x . '• i ( i i i ) K(x,s) i s a continuous function of x f o r 11 1 ad|a - a . a . s . / (iv) Y k > 0 a H = H(k) s.t. fx| <_ b implies ]k(x,s)| <_ H f o r .ad|i.- a.a.s . • ,(v) ) i f F ( f ( s ) ) = K(f(s),s)a(s) then F i s a trans-formation from Lp to L^ . Proof of the Lemma i s given In Lemma 4.12 next and condition (v) follows by u t i l i z i n g the conditions ( l ) and (2) of Definition] • 4. 3 . for F . 4.5 D e f i n i t i o n . Let (X,E) be a measurable space. An extended real-valued set function \i defined on Z i s c a l l e d a signed measure i f i t s a t i s f i e s the following: ( i ) . u assumes at most one of the values +», '-« ( i i ) n(<p) = 0 00 CO ( i i i ) n(U E. ) = I |i(E. ) f o r any sequence E. of d i s j o i n t .1 i = l 1 1 • measurable sets where the equality taken means that i • the series on the r i g h t converges absolutely i f n(U E^) < oo and that i t properly diverges otherwise; 4.6 D e f i n i t i o n . ' I f u i s a signed measure then |u|E = u +(E) + U~(E) i s called the t o t a l v a r i a t i o n of \i where p.+ • r and u are c a l l e d the p o s i t i v e and negative v a r i a t i o n s of u . 4.7 D e f i n i t i o n . A measure v i s said to be absolutely con-tinuous with respect to a measure u i f v(A) =0 V set A f o r which \i(A) .= 0 .. / . 4.8 Lemma. Y h e (-00,00) there exists a function K^(s) j which i s a measurable function of s and i s uniquely defined . i up to t-i-null sets s.t. (a) K Q(s) = 0 , s e X (b) P(hy B>=. J K h(s)dp , • B e S B Proof: Define, n n(B). =' P ( h ^ ) . Clearly uh(<p) = 0 . 33. Conditions (1.), (2) and (3) i n 4.3 imply that i s a signed measure and since ||ih|.(E).= ^ ( E ) + ^ ( E ) < 0 8 f o r e a c n s e t E e E | i h i s of f i n i t e v a r i a t i o n on E . F i n a l l y i f , u(B) = 0 , B e E ^ h ( B ) = J* K h ( s ) d u = 0 B | j h i s absolutely continuous w.r.t. \x . Hence by Radon- • i Nikodym Theorem, there exists a function K h that s a t i s f i e s (a) and (b) . 4.9 Remarks. (1) I f F i s l i n e a r then we have . u h(B) = F(hxB) = hP(xB) = h.fi 1(B) , B € E and hence K h(s) = hK-^s) , s e X . v 1 (2) Let Kh(-s) = K*(h,s) , then we have F(hxB) = J K # ( h x B s ),s)d U , B € E . , i • ! :'• 4.10 Lemma. • There exists a kernel K(x,s) and d ' s'atisfy-/ ing (1) - (iv) of Theorem 3-4. Proof: By Lemma 11 i n [ 4 ] , i t . can be shown that K^(x,s) i n 4.9 (2) i s continuous i n x f o r |i - a.a.s. Next by the proof of Lemma 12 i n [ 4 ] , we can obtain K and a to f u l f i l the requirements. -4.11 Remark. I f F i s l i n e a r , then ;}4 F(hX-g)'•'=' h F ( X g ) a n d we h a v e t h a t K ( x , s ) = x a n d a ( s ) = K - ^ s ) Now t h e f o l l o w i n g lemma y i e l d s t h e p r o o f o f t h e T h e o r e m . F o r e a c h f e L • d e f i n e P ' p ( f ) = f K ( f ( s ) , s ) a s d n . i 4 . 1 2 Lemma. F ^ ( f ) = F ( f ) , f e L . P r o o f : S i n c e a s i m p l e f u n c t i o n f i s a f i n i t e l i n e a r c o m b i n -• . ' ' ' • a t i o n o f c h a r a c t e r i s t i c f u n c t i o n s , b y t h e a d d i t i v i t y o f F ,^ we h a v e t h a t , i f f i s a s i m p l e f u n c t i o n , t h e n \ F ( f ) = J K ( f ( s ) , s ) a ( s ) d p = F ( f ) Now s u p p o s e t h a t 3 b > 0 s . t . | f ( s ) | <_ b . T h e r e e x i s t s a s e q u e n c e C f n l o f s i m p l e f u n c t i o n s , . [ f n ( s ) | <_ b , s . t . f „ - f a . e . a n d l i m II f - f II = 0 . H e n c e b y t h e i n n P c o n t i n u i t y o f F , we h a v e l i m F ( f n ) = F ( f ) a n d s i n c e K ( x , s ) n • ' ; i s ' a c o n t i n u o u s f u n c t i o n o f x f o r >,adu - . a . a . s , we h a v e t h a t l i m K ( f n ( s ) , s ) a ( s ) = K( f ( s ) , s )a ( s ) f o r | i : ' - a . a . s . a n d s i n c e ) l f nI 1 b !K(fn(s)>s)I 1 H ' B H ( b ) . H e n c e b y b o u n d e d c o n -v e r g e n c e t h e o r e m , we g e t t h a t i l i m F L ( f n ) = l i m -J K ( f ( s ) , s )a( s ) d u . = J K ( f ( s ) , s )a ( s )du L 4(f) n n X X : ii, 35-and since ^ ( f ^ ) = F(f) , i t follows that F- J f ) = F(f) for bounded f . , F i n a l l y consider f e L p and l e t E = {s : K(f ( s ) , s ) <x(s) > 0} and G = {s : K(f( s ) , s ) a ( s ) < 0} . To make f bounded, we define n f n ( s ) = f ( s ) i f |f ( s ) | < _ n and f n ( o ) 0 i f | f ( s ) ! > n and hence l i m || f - f || = 0 . n . Hence by condition (1) i n 3.3, we have lim F ( f n ) = P(f) , and since f i s bounded, F l ( f n ) = F ( f n ) . Let A n = {s : : [ f ( s ) | <_ n} , E n = EHA^ F n = GHAn f n , l = f*>2 = V n " We have || f n ||p < || f ||p , hence l l f ^ ' l f p < If f ||p , i = 1,2 By condition (.2) i n 4.3, we have X F < f n , i ' = . r x K ( f n , l ) l B ( H f llp) 1 = 1 , 2 Therefore F ( f ,) = f K(f , (s),s)a(s)dp i .= 1,2 are •uniformly bounded i n n and we can write 36 F ( f n ) = J K(f (s),s)a(s)dp = J* K(f(s),s)a(s)du • X * E and by Lebesgue monotone convergence theorem, we have li m F (f . ) = r..K(f(.s),s)a(s)du n n ) X E Si m i l a r l y l i m F(f ) = I* K(f(s)','s)a(s)dp n 1 1 G T h e r e f o r e P(f) = l i m F(f ) = lim{F(f -,)+F(f -)} = P. (f) Q.E.D. Now i n the following theorems we again prove the in t e g r a l representation of an additive, functional on 'L spaces, 1 <_ p < co , under d i f f e r e n t continuity conditions on F , when the underlying measure space i s atom-free and f i n i t e or cr-finite. For this purpose we s h a l l follow [ 2 ] . 4.13 Theorem. Let (X,E,u) be a f i n i t e atom-free measure space. F i s an additive functional on L (u) , 1 _< p < » t 'ST then the following are equivalent: (3 ) : x n - x a.e. =29* F(x n) F(x) and (*) . F(x) = / (f.x)du V x € L (u) X • - p where f s a t i s f i e s the conditions: 37. (a) f i s continuous and f(o) = 0 . (b) range (f) i s bounded. Proof: Let f s a t i s f y the continuity condition (a)' and also condition^ (b). — Then-as-in Section 2, i f . F .: L (u) -* R i s a functional defined by F(x) = [ (f«x)du V x e L (u ) j , i t is X • P • • I a well-defined additive functional and i t s a t i s f i e s condition ( 3 ) : x n x a.e. F ( x n ^ ~* F ( x ) • Conversely, i f , F i s an additive functional on L (u) which s a t i s f i e s the contin-u i t y condition (3) then f o r a sequence c ^ ( l - 1 ) s'. t-. x n x a.e., x € L c o(u) and l x n L l x l •.<. b f o r some constant b > 0 , we have that F^ = F/ L ^ ^ also ..satisfies (3) . Then there exists a sequence {r n} s.t., 'fz*n|.-» 0 0 and v 1 <_ | f ( r n ) | / y co By : the strong intermediate value property 3 a decreasing sequence £B n) of measurable sets s.t. ] U ( B n } = Iffr-TT * L e t x n = rn*B n • x n e M a n d x n "* 0 a ' e -However, since ' x n e ^ ( u ) > w e have ^(x^) = J*(f )<^ u = _+ U-(X) which contradicts that F ( x n ~* Thus range (f) i s bounded. Now as usual, for x e M , there exists a sequence 1 {Sn} of simple functions s.t. ^ n "* x a « e - a n d by the con-t i n u i t y of, f , f«S n - f°x boundedly a.e. Thus f«x e L 1 ( p ' ^ ^ and J(f«x)dp = l i m J(f.S )du = l i m F(S ) = F(x) . The n • . : \ . • uniqueness of f follows from Theorem 3-8 by applying 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. implies convergence i n measure, we have i 4.14 Corollary. I f Condition (3) i n Theorem 4.13 i s replaced by a.condition (4) x n "* x i n measure F(x n) -'F(x) , then the above theorem Is s t i l l true. I f the underlying measure space (X,E,u) i s atom-free a - f i n i t e and u(X) = co then we prove that F = 0.. 4.15 Theorem. I f (X,E,n) i s a - f i n i t e , u(X) = » and F . i s an additive functional on L_(u) , 1 <. P < °> , then s a t i s f i e s condition (3) i n Theorem 4.13 i f f F = 0 . F Proof: Let = x 1 x A , i f . e E and 0 < u(A) < » . Ii: x l e V u ) "•' » Since (X,E,u) Is atom-free, we can f i n d a sequence £An}n>;i_ A±C\Aj = <J> , -i =}= j and A± e E ' V i >_ 1 , s.t. • • . . u{An) = u(A 1) V n >" 1 Thus x n = x n X A f o r n >_ 2 and x^ are equimeasurable and fo r this sequence ( ^ l w e have that x n "* 0 a.e. However since F Is additive, we have that F(x n) = F ( x 1 ) Is constant. 39 F('x) = 0 f o r functions x where u(supp x) < » |j By (3) of 3.12, i t follows that F(x) =0 Y x e L (p) j ! The converse i s vacuously true. Q.E.D. 4.16 Remark. M. M. Day has proved that any l i n e a r functional! on L P(X) , 0 < p < 1 i s i d e n t i c a l l y zero, where u .is Lebesgue measure and (X,E,u) i s atom-free^ but from Theorems 3.14 and 4.15 we have seen that f o r 1 X p _< any nonlinear additive-functional F on 'L (p,) that s a t i s f i e s (3) of |. Theorem 3.12 i s i d e n t i c a l l y zero when the underlying measure! space (X,E,p) i s atom-free -ex-finite and p(X) = » . Now f o r analogue of Corollary 4.14 i n the c f - f i n i t e case, we have 4.17 Theorem. Let (X,E,p) be a cr-finite atom-free measure space and suppose p(X) = » . Let F be an/additive functional on L (p) , 1 <_ p < » . Then the following conditions are equivalent; (4) x n -* x i n measure =?? F ( x n ) r* F(x) and i (*.) F(x) = f (f'x)du Y x € L (p) with an f s a t i s f y i n g : 1 X p ' i (a*) f i s continuous and f[-h,h] = 0 f o r some h > 0 . 40. Proof: Let B e S , 0 < u(B) < »' and l e t ^ be the • r e s t r i c t i o n of u to B . Let y' = yx B > Y* e L p(u) . | Define P B on L p ( u B ) by F-B(y) = F(y') where P i s an . i additive functional on L (u) •", 1 < p < » . p i s a well-• p — — B defined functional and s a t i s f i e s (4). Hence by Corollary- 4.14 we have P B(y) = J f«y dp where f : R - R i s continuous, X f(o) =0 and range (f) i s bounded. Now we claim that t h i s f determined by F B i s independent' of B e E . For i f C c E and 0 < u(C) '< u(B) <" » \ we have "by the. strong intermediate value.property that, there exists B l e S ' B l C B s " t " = u(C) . Since f o r any r e a l number r , r x B , rx^ are equimeasurable, we have that i f f,g represent P B and P^ then F ( r x B .) = P ( r x c ) = ^ F B ( r x B ) = P c ( r x c ) / / • "" f(r) fa(B 1) = g(r)|i(C) f ( r ) = g(r) i . e . f = g since u(C) = u(B x) ^ 0 Hence i f x e L , p(supp x) < » then with f determined above we have \ . I 41. 'P (x ) = J ( f - x ) d u = J ( f . x ) d n V x e L p ( u ) , s u p p ( x ) | i ( s u p p X) < oo . Now i f f d o e s n o t s a t i s f y c o n d i t i o n ( a ) * , t h e n t h e r e e x i s t s a n u l l s e q u e n c e £ a n5 o f r e a l s s . t . f ( a n ) ^ 0 v ' n ' , w h e r e a s e q u e n c e (an5 i - s c a l l e d n u l l i f i t c o n v e r g e s t o z e r o . As- i n T h e o r e m 4.15, l e t £ A i ^ > i b e a s e q u e n c e o f !• p a i r w i s e d i s j o i n t s e t s A ^ e E s . t . | J [i(A±) = p ( A 1 ) < co 9 T i >_ i . T h e r e f o r e b y t h e a d d i v i t y o f F , f o r a n y i n t e g e r m , we h a v e I L e t x = a v m w h e r e we c h o o s e m s . t . U A . • 1 1 ( i ) . . | F ( x n ; ) | . > l i ( i i ) e s s . sup IxJ = | a n [ / w h e r e e s s . s u p f x n ( t ) f = i n f {M : n ( t : ; fx ( t ) f > M) = 0} a n d s i n c e x n — 0 i n m e a s u r e we h a v e c o n t r a d i c t i o n , f o r ' f ( x n ) / 0 = F ( o ) . H e n c e f s a t i s f i e s ( a ) * . L e t E 1 =•. {t e X : ( s ( t ) f >_ a n d f o r a r b i t r a r y x e L p ( ^ ) n X l e t x n = x x E . T h u s | j [ s u p p ( x n ) ] ' = \i(E1) < » a n d x n - x I ; " - •; ' ^  ' i n m e a s u r e 42. b y L e b e s g u e ' s l i m i t t h e o r e m , we h a v e • F ( x ) = l i m P ( x n ) = J f.(xx E )d^ i n a n d s i n c e f o r h c h o s e n i n ( a ) * we h a v e f o r a l l h- . w i t h • i <• h b y ( a ) * t h a t f»xx E = f»x , . i t f o l l o w s t h a t n. F ( x ) = < f ( f o x ) d u . C o n v e r s e l y , l e t f s a t i s f y ( a ) * a n d ( b ) . F o r x e Lp (u ) - we h a v e as a b o v e t h a t V i n t e g e r m , U ^ E ^ ) < » \ . • • ' ' I and ( a ) * g i v e s t h a t . V m , < h , f « x = f o(xx 1 ? ) w h i c h 1 m t o g e t h e r w i t h (b ) i m p l i e s t h a t f « x i s : d o m i n a t e d b y a b o u n d e d f u n c t i o n w i t h s u p p . c ' E ^ . m f ( x ) e L ^ u ) a n d so F ( x ) = J ' ( f « x ) d p i s d e f i n e d V x e L-nd-1) . a n d I s a d d i t i v e . . Now s u p p o s e x - x i n m e a s u r e , we h a v e V m > 0 ' ( i} xn\ ^  x \ m . in (i i) x n x E H—» xxE w h e r e KJ! = X -. m in m m i43. Also we see that conditions (a)* and.(b) imply conditions (a) and condition (b) that range (f) i s bounded. We have by Corollary 4.14 and with the choice of m that ( i ) implies ( i i i ) J*(f•V (E 1 ) d u - K f e x X E i ) d U = J(f»x)dn - P(x) , m m and i f we take C ^ h = (t e ^ : | x n ( t ) f <> h}c{t : |x n(t)-x(t)|>|} m we have by (a)* that supp (f°x \„ ) <= C , and by ( i i ) we have m that u(C n h ) - 0 . Hence the boundedness of range (f) implies that f o r n > 1 } the functions f.x x„ are dominated uniformly by n m bounded functions with support ( = c n ^ which implies (iv) r.f.(xx v )du - 0 • • 1 / m Thus by ( i i i ) and (iv) we have that l i m F(x n) = J [ f-(x n x E i ) + f o(x n x E i)]du . m m • • • = xlim J[f.(x x p )]du +lim f f . ( x v . )d|i n ^1 n • 1' . " m m = F(x) . Q.E.D. 44 4 . 18 T h e o r e m . L e t (X ,E,n) b e a f i n i t e a t o m - f r e e m e a s u r e >; s p a c e and F a n a d d i t i v e f u n c t i o n a l o n L (u ) 1 <_ p < » . ! ; T h e n F s a t i s f i e s (5): x n - x i n L p n o r m -=4> F ( x n ) "* , i f f . ( * ) h o l d s w i t h a n f s a t i s f y i n g c o n d i t i o n s . ( a ) f i s . c o n t i n u o u s a n d f ( o ) = 0 ( c ) | f ( r ) | < k ( l + [ r | ) p V r e R and some K > 0 . P r o o f : S i n c e c o n v e r g e n c e i n L - n o r m i m p l i e s c o n v e r g e n c e i n m e a s u r e i n a f i n i t e m e a s u r e s p a c e , f o r F an a d d i t i v e f u n c t i o n a l on L p ( ^ ) a n d F ]_ = F / L ( u ) w e h a v e as i n T h e o r e m 3-8 t h a t CO F ( x ) = F^-(x) = f ( f - x ) d p V x c L o o ( u ) . w h e r e f s a t i s f i e s ( a ) . Now s u p p o s e f d o e s n o t s a t i s f y ( c ) . •' T h e n t h e r e e x i s t s a j s e q u e n c e {r_} c R s . t . | f ( r ) f > n ( l + | r l ) p . L e t { B n ) ' b e a s e q u e n c e o f s e t s . I n X , s . t . B e S V n a n d S i n c e J f r x B ! Pdu = | r n [ P p ( B ) = / | r | p 1 W(X) n . • n n n | f ( r n ) [ r p u ( X ) < =r- < ^ H('X) - 0 n ( l + | r n [ ) p " n we h a v e t h a t r n X B - 0 i n L ^ - n o r m . B u t F ( r v ) = J!(f'y B )dp = f ( r )uB ) = + u ( X ) > 0 w h i c h n n c o n t r a d i c t s t h e . . . c o n t i n u i t y ' o f F T h u s f s a t i s f i e s ( c ) . 45.. Now i f B = (t :' |x(t)| < c^} . and B = X~B condition (c) implies that there exists constants c-^  and s.t. f o r |r[ > c± | f ( r ) | <_ k | r | p . Thus |f-xf i s bounded on B and |f«(xxg)| <. K|xxg| P , and since x e L (|a) we can select a sequence c L e o ( , a ) s.t. x -• x a.e.' because i L ^ u ) i s a dense subset of Lp(u) . . . j Thus by the continuity of ' P and by previous theorems we have that F(x) = l i m F(x n) "= l i m f ( f . x )du i f n n f •xn. - f .x e L ^ u ) Now by the continuity of f and the f a c t that x n - x a.e. we have . fox^-* f°x a'.e. Also by (c) 3 constants K and M-L s.t. f o r | t | > K , | f ( t ) | < M 1 | t | p . Thus from ( i ) we have ( i ) ' : V .€ > 0 3 6'(€) > 0 s.t. for u(B) < 6 '(f) we get f |f«x |du < € V n > 1 • which B - !: i s v a l i d f o r 6 ' ( e ) = min {6(-^-)> 2K~3 " w n e r e w e l e t I sup ( f ( t ) | '• ' • / K l = | 11 < K Thus for the case p = 1 , by V i t a l i ' s Theorem, we have, f o r f i n i t e measure space (X,E,|a) that f«x n -• f»x e L^('u) Hence we have the required representation .(*")'. ! Conversely i f f^: R R s a t i s f i e s (a) and (c) then the functional P(x) = f (fox)dii i s well-defined and has the additive property. X he Now i f a sequence £ x n 3 n > i ^ -^(u) i s s u c n that II 3^ - x || •-• 0 , then as above, f o r every subsequence ixm^> of t x n 3 which converges pointwise as well as i n norm, we have that ~* f o X € L i and since every norm convergent sequence i n L (u) converges In measure, i t contains an a.e. convergent subsequence. It follows that every subsequence of ^ o X n ^ n > l contains a subsequence which converges i n norm to f«x . Hence { f i xn}ri>i i ^ 3 8 1 ^ converges to f»x i n ' norm and hence l i m F(x ) = l i m f f«x du = f f i x dp = F(x) n n n VX n X An analogue to the above .theorem, when the space i s a - f i n i t e , i s 4.19 Theorem. . Let (X,£,p) be a d - f i n i t e atom-free measure space with u(X) = » and l e t F/ be an additive functional on Lp(u) . Then the following conditions are equivalent: 5: x n -* x i n L p norm -•—^ Ftx^) F(x) and (*) F(x) = I* "(f «x)du V x e. L(\x) with f s a t i s f y i n g X p conditions '• .'. (a) f i s continuous and f(o) = 0 (d) )f (r) | <_ k | r | p 'V r e R and some K > 0 . P r o o f : I f F i s a n a d d i t i v e f u n c t i o n a l on L (|i) w h i c h s a t i s f i e s c o n d i t i o n (5) t h e n b y T h e o r e m 4 .17 f o r a l l f u n c t i o n a l F B o b t a i n e d f r o m F d e f i n e d b y F B ( y ) = F(yx B) , 0 < p ( B ) < » t h e r e e x i s t s a ' . u n i q u e c o n t i n u o u s f u n c t i o n f : R - R w h i c h s a t i s f i e s ( a ) a n d ( c ) b y T h e o r e m 4 . 18 . We c l a i m t h a t f s a t i s f i e s ( d ) . S u p p o s e n o t . T h e n t h e r e e x i s t s a n u l l s e q u e n c e { a n l o f n o n - z e r o r e a l s s . t . \ | f ( a ) > n | a n | p . L e t fB^} b e a s e q u e n c e i n X s . t . B e E a n d s i n c e ( X , E , n ) i s a t o m - f r e e , we h a v e p ( B ) = 1 n I f ( 0 .nee I| a n X BjPa U = , : a j P u ( B j y , a J I ^ _ < J ^ ! . ,J n 1 n 1 we h a v e t h a t . ' { a n x B •'-}.<= L p ( u ) . a n d [J a n x B |!p -. 0 . . H o w e v e r F ( a n x B ) / 0 = F ( o ) w h i c h c o n t r a d i c t s (5). T h i s e s t a b l i s h e s ( d ) . Now f o r t h e r e p r e s e n t a t i o n o f F , we h a v e f o r x e LpCt-1) a n d f o r E c d e f i n e d i n T h e o r e m 4.17 t h a t i f ^ = x x ^ t h e n || x n - x | | p - 0 a n d | x n ( <_ | x | . T h u s f o r t h i s s e q u e n c e { x n l , f"*^ "* f o X a . e . a n d b y ( d ) , J f . x J < K [ x | p e L 1 ( u ) . B y L e b e s g u e N d o m i n a t e d c o n v e r g e n c e t h e o r e m , we h a v e t h a t F ( x ) = l i m F ( x n ) «= l i m J f-(xx E = J(f .x)dn . n n n 48. F o r t h e c o n v e r s e we n e e d o n l y show t h e c o n t i n u i t y P r o p e r t y (5 ) o f F , f o r t h e c o n d i t i o n s ( a ) a n d ( d ) on- . F d e f i n e F ( x ) t o b e a n a d d i t i v e f u n c t i o n a l on L (u ) . A p p e a l i n g a g a i n t o t h e V i t a l i ' s T h e o r e m , i f x n -» x i n L p n o r m , t h e n t h e r e e x i s t s u b s e q u e n c e s ^ x m ^ s . t . || x^. - x Hp -* 0 as w e l l a s x m -• x a . e . a n d f o r s u c h s u b -s e q u e n c e s , b y t h e c o n t i n u i t y o f f we h a v e t h a t f o X m "* f ° x a . e . T h u s V i t a l i ' s T h e o r e m e n s u r e s t h a t e v e r y s u b s e q u e n c e o f C f ° x n ] c o n t a i n s a s u b s e q u e n c e { f « x } w h i c h ' c o n v e r g e s ' t o f o x i n . ' L-^ n o r m a n d h e n c e t h e s e q u e n c e ^ o X n ^ n > l i t s e l f c o n v e r g e s t o f « x i n , n o r m . ' H e n c e l i m P ( x n ) = l i m J ( f « x ) d u = J ( f « x ) d u = F ( x ) . n n SECTION 5 EXAMPLES AND COUNTER EXAMPLES ON REPRESENTATION OF ADDITIVE FUNCTIONALS Let B = L (n) be the set of a l l e s s e n t i a l l y bounded CO real-valued measurable functions on X . Let F : R -* R be continuous f o r which 'f(o) = 0 . For every x e B , consider F(x) = J (f x ( t ) d U ( t ) — — (*) X F s a t i s f i e s 5-1 (a) I f x,y have d i s j o i n t support, then since n(supp x D supp y) = 0 and f(o) = 0 , we have F(x+y) = F(x) + F(y) . 5-1 (b) I f C x n} i s a sequence i n B s.t. x n xja.e. ahq 1^ 1 , |x| <_ M^ 'for some, p o s i t i v e constant- M-^. then' 'F(x-). - F(x) since f ( x ) - f(x) a.e. and |f(x )[" <_ sup f (y|) • • - • - Ty|<c by Lebesgue'd dominated convergence theorem that l i m F(x n) = l i m J*f .(x^ t ) )d|i(t) = Jf(x(t))dn(t) =F(x) • n , . ... F- i s continuous. 5 .1 (c) I f x,y e B s.t. x',y are equimeasurable, th en : 50. 'F(x') = F(y). . For i f x,y are equimeasurable then so are f(x) and 'f(y) and hence | j P(x) = J f'(x(t))du(t) = J f(y(t))d,i(t) = F(y) . X; X X In this section we w i l l be concerned to the extentj to which Properties 5-1 {(a), (b), (c)} characterize functionals of the type (*)'.'. Our f i r s t example shows that Theorem ~5.8 i s f a l s e i f the underlying measure space (X,i:,|i) i s atomic and we. s h a l l follow mainly V... J. Mizel and A. D. Martin [1] i n this section. 5.2. Example. Let X - {1,2} , S = { a l l measurable subsets of X} . Let u be the measure o n . (X,E) defined by ja(l) = m1 , u(2) = m2 , =(= m2 . For x e-B(XyS) , l e t x ( i ) = x.. , i = 1,2, i ' j' For each . x e B(X,E) define the functional F by F(x) = f- L(x^)m 1 •+ f 2 ( x 2 ) m 2 where f ^ : R - R are continuous and f^'o) = 0 f o r i = 1,2. !i : Thus by. the continuity condition of R , i f x^ n^ -• x^°^ boundedly then x i n ^ x i ° ^ and x^ n^ -» x^°^ .implies that F ( x ( n ) ) = f 1 ( x [ n ) ) m 1 + f 2 ( x ^ ) m 2 - f ^ x j 0 ^ + f 2(x^° )' )m ?> ' P ( x ^ o ) ) . \ • • For a d d i t i v i t y of F : . -51. < (!)• i f x-^x^ e B s.t. supp x.^  n supp x^ = <p then i f x-^  ^ 0 , and x^ ^ 0 then x^ = ) a n d ( 2 ) 1 x2 = X2 x ( 2 ) w h e r e ^(1) = m i > ^ ( 2 ) = m 2 ' • implies that F ( x 1 + x 2). = F(x 1)'+ F(x 2) . ( i i ) Since m^  ^ m^  , two proper subsets of X have equal measure i f f they are equal, which implies' that x,y e B are-equimeasurable i f f x = y . . And since m^ ' = |a{V : x(w') "= x^}. = |i{w .: y(w) = x^} i . e . {w .:, oc(w) = x- } = {w' : 'y(w) = x, } implies that x = y . It follows that x,y are equimeasurable i f f x =y=^> F(x) = F(y) Now i f the theorem i s , to hold true we would have f o r some con-tinuous function_ f : R -R , f(o) = 0 that F(x) = f 1 ( x 1 ) m 1 + f 2 ( x 2 ) m 2 = f ( x 1 ) m 1 + f ( x 2 ) m 2 .' i: But i f x^ =0 f o r some x then F(x). = f 2 ( x 2 ) m 2 =.f(x 2)m 2 . i . e . ' f g = f . Si m i l a r l y f ^ = f . .But i t i s not necessary that f l f2 a l w a y s - Q.E.D. So we have Seen i n the above example that, presence of atoms i n the..measure space makes the' Representation (*) f a l s e . !52 5.3 Theorem. If X =T (X,E,u) i s a f i n i t e measure space and X = AUC Is the decomposition of X into atomic and atom free parts then the s u f f i c i e n t condition f o r Theorem 3-8 to hold true i s that, f o r every atom • A i of X , u(A i) <_ ia(C) . Proof: I f x e B(X,E) , l e t x = xx f l' and x = x x n and ————— , a* J\ C L/ hence x = x 0 + x o which by the a d d i t i v i t y of P implies that £L C F(x) = P(x & ) + [ supp x a ' n supp x c = cp] Now C = (C , E , n ) 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 Q as a member of ' B = B(C,E,ia) . . Hence as i n Theorem 3-8, 3 >f• : R R continuous, s.t. . P ( X C ) = j f(x c(t»du(t) = ; f ( x ( t ) ) d p ( t ) . c c Now A-^,Ag,...' are atoms of X and any measurable function i s constant on each A^ with value x^ , I = 1,2,... I f \s(A±). = m1 , i = .l'32,... , then V x.e B(X,E) , CO x e B(A,E,|_i) and since A = U A. a • 1 1 W OJ CO P(x a) = F(E x ± x A ) = P(lim E x x A ) = l i m F(E x x A ) • 1 i • n-»» 1 i n-*o>. 1 i • n and since E x, x . -• E x. x A boundedly we have 1 1 A i 1 1 A i n n li m F(£ x ± t ) = l i m E F C x ^ ) "= "E f ) m . 53 ?(yx A ) P(o«xAt) where f, (y) = — — — — . and f. (o) = — — — — = 0 as i n Theorem 3-8 . Now i f we can show that f ^ = f V i = 1 , 2 , . . ' . , then' F ( x J = S f( x ) m, = J* f(x(t))dn(t) ' a 1 A . . • i . e . i f x e B = B(X,S,u) t h e n F( x ) = F ( x a ) + F ( x Q ) = J f(x(t))du(t) +Jf(x(t) )dp(t) = j f ( x ( t ) )d U ( t ) ' ; X and hence the theorem follows . 1 5 0 we now show that f = f^ , i = 1 , 2 , . . . . Since ^(A^) < u(C) , by nonatomicity of C , S S± e Y, , 5 1 <= C s.t. n(S i) = \ ± { A ± ) , i = 1 , 2 , . . . , and hence f o r every rea l number a , axg_ and ax A > are equimeasurable and i t follows that F(ax s > ) = F(ax A^) = f i(a)m i . ' • . 1 = 1 , 2 , . . . . '. So J'f(axg>)du = J f(a)xs>d|i' = f(a)u(S i) = f ( a ) p ( A i ) = f(a)m;. C l C l -. and since m^  > 0 , X f ^ a j n ^ - = f(a")m±. f ± ( a ) = f(a) V re a l a . Q . E . . D . 54. We give an example to show.that the condition given i n Theorem 5.3 i s not necessary for the theorem to hold true. It may be possible that the atom-free part of X i s empty while i t s atomic part i s nonempty. 5.4 Example. Let X = { 1 ,2 ,3 , . . . } , E = a l l measurable sub-sets of X and l e t the measure |i be defined on E by li(n) = , n = 1 ,2 ,3 , . . . • 2 So here . C = the atom-free part i s empty but we show that the theorem s t i l l holds, true. For x e B , take x(n) = x n . The functional F ' which s a t i s f i e s 5-1 (a) and 5.1 (b) can be defined by f (x ) n v n'-F(x) = E n=l 2 1 • 0 0 1 Now since —- ='•• E ' —^ , the sets- S„ = {n} and . 2 k=n+l 2 T n '= {n+1, n+2,.".. } have the same measure and hence f o r any re a l a e (-eo>ai) aXc a n d ' a X m a r e equimeasurable and Y n f oo f F ( a X ) = F(ax T ) -£ = S ~4 • . n . V 2 n k=n+l 2 • f 1 00 fk Also then —S-v = £ '-TT which implies by subtraction that 2 n _ 1 - , k=n 2 k f f f n •. n-1 n . ^ . ~ „ ^ v _ —rr - „ H = • — x.e. f = f , Y n > 2 2 n 2 2 ~ 55. a n d h e n c e t h e T h e o r e m 5 - 3 ' I s s t i l l t r u e . 5.5 Lemma. L e t X = {1,2,3,.... } a n d d e f i n e u(n.) = m. n whe r e m^. i s a p o s i t i v e n u m b e r s . t . f o r e v e r y n = 1,2,... CO ' B I > E m, , t h e n E m. = E m . — ^ I = J a n d h e n c e 7 1 k=n+ l . i e l 1 j e J 3 two s u b s e t s o f X = ( X , E , p ) h a v e t h e same m e a s u r e o n l y i f t h e y a r e e q u a l . P r o o f : A s s u m e t h a t I f l J = tp a n d c l a i m t h a t I U J = cp . S u p p o s e I U J 4= <P > l e t n Q = m ^ n a n d t h u s n Q e I . o r n Q e J a n d n o t - t o b o t h . L e t n e I , T h e n CO ' E m. > m > E m, > E m . a n d t h e r e f o r e E m . > E m . i e l 1 n o k=n + 1 j e J 3 I e l 1 .. j e J 3 o ;. w h i c h i s a c o n t r a d i c t i o n . • ' . • , „ „ : H e n c e I = J = cp . So two s u b s e t s o f X h a v e e q u a l m e a s u r e o n l y i f t h e y a r e e q u a l . ' . I t ' - f o l l o w s ' t h a t two r e a l m e a s u r a b l e f u n c t i o n s a r e e q u i m e a s u r a b l e o n l y i f t h e y a r e e q u a l and a s i n E x a m p l e . 5 . 2 , C o n d i t i o n 5 .1 ( c ) i s v a c u o u s l y s a t i s f i e d a n d h e n c e e v e r y f u n c t i o n a l F s . t . ' ' I ' F ( x ) = s f i ( x i ) m . = J > f ( x ( t ) ) d p ( t ) (1) N . . . . s a t i s f i e s T h e o r e m 3.8 p r o v i d e d f^( 'x) . = f ( x , t ) i s c o n t i n u o u s I n x V t e T a n d f ( o , t ) = 0 a n d the- s e r i e s ( l ) c o n v e r g e s 56. u n i f o r m l y a n d a b s o l u t e l y , ' f o r | x n J , V n , t o b e u n i f o r m l y b o u n d e d . H e n c e f o r t h i s ( X , E , p ) , T h e o r e m 3-8 i s n o t t r u e . T h e n e x t t h e o r e m , 5-6, g i v e s a n e c e s s a r y c o n d i t i o n . , f o r T h e o r e m 3.8 t o h o l d ' t r u e f o r a c o u n t a b l y i n f i n i t e d i s c r e t e m e a s u r e s p a c e . 5.6- T h e o r e m . L e t X = ( X , E , p ) be t h e c o u n t a b l y i n f i n i t e , ' d i s c r e t e m e a s u r e s p a c e and m^ >_ m 2 >_ m-^  >.. . b e t h e measures ' j o f t h e a toms o f X . T h e n a n e c e s s a r y c o n d i t i o n f o r T h e o r e m ! 00 3.8 t o b e t r u e i s t h a t f o r . i n f i n i t e l y many n , in < 2 E m, ^ ~ k=n+ l k (2) ' P r o o f : S u p p o s e c o n d i t i o n (2). i s f a l s e f o r a l l b u t a f i n i t e number o f n a n d . t a k e n > 1 ( w h e r e n . c h o s e n i s t h e same o v o CO as i n Lemma 2.39) s . t . V . n i>' n , m > 2 £ m, (3) ° n k=n+ l L e t ??'= { a l l f u n c t i o n a l s F on / B ( X , E , p ) w h i c h s a t i s f y 5.1 ( 5.1 ( b ) , 5-1 (c)} . • • ' !: i. T h e n f o r e v e r y r e a l - v a l u e d f u n c t i o n f . : R -• R ; F = £ f , m (4;) i = l 1 1 s a t i s f i e s 5.1 ( a ) i f f f ± ( o ) = 0 V I = 1 , 2 , . . . a n d s a t i s f i e s CO - j 5-1 ( b ) i f f • £ f . ( a . ) m . c o n v e r g e s u n i f o r m l y and a b s o l u t e l y ' 1 i = l x . • 1 57. f o r a^ i n any compact s e t o f R . And l a s t l y F s a t i s f i e s 5.1 (c) i f f ' V a e (-co,00) , the sequence ^ ± ^ a ^ ± y ± s a t i s f i e s — i t h a t £ f.m. = S f.m. f o r (I,«T) e H — — - ( 5 ) i e l 1 1 J e J 3 3 • ' - ' I I 1 as i n Lemma 2.39. i \ L e t Q <= 3 be the s e t o f thos e f u n c t i o n a l s F f o r wh i c h f •.= f 2 = • • • • We c l a i m t h a t 5 = Q f o r the above theorem' t o be t r u e . L e t K be a f i n i t e s u bset o f the p o s i t i v e i n t e g e r s c o n t a i n i n g k+1 i n t e g e r s where k, i s the number o f e q u a t i o n s i n ' 2 . 3 9 ('c) . L e t £Sj_}j_>i b e t n e s ° l u t i o n o f 2.39 ( c ) whos s u p p o r t i s K and l e t g : R -» R be a c o n t i n u o u s f u n c t i o n w h i c h v a n i s h e s o n l y a t z e r o . L e t • f i = Sj.S f o r 1 = 1 ,2 , . . . so f ± ± f j f o r 1 ={= j ______ ( 6j Then F = _ f.m. s a t i s f i e s (5 )'above and a l s o 5.1 (a),-1 5.1 (b) and 5-1 (c) and hence F e 3? and by ( 6 ) , F | Q wh i c h c o n t r a d i c t s our assumption. Q.E.D. The f o l l o w i n g example shows t h a t t he 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 the s t r o n g c o n d i t i o n o f Theorem 2.31 w h i c h i s e q u i v a l e n t t o 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 , i s s u f f i c i e n t i f f r > 1 where 0 < r < 1 and 58 = r 1 for i = 1,2,... 5.-7 Example. Let p > 1 be an Integer.. (_) p = r l i e s between \ and 1 . Since r also s a t i s f i e s 2x p - 1 = 0 — ( 7 ) i t i s an algebraic number of degree p . ' Si n c e ' E r p + k p = • = 1 f o r r p = \ — — ( 8 ) k=o l - r p we have for any i = 1,2,... that 1 = _. rP+kP-hi. (9) k=o Let X = {1,2,.-.. } and = r 1 = ( i ) 1 / / p = mi , we have that m-^  m2 _L ••• a n d also that m < E m, • ^ k=n+l k So X s a t i s f i e s t h e weak intermediate value property provided that p _> 1 . / We claim that Theorem J5..8 does not hold for this X = (X,E,n) . i For i f F = E f . r i s a functional on B which i = l . s a t i s f i e s conditions 5-1 (a), 5-1 (b) and 5.1 (c) then by (9) above, i f g± = -,£±r^= f ^ , i = 1,2,... , " then J 59 S i =k = o g p ^ p + i % i g k p + i — " ( 1 0 ) gi+P = k f 2 g k P + i W and subtracting (11) from ( 1 0 ) , we get g i " gi+p ~ Si+p ' = ±? gi+p ~ 2 i . e . f . + p = f ± f o r i = 1 , 2 , . . . . Thus fp +]_*^p+2*""" a n d hence F can be. completely determined i f f n , f 0 , . . . , f are known. 1 2 p We claim now that f ^ , f 2 , . . . , f p are. a r b i t r a r y con-tinuous functions'on R and f^(o) = 0 f o r , i = l , 2 , . . . , p . Now V (I,J) € H as i n Lemma 2.39,* E r 1 = £ r J _____(12) i e l ' j e J . implies -E f . r 1 = E f .r J' . (13) i e l j e J J Dividing by the lowest power r h of r , (12) and (13) can be written as 1 = E c r k (14) k=l K and • / f h = ^ f h + k r k " ,- ^ — - ( 1 5 ) 6 0 . ? k F u r t h e r m o r e (14) b e c o m e s 1 = E d, r . _ _ ( i 6 ) . k = l • w h e r e d ^ = E k " ^ p r ' K f o r k =• 1 , . • . . , p w h e r e we u s e t h e n=o 2" f a c t t h a t k = •'a^-p + b^. - and i n t h e .same way b y . u s i n g t h e p e r i o d -i c i t y o f . f u n c t i o n " s e q u e n c e f ^ f ^ . . . . , (15 ) c a n b e w r i t t e n a s . -S i n c e r s a t i s f i i . e s a u n i q u e i r r e d u c i b l e p o l y n o m i a l o f d e g r e e p t b y c o m p a r i n g (2 ) a n d (16 ) we g e t t h a t d n = d_ = . . . ' = d, •_ = 0 and 1 2 k = l . .dp - 2 . r p = 1 i . e . d r p P r h = 2 r h + P a n d (17) r e d u c e s t o a n d s i n c e 2rp r p = 1 , r e l a t i o n (12.) r e d u c e s t o f h ' ~ f h + p ' h ~ 1 , 2 ^ • • * » / • Q . E . D . SECTION 6 REPRESENTATION OF BIADDITIVE FUNCTIONALS 6 . 1 D e f i n i t i o n . I f X,Y a r e two s e t s then XxY = { ( x , y ) : x e X , y e Y] i s c a l l e d the c a r t e s i a n p r o d u c t o f X and Y . 6.2 D e f i n i t i o n . I f . A c X' and B c Y then AxB c XxY .and any s e t o f t h e form AxB i s c a l l e d a r e c t a n g l e . L e t -(X,S^ji_^) and ( Y ^ S ^ , ^ ) be measure spaces where and S 2 a r e a - a l g e b r a s o f s u b s e t s o f • X and Y r e s p e c t i v e l y . 6.3 D e f i n i t i o n . • A s e t o f the form- AxB where A e S^ / B e Sg i s c a l l e d a measurable r e c t a n g l e . 6.4 D e f i n i t i o n . ' ' I f E a XxY f o r x e X and y e Y , / . d e f i n e E x - : ( x ^ y ) e E} . E y = {x : ( x , y ) € E} . a r e c a l l e d the x - s e c t i o n s and y - s e c t i o n s r e s p e c t -and E Y c Y , E y c X . E v • and E' y i v e l y o f E 62 6.5 Theorem. I f E e S-^Sg , then E x € S 2 and E y e• S 1 for x e X and y e Y . 6.6 D e f i n i t i o n . 'With each function f on XxY and with each x e X we associate a function f defined on Y by f (y) = f (x,y.) . S i m i l a r l y ' f y i s the function defined on X by f y ( x ) = f(x,y) : 6.7 Theorem. Let f be an S^xS^-measurable function on XxY . Then (a) .1 x e X , f i s an S 0-measurable function. (b) V y e Y , ' f y i s an S-^-measurable function. '6.8 Theorem.. ' Let (X^S-^i^) and (Y.,S2,n2) be a - f i n i t e measure spaces.. Suppose E e Sn xS 0 . I f cp(x) '=. |_ (E )' , I); (y) = E y) V x e X , y e Y , then cp - i s S-^-measurable and \Jr .is S p-measurable and J cpd|_ = P d|_i and since * X Y . _U 2(E X) = ..J X E(x,y)du p(y). V x e X we have that . J ^ ( x J J * X E ( x,y)du 2(y) '= J d|i_(y)J X E (x ,y ) 'd^(x) X Y. Y X l • " 5.9 D e f i n i t i o n . I f - ( X ^ S ^ ^ ) f o r i =1,2, are a - f i n i t e measure spaces and i f E e S^xS 2 then define 6 3 , ( U l x p 2 ) ( E ) = J* p 2 ( E x ) d U l ( x ) = J U l ( E y ) d u 2 ( y ) . X Y The p r o d u c t U^xUg o f m e a s u r e s a n d i s a l s o a m e a s u r e . ( F o r P r o o f s e e P . R. H a l m o s ) a n d a l s o t h a t u-^X!-^ 1 S ' ' C T ~ f i ^ ^ e . I n t h i s s e c t i o n we p r o v e t h e i n t e g r a l r e p r e s e n t a t i o n o f b i a d d i t i v e f u n c t i o n a l s when t h e a s s o c i a t e d m e a s u r e s p a c e i s f i n i t e n o n - a t o m i c a n d f o r t h i s p u r p o s e we s h a l l f o l l o w c l o s e l y t h e p r o o f s i n [2]. S i m i l a r r e s u l t s h o l d when the u n d e r l y i n g m e a s u r e s p a c e i s c r - f i n i t e n o n - a t o m i c b u t we s h a l l r e s t r i c t o u r s e l v e s t o • t h e f i n i t e n o n - a t o m i c m e a s u r e s p a c e o n l y . • ' 5.10 D e f i n i t i o n . L e t . ( X ^ , S ^ , ^ ) , i = 1,2 b e two m e a s u r e s p a c e s and l e t b e t h e s p a c e o f m e a s u r a b l e f u n c t i o n s o n X ^ I f B^ i s a v e c t o r sub s p a c e o f f o r i = 1,2, t h e n a . ' f u n c t i o n a l N o n B ^ x B 2 i s s a i d t o b e b i a d d i t i v e i f N ( * , . y ) and N ( x , •') a r e a d d i t i v e f o r e v e r y f u n c t i o n y e B p a n d . I x e B-, r e s p e c t i v e l y . . / T h e r e s u l t s f o r b i a d d i t i v e f u n c t i o n a l s a r e a n a l o g o u s t o t h e ones p r o v e d f o r a d d i t i v e f u n c t i o n a l s i n S e c t i o n 2. We s h a l l e s t a b l i s h t h e n e c e s s a r y a n d ' s u f f i c i e n t c o n d i t i o n s t h a t a b i a d d i t i v e f u n c t i o n a l N d e f i n e d o n t h e ' p r o d u c t ' B-^xB^ o f p r e s c r i b e d s u b s p a c e s B^ c M^- , cz p e r m i t a r e p r e s e n t a t i o n o f t h e f o r m 64. (**) N(x 1,x 2) = J* tp(x1,x2)d(vjuLxu2) f o r a l l x± e B± , i = 1,2 . I |[ p where cp i s a unique continuous real-valued function on R 6.11 D e f i n i t i o n . A function cp i s said to be separately-continuous i f cp(x, • ) and cp(«,y) are continuous V x,y e R 6.12 Remark. If ep(x,y) i s continuous i n both variables separately then i t may not be continuous i n both variables I j o i n t l y . r x v Y2 2 i n I Example. Consider . cp(x,y) = -I ^2 2 '  + y *  u 0 , x^,+ y^ = 0 Now cp(o,y) = 0 , y 4= 0 • Therefore ' lim . cp(o,y) = 0 = cp(o,y ) and since cp(o,o) = 0 '; y - » y • . • . i o. . \ lim cp(o,y) = 0 = 9 ( 0 , 0 ) . y-o | ; Hence 9 i s continuous for a l l y and by symmetry;, i t i s continuous f o r a l l x also. S But l i m cp(x,y) ^ 9 ( 0 , 0 ) = 0 x-»o .. ... Y-o For l e t (x,y)—• ( 0 , 0 ) along the l i n e y = mx . . Then 65. l i m cp(x ,y) = — ^ + 0 ' . x-»o 1+m y - o H e n c e t h e g i v e n f u n c t i o n i s d i s c o n t i n u o u s , a t ( 0 , 0 ) . 5.13 T h e o r e m . L e t ( X ^ S ^ , ^ ) i = 1 , 2 , b e . . f i n i t e n o n -a t o m i c m e a s u r e s p a c e s and l e t N 'be a b i a d d i t i v e f u n c t i o n a l | on L (m ) x L ( u 0 ) . T h e n N s a t i s f i e s c o n d i t i o n : 00 J_ 00 (1,1) ^ .x n - x b o u n d e d l y a . e . N ( x ^ , y ) - N ( x , y ) V y e B g y n . - y b o u n d e d l y a . e . . N ( x , y n ) - N ( x , y ) V x e B 1 i f f ( * * ) h o l d s w i t h a r e p r e s e n t i n g f u n c t i o n cp s a t i s f y i n g ' ; ( a , a ) : cp i s s e p a r a t e l y c o n t i n u o u s and cp (c ,o ) = cp(o ,d) = 0 V c , d € R , a n d • ' ' ( b , b ) : cp i s b o u n d e d o n e v e r y s u b s e t o f R . P r o o f : L e t x / e L < a ( i i 1 ) . T h e n N ( x 1 , « ) i s an a d d i t i v e f u n c t i o n a l . w h i c h s a t i s f i e s c o n d i t i o n ( l ) : N ( x ^ , - ) i s c o n -t i n u o u s and N ( x 1 , o ) = 0 and h e n c e b y T h e o r e m 3 . 8 , t h e r e e x i s t s a u n i q u e c o n t i n u o u s f u n c t i o n f : R - » R , f ( o ) = 0 , s . t . x l x l . . . N ( x 1 , x 2 ) = I ( f x ^ o x 2 ) d n 2 V x 2 e L t t(vi 2) ( l ) . L e t v- denote-- x Y , i = 1,2 . A i 66. Define cp : R 2 - R by , ' \ 1 r. ^(cx^dXp). ^>^=^VqjS p l ( X l ) ^ ( x 2 ) T — - ( g ) c * l Since N i s b i a d d i t i v e , (1 ,1) Implies that cp i s separately continuous. Let E^ be the d i s j o i n t measurable sets i n . For a . . k fixed X g e L ^ d i g ) and for each simple function x^ = S c.x E 0 0 • 1=1 1 I. we have by the b i a d d i t i v i t y of N that k • k N( S c v . , x ) = S N . ( c x , x ) _ _ (3) .1=1- 1 * i ^ . i = l 1 . . If each c± = c 1 and u 1 ( E i ) = ^ ( E ^ ) then we get N ( c l x ft E . ^ 2 ^ = ^ ( ^ E ^ X 2 ) • 1=1 1 1 and hence- i f ^ E i ^ i _ ] _ i s a . p a r t i t i o n of X^ then 1 Un (E-, ) (5) If ^ ( F ) i s an i n t e g r a l multiple of u 1 ( X 1 ) then from ( l ) we have N(cxp,^.) = J f Q X «(-)du 2 •. ;. x 2 F • • By (5).and ( l ) 67. i.e. • ' a (?) ' • ' n(F) N ( c ^ - . ) = ^ ^ ) N ( c X i , 0 ='t^\) U 1 ( X 1 ) ! a 2 ( X 2 ) c p ( c J . ) . •N(cx F,- ) = i a 1 ( F ) p 2 ( X 2 ) c p ( c , . ) P 1(P ) | i 2(X 2)«p(c,.) = J f .(v)dn 2 ' ; . x2 i . e . . f c x (•) = ! a 1(F )cp(c , . ) (6) f o r F e S-^ . , s . t . ia-]_(F) i s a n i n t e g r a l m u l t i p l e o f p ^ ( X ^ ) . As i n . 3 . 8 ( g ) , b y a p p l y i n g t h e a d d i t i v i t y a g a i n , (5 ) i m p l i e s MF) • t h a t (6) i s t r u e - w h e n e v e r ^ i s a r a t i o n a l number a l s o and h e n c e t h e c o n t i n u i t y o f N i m p l i e s t h a t (6) i s t r u e i n g e n e r a l . I Now (3) c a n b e w r i t t e n as k k ^ (S ) k U (E. ) ' . N ^ C i \ , X 2 ) ~f i ^ " N < c i * i ' x 2 > " J P ^T T ( fc A"2J^ .. X 2 ! = E p 1(E.) J/Cp(ci,x2)du2 (7) i = l X g .. k ' = J cp( E c ± x E ,x2)d(u1xp2) X 1xX 2 1 w h i c h p r o v e s t h e r e p r e s e n t a t i o n ( * * ) when x^ e L ^ p ^ ) i s a s i m p l e f u n c t i o n . 68. To p r o v e t h a t ( * * ) h o l d s i n g e n e r a l , we p r o v e f i r s t l y t h a t cp s a t i s f i e s ( b , b ) . S u p p o s e cp does n o t s a t i s f y ( b , b ) . T h e r e e x i s t s .a r e c t a n g l e Q = { ( c , d ) : | c | <_ K^ , ' [ d | <_ Kg} s . t . , cp (c ,d ) i s u n b o u n d e d o n Q, . F o r f i x e d c * , d * , l e t K * = max | c p ( c * , d ) | a n d = ' max | c p ( c , d * ) | " . S i n c e cp i s s e p a r a t e l y c o n t i n u o u s , K * , a r e w e l l - d e f i n e d . B u t b y a s s u m p t i o n b o t h . A 1 = {K c : [ c | <_ K^} a n d A g =• {i^ : | d | <_ Kg} a r e u n b o u n d e d . .'. ' • L e t {0.1 b e a s e q u e n c e o f p o s i t i v e n u m b e r s s . t . M l 3 ' ]' E 6 . = 1 a n d E 6 . < \ 0 " f o r n > 1- . J>1 J j > n + l 3 n  :r C h o o s e a s e q u e n c e o f p o i n t s . { ( c^ , d^) } i n Q b y i n d u c t i o n as f o l l o w s : " C h o o s e Or s . t . . K c > 40""1" a n d . d^ s . t . | c p ( c 1 , d 1 ) [ = K c . I n g e n e r a l , h a v i n g c h o s e n ( c ^ d ^ ) f o r 1 <_ i <_ n - 1 c h o o s e c n so t h a t Kn > 2 6 " i n E 1 i, 6. + 2 n + 1 0 ~ 1 a w h e r e c n ~ ' n i = l a i 1 n n -, n - 1 ' _ . n - 1 _ . i - 1 , d _ > (f) E K 2 0. + E 2 1 E I, 0 , . + n n ~ . • 1=1 . C i 1 1=1 J = l a j J a n d t h e n c h o o s e d n s . t . l c P ( c n , d n ^ = ^ c L e t ^ j [ ^ i > l a n d ' ^ j ^ j > i ^ e s 6 0 ! 1 1 6 1 1 0 6 ° f d i s j o i n t m e a s u r a b l e s e t s i n X-^ and X ^ r e s p e c t i v e l y s . t . . 69, p 1 ( E i ) = 2 1 | _ 1 ( X 1 ) _ a n d H - ( F j ) = 6 .^(X.) . • L e t . X2 = j ^ d J X F j X 2 £ L > 2 } ' •' n n - " [ A l s o , t h e s e q u e n c e o f f u n c t i o n s x , = E c.x-c- a n d t h e f u n c t i o n x • 1=1 1 ^ i • x x l = S c i ^ E a r e i n """a, (l-1^) a n d x l "* x l b o u n d e d l y a . e . " i>_l i T h u s N ( x ^ , x 2 ) - N ( x 1 , x 2 ) as . n - oo . Now c o n s i d e r t h e I n t e g r a l r e p r e s e n t a t i o n f o r . N ( x - ^ , x 2 ) w h i c h we h a v e e s t a b l i s h e d when x^ o r x £ i s a s i m p l e f u n c t i o n , - we h a v e t h a t n N ( x £ , x 2 ) = J<P( . S ^ x - ^ , . ^ 1 d j X p . ) d ( u u L x n 2 : n - • • - | If = S E cp(c , d )vi ( E )n ( F ) , - - (8) i = i : > i . J - . 1 • - . J -A l s o we h a v e t h a t f o r e a c h 1 < i < n , ^ % ( c i , d . ) p 1 ( E . ) p 2 ( F . ) | ^ . 2 - 1 e ^ 1 ( X 1 ) p 2 ( X 2 ) , 6, , 1 2 - 1 ( K c ^ - 2 1 d i ) u 1 ( X 1 ) n 2 ( X 2 ) (9) a n d | E cp(c y d W ( E )^ ( F )[ < S K 2 ~ i e . H 1 ( X ) p ? ( X ) j > i + l . 1 J J L . . - . J - - j > i + l c i J ± . • .  •. 1 _ , e . < 2~ X(-±) K c . p 1 ( X 1 ) p 2 ( X 2 ) — ( 1 0 ) 7 0 . H e n c e f r o m . ( 8 ) , ( 9 ) . a n d (10 ) we h a v e t h a t | N ( x ^ x 2 ) | > | _S cp (c n,d J)e j.|i 1 (E n)p 2 (X 2 ) | n-1" Z 1 _S 9 ( c 1 , d . ) e , y j u L ( E i ) n - ( X 2 : i = l j X L 1 3 3 n - 1 . e. s e. - K -^ [ K c „ 6 n " X^J'i " K c ^ ] 2 - n l i 1 ( X 1 ) u 2 ( X 2 ) X — J- -L J —-_L J _L > n w h i c h c o n t r a d i c t s t h e f a c t t h a t N ( x ^ , x 2 ) -• N ( x 1 , x 2 ) . H e n c e 9 h a s p r o p e r t y ( b , b ) . Now f o r . ( * * ) t o h o l d i n g e n e r a l , l e t x^ e L^n^) •. T h e r e ' e x i s t s a s e q u e n c e {x^} c L ^ f ^ ) o f s i m p l e f u n c t i o n s , s . t . -• x^ b o u n d e d l y a . e . H e n c e we h a v e t h a t N(x 1,x 2) = l i m N ( x £ , x 2 ) = l i m J c p ( x ^ , x 2 ) d ( t i 1 x | i 2 ) . - — ( 1 1 ) 1 2 V x 2 e -LJM2) . Now b y t h e ' s e p a r a t e c o n t i n u i t y o f cp a n d t h e p r o p e r t y ( b , b ) we h a v e t h a t t h e f u n c t i o n s = cp(x^, .x 2 ) : X ^ x X g -> R • c o n v e r g e b o u n d e d l y p o i n t w i s e t o ' h = cp(x-^,x 2 ) o u t s i d e a s e t ' 7 1 l i o f t h e f o r m ( T i ^ x X ^ U ^ x N g ) w h e r e . N ± a r e n u l l s e t s i n X i : Hence, b y t h e L e b e s g u e d o m i n a t e d c o n v e r g e n c e t h e o r e m we h a v e f r o m . (11 ) t h a t i N ( x 1 , x 2 ) = J c p ( x 1 , x 2 ) d ( > x L x i i 2 ) V x± e L ^ ^ ) a n d X x x X 2 x 2 € a n d t h i s p r o v e s ( * * ) . C o n v e r s e l y , l e t N ( x ^ , x ) = J c p ( x 1 , x 2 ) d ( u 1 x i a 2 ) i X l x X 2 , i ; w h e r e . cp s a t i s f i e s ( Si y St ) a n d ( b , b ) . F r o m t h e s t e p s l e a d i n g t o (11')' we h a v e t h a t a f i n i t e v a l u e d ^ N a b o v e i s w e l l - d e f i n e d a n d t h a t i t s a t i s f i e s ( 1 , 1 ) f o l l o w s f r o m t h e p r o o f s i n S e c t i o n 2 . 5 . 1 4 Remark . " I t i s c l e a r f r o m the p r o o f o f t h e a b o v e t h e o r e m t h a t i t s t i l l h o l d s i f ( 1 , 1 ) i s Replaced b y c o n d i t i o n ( 2 , 2 ) x n x b o u n d e d l y • a . e . , y n -» y b o u n d e d l y a . e . N ( x n , y n ) -»' N ( x , y ) a n d ( a , a ) i s r e p l a c e d b y c o n d i t i o n : ( a , a ) ^ •, cp i s j o i n t l y c o n t i n u o u s a n d . c p ( c , o ) = cp (o ,d ) = 0 : V c , d e R . 72. S i n c e i n a t o t a l l y f i n i t e m e a s u r e s p a c e , c o n v e r g e n c e b o u n d e d l y a . e . i m p l i e s c o n v e r g e n c e b o u n d e d l y i n m e a s u r e , we • ' 1 h a v e j i i i 5.15 C o r o l l a r y . L e t ( X . , S ) b e a s i n T h e o r e m 5-13, a n d j i . I l e t N -be a b i a d d i t i v e f u n c t i o n a l o n L ( u , ) x L ( u 0 ) . T h e n CO x " j _ CO v ^ J J I N s a t i s f i e s , c o n d i t i o n : I (3.3) x n "* x b o u n d e d l y i n m e a s u r e N ( x , y ) N ( x , y ) V y e B 2 y n • * y b o u n d e d l y i n m e a s u r e ^ > N ( x , y ) - N ( x , y ) Y x e B 1 o r (4.4) x n ' " * x b o u n d e d l y i n m e a s u r e , y n - y " b o u n d e d l y i n . m e a s u r e ^ > N ( x n , y n ) - N ( x , y ) , i f f ( * * ) h o l d s w i t h a 2 cp : R •-• R h a v i n g t h e p r o p e r t i e s ( a , a ) o r ( a , a ) 1 a n d ( b , b ) . .•' P r o o f : We n e e d o n l y show t h a t i f cp s a t i s f i e s ( a , a ) a n d (b,-,b t h e n N s a t i s f i e s (.3,3)- S u p p o s e x ^ - x^ b o u n d e d l y i n m e a s u r e t h e n t h e r e e x i s t . s u b s e q u e n c e s {x^} o f {x^} s . t . x m - - » x i b o u n d e d l y a . e . a n d T h e o r e m 5.13. I m p l i e s t h a t f o r a l l s u c h s u b s e q u e n c e s c p ( x ^ , x 2 ) -• c p ( x ^ , x 2 ) i n . L^ ( I J^XU 2 ) - n o r m a n d b y t h e same a r g u m e n t e v e r y ' s u b s e q u e n c e o f ( c p ( x ^ , x 2 ) } c o n t a i n s a s u b s e q u e n c e c o n v e r g i n g t o c p ( x ^ , x 2 ) i n L-^IJU^XIJ ) n o r m a n d t h i s i m p l i e s t h a t £ c p ( x ^ , x 2 ) } . i t s e l f c o n v e r g e s t o c p ( x ^ , x 2 ) i n L-L(p-^xi-ig) n o r m a n d h e n c e b y L e b e s g u e l i m i t t h e o r e m . N ( x n , x r ) _ N { . X I ; X 2 ) . S i m i l a r l y N^x^xlJ) -» N(x1,x-) . Q . E . D . 5.16 T h e o r e m . L e t ( X ^ S ^ , ^ ) b e a s i n T h e o r e m 5.13. T h e n a b i a d d i t i v e f u n c t i o n a l N o n L a ( ) x L ^ ( ) s a t i s f i e s c o n d i t i o n : • ' (5,5) x n - x a . e . ^ N ( x n , y ) -» N ( x , y ) f o r a l l y e X - v y n - y a . e . =£p N(x,y n) - N ( x , y ) f o r a l l x e I . O i i f f ( * * ) h o l d s w i t h a cp : R R s a t i s f y i n g ( a , a ) a n d ;' ( b , b ) ^ t cp i s b o u n d e d o n f i n i t e s t r i p s o f t h e f o l l o w i n g t y p e s f o r a l l h >_ 0 , S ^ = { ( c , d ) : | d | < h } , S 2 = { ( c , d ) : [ c | < h} P r o o f : L e t cp s a t i s f y ( a , a ) a n d ( b , b ) ^ a n d l e t N ( x 1 , x 2 ) = J c p ( x 1 , x 2 ) d ( ^ 1 x u 2 ) X ^ x X 2 I t i s j u s t a r o u t i n e v e r i f i c a t i o n , t h a t t h e b o u n d e d c o n v e r g e n c e t h e o r e m i m p l i e s t h a t N s a t i s f i e s ( 5 ,5 ) . 1 C o n v e r s e l y , l e t N b e a b i a d d i t i v e f u n c t i o n a l o n '. L £ a ( u 1 ) x L o o ( n 2 ) s a t i s f y i n g . ( 5 ,5 ) . T h e n N a l s o s a t i s f i e s 2 (1 ,1) . H e n c e t h e r e e x i s t s a u n i q u e f u n c t i o n cp : R -» R s a t i s f y i n g c o n d i t i o n ( a , a ) a n d ( b , b ) s . t . ( * * ) h o l d s f o r a l l ( x 1 , x 2 ) e L o o ( u ) x L e o ( u 2 ) . 74. Now we c l a i m t h a t cp s a t i s f i e s ( b , b ) ^ . ! S u p p o s e n o t . T h e n t h e r e e x i s t s a s t r i p = { ( c , d ) : | d | <_ h} o r a s t r i p S^ , = { ( c , d ) : | c | < h|'.} 1 • 2 s . t . cp i s u n b o u n d e d o n S^ o r S^/ . S u p p o s e t h a t cp i s , u n b o u n d e d on S^ . A s b e f o r e , l e t ^ i ^ > ] _ b e a s e q u e n c e ^ o f p o s i t i v e r e a l s s . t . E 9 . = 1 a n d £ 9. < | 9 , n > 1 . i X L . i > n + l n ~ i We c h o o s e a s e q u e n c e o f p o i n t s ^ ( c i ^ d i ^ i > i ^ n ^ i n d u c t i v e l y , a s f o l l o w s : . " . L e t K = max | c p ( c , d ) [ , I, = sup | c p ( c , d ) | . c d <h . a _ r o < c < c 0 S i n c e cp i s s e p a r a t e l y c o n t i n u o u s , * K c i s w e l l d e f i n e d a n d b y t h e c o n t i n u i t y c o n d i t i o n (3) o f T h e o r e m 4.13 o n N(-,dx 2) > i s f i n i t e . S i n c e cp i s u n b o u n d e d o n S?" , K a n d £ , a r e ^ h 3 c d u n b o u n d e d f u n c t i o n s o f c a n d d , ' | d | < h . C h o o s e c-, s . t . K c >_:4 S^1 a n d t h e n t a k e d^ e [ - h , h ] s . t . fcp(c-, ,d-, ) | = K . H a v i n g c h o s e n { ( c . , d . ) } 1 < i < n - 1 , ; 1 s e l e c t c s . t . K. > 2 9„ S ' 9 . 1 . • + n • 2 n + 9~ 1 a n d t h e n -c l ~ n - i = l 1 a i n c h o o s e d n s.t^. M 0 ^ ^ ) I" = K c • F o r ^ F i ^ i > i a d i s:j . o i m ; s e q u e n c e o f m e a s u r a b l e s e t s s u c h t h a t u p ( F . ) = 9 . H p ( X p ) , l e t x p = £ d x p / x 2 e L (i jp) . • - • 7 5 . L e t {E^} b e a n e s t e d s e q u e n c e o f m e a s u r a b l e s e t s j i n s . t . ^ ( E . . ) = 2 ~ i u 1 ( X 1 ) • ' A s i n T h e o r e m 5 . 1 3 , i f x ^ = c n x E t h e n ! i . N ( x ^ , x 2 ) > n p 1 ( X 1 ) u 2 ( X 2 ) . • • i i A n d s i n c e x ^ -• 0 a . e . , t h e c o n t i n u i t y o f N i m p l i e s t h a t j , N ( x ^ , x 2 ) -» 0 . H e n c e we h a v e a c o n t r a d i c t i o n w h i c h e s t a b l i s h , t h e p r o p e r t y (.b,b)-^ f o r cp . 5.17 C o r o l l a r y . S i n c e i n a t o t a l l y f i n i t e m e a s u r e s p a c e , 1 -c o n v e r g e n c e a . e . i m p l i e s c o n v e r g e n c e i n m e a s u r e , t h e a b o v e t h e o r e m i s t r u e i f c o n d i t i o n ( 5 , 5 ) i s r e p l a c e d by. c o n d i t i o n : (6 ,6) x n - x a . e . , y n - y a . e . N ( x n : , y n ^ " * ' N ( x » y ) .• S i n c e t h e a b o v e t h e o r e m s a r e t r u e f o r n = 2 , t h e s e • a r e t r u e f o r n - a d d i t i v e f u n c t i o n a l s f o r n f i n i t e . | S E C T I O N 7 R E P R E S E N T A T I O N OP NONLINEAR.. .TRANSFORMATIONS ON L P - S P A C E S L e t T b e a s u b s e t o f t h e n - d i m e n s i o n a l s p a c e R n s . t . | i ( T ) < eo } • w h e r e \i i s t h e L e b e s g u e " m e a s u r e . 7.1 D e f i n i t i o n . A r e a l v a l u e d f u n c t i o n cp : TxR -> R . i s s a i d t o b e o f . C a r a t h e o d o r y t y p e f o r T , d e n o t e d b y cp e C a r ( T ) i f i t s a t i s f i e s . ( a ) c p ( t , - ) : R -* R i s c o n t i n u o u s f o r a . a - t € T . (b ) cp(--,c) : T -* R i s m e a s u r a r . a b l e f o r ' a l l c e R . 7 . 2 R e m a r k . I f T = (T,Y,}\i) i s a a - f i n i t e m e a s u r e s p a c e a n d i f M ( T ) d e n o t e s t h e c l a s s , o f r e a l - v a l u e d m e a s u r a b l e f u n c t i o n s on X t h e n s i n c e f o r e a c h s i m p l e f u n c t i o n x , t h e f u n c t i o n c p ° x d e f i n e d b y (cp<>x)(t) = c p ( t , x ( t ) ) b e l o n g s t o M ( T ) and s i n c e f o r e a c h x e M ( T ) , t h e r e e x i s t s a s e q u e n c e {Xj^} o f s i m p l e f u n c t i o n s c o n v e r g i n g t o x , i t f o l l o w s b y u s i n g 7 . 1 ( a ) t h a t c p « x € M ( T ) . 7-3 D e f i n i t i o n . . We d e n o t e b y A , t h e o p e r a t o r d e f i n e d o n t h e s e t o f r e a l f u n c t i o n s on ' T b y , '- : A u ( s ) = c p [ s , u ( s ) ] w h e r e cp e CAR (T ) 77. I n t h e b e g i n n i n g o f t h i s s e c t i o n , we s h a l l be m a i n l y i n t e r e s t e d i n t h e p r o p e r t i e s o f t h e . o p e r a t o r A f o r t h e c a s e , when i t a c t s f r o m a s p a c e iF^- t o L ^ 2 f o r p - ^ , p 2 >_ 1 a n d -f o r t h i s p u r p o s e we s h a l l f o l l o w M . A . K r a s n o s e l s k i i [ 1 1 ] . T o w a r d s t h e end o f t h i s s e c t i o n we s h a l l s t a t e two t h e o r e m s b y V . J . M i z e l [8] w h i c h p r o v e t h e i n t e g r a 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 t r a n s f o r m a t i o n s on . L p - s p a c e s a n d a l s o i e x t e n d o u r e a r l i e r r e s u l t s o f i n t e g r a l r e p r e s e n t a t i o n o f non-l i n e a r f u n c t i o n a l s d e f i n e d e s s e n t i a l l y f o r a t o m - f r e e o r a - f i n i t e . m e a s u r e s p a c e s . • f i n i t e J A Lemma. ( V . V . N e m y t s k i i [ 1 2 ] ) . ; L e t G b e a s e t o f f i n i t e m e a s u r e . T h e n t h e o p e r a t o r A t r a n s f o r m s e v e r y s e q u e n c e ( u n ( s ) } , s e G - - — ( l ) w h i c h c o n v e r g e s i n m e a s u r e i n t o a s e q u e n c e o f f u n c t i o n s w h i c h a l s o c o n v e r g e s i n m e a s u r e . P r o o f : Suppose, t h a t the . s e q u e n c e { u n ( s ) } c o n v e r g e s i n m e a s u r e t o u ( s ) f o r s e G . / L e t , . j G K = ( s e G s . t . f o r g i v e n 6 > 0 , | u ^ ( s ) - u ( s ) | < ^ | c p [ s o u ( s o ) ] - cp ( s ,u ) | " < € _ i G x c G 2 c . . . , a n d t h e c o n t i n u i t y o f t h e f u n c t i o n c p ( s , u ) w . r . t . u f o r a . a . -s e G i m p l i e s t h a t 00 'V( U G K ) = U ( G ) ;;-=_*> l i m | i ( G K ) = u (G) ( 2 ) . K = l K-*co . * '78. a n d h e n c e / g i v e n n > 0 / 3 k-j^  s . t . p ( G k ) > u(G) - r i /2 . ' L e t P = ( s 6 G : f u o ( s ) - u v , ( s ) < TT1"} • Choose ' N s . t . n o n K. o |i(F n)__> (i(G) - f o r a l l n > N . ' C o n s i d e r t h e s e q u e n c e o f f u n c t i o n s C A u n ( s ) } w h e r e i A u ^ s ) = c p [ s > u n ( s ) ] • . f a n d l e t D n . = .{s € G : | c p [ s , u Q ( a ) ] - c p [ s 3 u n ( s ) ] | < f . T h e n we h a v e t h a t G, f lF^ c D „ a n d i t f o l l o w s t h a t o l-i(D n ) > u ( G ) - r\ a n d s i n c e g a n d t) a r e a r b i t r a r y , t h i s c o m p l e t e s t h e p r o o f . 7.5 D e f i n i t i o n . L e t d ( x , y ) = || x - y || f o r x , y e L p , t h e n a s e q u e n c e ( x ^ " i n k P c o n v e r g e s s t r o n g l y t o x e L p i f l i m d ( x n , x ) = || x n - x || = 0 . r' P l P 2 7.Q T h e o r e m . I f A : «L -» L t r a n s f o r m s e v e r y f u n c t i o n i n L i n t o a f u n c t i o n i n L ( p - ^ P g >_ l ) t h e n A i s c o n t i n -u o u s . P r o o f : C a s e ( i ) . S u p p o s e | i (G) < <» . L e t @ b e t h e . z e r o f u n c t i o n i n t h e s p a c e L ^ . A s s u m e t h a t A8 = 6 and we show t h a t t h e o p e r a t o r A i s c o n t i n u o u s at . t h e p l 79 z e r o • 0 . S u p p o s e i t i s n o t c o n t i n u o u s a t 0 . T h e n t h e r e P l • e x i s t s a s e q u e n c e c p n ( s ) e L ( n = 1 , 2 , . . . ) a n d <P n ( s ) c o n v e r g e s s t r o n g l y t o 0 s . t . J |Acp ( s ) | P 2 d s > a ( n = l , 2 , . . . )' (i) G f o r some p o s i t i v e number a . co p ' Assume t h a t _ f |cp ( s ) | ds < « (4 ) n = l G n We c o n s t r u c t b y i n d u c t i o n , a s e q u e n c e o f n u m b e r s £ ^ , f u n c t i o n s cp ( s ) a n d s e t s - G, c=. G (k = 1 , 2 , . . . ) s . t . t h e n k ' . f o l l o w i n g c o n d i t i o n s a r e s a t i s f i e d : ; £ k + l < * \ (b) . t i ( G k ) < e k ... • P 2 v 2 a (c) - J |A cpn ( s ) | .- >4 - G k . n k . 5 pi (d ) F o r a n y s e t D c G , |i(D) <_ 2 e. k+1 J |A cpn ( s ) | P 2 d * < % D ' k * We s u p p o s e t h a t t-^ = |i(G) , cpR ( s ) = c p 1 ( s ) , G^ = G I f 6, , cp^ ( s ) a n d G, h a v e b e e n c o n s t r u c t e d , t h e n f o r * n k ' - k ^ k + l 5 w e s e l e c ' - t a n u m b e r s . t . c o n d i t i o n (d ) i s s a t i s f i e d , w h i c h i s p o s s i b l e . b y v i r t u e o f t h e a b s o l u t e c o n v e r g e n c e o f t h e 8o. • P 2. i n t e g r a l P IA cp ( s ) | ds ', a n d h e n c e c o n d i t i o n ( a ) i s a l s o G k . . s a t i s f i e d , s i n c e t h e f u n c t i o n cp ( s ) s a t i s f i e s c o n d i t i o n ( c ) . k • B y Lemma 7 .4, i t i s p o s s i b l e t o f i n d a n u m b e r n k + - i _ a n d a s e t F k + - ^ c G s . t . , A % + 1 ( 3 ) I < [ W I ] 1 / P 2 f ° R 8 £ V — — ( 5 ) w i t h . U(G) - u ( P k + 1 ) < e k + 1 -(6) L e t G k + i = G ~. F i c + i ' T h e n . c o n d i t i o n ( b ) i s s a t i s f i e d b y (6). A l s o b y (3) a n d (5). I fA cp ( s ) | P 2 d s = J | A c p n ( s ) | P 2 d s . - "\ r k+1 r n k + l M • • . . . . J • U ^ ( ^ l ^ 8 >- a - : W T ^ ( G ) 3 . • and t h i s s a t i s f i e s c o n d i t i o n ( c ) . | CO . C o n s i d e r t h e s e t s D, = G, - . U G . (k = 1,2, . . . ) j . i -k+1 1 B y ( a ) _and_-_(b )__ we h a v e - t h a t - - - - . n( u G±) < s e ± < 2e . . ( k = 1,2,.. . ) (7) i = k + l 1 i = k + l 1 K + 1 811 Define the function i|r(s) by cpn (s) i f s. e D^ (k=l,2,. ....). ... t ( s ) = „ — — ( 8 ) 0 i f s | U D. . 1=1 1 From (c), (d) and (7) we have for k = 1 , 2 , . . . that J | A | ( s ) | % s = J ]A tpn ( s ) | P 2 > J |A cpn ( s ) | P 2 d s -J | A tpn (s)| P 2as > § (9) Pn Po " By ( 4 ) , i|f e L and by hypothesis A f e L . But (9) shows that A \|r f L . Since. D. (ID. = cp , i '4= j , i t follows p CO P . that J |A IJJ(S). | 2ds >_ ' E f | A I|J(S) | 2ds = » , and this C- . k=l D k ' contradiction proves that the operator A s.t. A0 = 9 , i s continuous at 6 . Now we prove.in general 7that A is continuous at p u Q e L . Consider the function g(s,u) = f[s,uQ(s)+u] -f[s ,u Q (s)] for s e G and u e . The operator A-^  . determined by the function g(s,u) where A-Lu(s) = g[s,u(s)] satisfies the condition that A-[_6 = 9 and we have proved that i t i s continuous at the point P i - ... % £ L ' -8 2 . C a s e ( i i ) . S u p p o s e t h a t n (G) . .= <» . Assume, w . l . o . g . t h a t t h e o p e r a t o r A i s d i s c o n t i n u o u s a t 8 a n d A3 = 9 . !» • • • 1 P i A s i n C a s e ( i ) , " a s sume t h a t • c p „ ( s ) e L (n = 1 , 2 , . j. . ) • ' • i 'f i s a s e q u e n c e o f f u n c t i o n s s . t . ' 11 ,P J | A c p n ( - s ) | d d s > a ( n = l , 2 , . . . ) — — - ( I d ) G f o r some p o s i t i v e ' n u m b e r a . • p l A s s u m e a l s o t h a t S T |cp' ( s ) | ds < » ( l l ) n = l G n . A g a i n we c o n s t r u c t b y i n d u c t i o n , a s e q u e n c e o f f u n c t i o n s I cp ( s ) a n d s e t s D. c G- ( k = 1 , 2 , . . . ) s . t . k . (a ) u ( D k ) < » , D inD J.= cp I + j . ' I (b ) . J | A cpn ( s ) | P 2 d s > f (k = 1 , 2 , . . . ) ' ' . D k . k I We l e t cpn ( s ) = c p 1 ( s ) . / a n d c o n s t r u c t b y 1 i v i r t u e o f ( 10) . H a v i n g c o n s t r u c t e d cp ( s ) a n d • D, , we s ee b y (a ) n k . K . . . | CO I t h a t \s( U D . )•< co a n d . b y C a s e ( i ) i n w h i c h we rave p r o v e d t h e \ .1=1 • • . . : ' ' • ' ' • ' • | c o n t i n u i t y o f A- f o r u ( G ) < co , we c a n f i n d a n i n t e g e r n k + - J I s . t . 83. J | A cp ( s ) | 2 d s < | (12) J n k + l . 2 U D I 1=1 • and t h e n we can f i n d a set. > ^ ^ k + l ^ < °° ' s'^" J |A cp • ( s ) | 2 d s > a - (13) Gk+1 k L e t LV - = G. _ - U D. , c o n d i t i o n (a) i s s a t i s -k+1 k+1 ±-j_ f i e d and sine e. . J |Acp n ( s ) | P 2 d s = j * |Acp n ( s ) | P 2 d s - J |Acp n ( s ) i p 2 d s 1=1 Dk+1 Gk+1 U _ D i - a 2 ~ 2 and i t f o l l o w s f r o m t h i s t h a t t h e f u n c t i o n cp ( s ) s a t i s f i e s n k + l c o n d i t i o n ( b ) . .. / / A g a i n d e f i n e a f u n c t i o n \jf(s) . as i n (8), t h e n by /' . - - - - - - - - p - p (11) we see t h a t i|r e L x and by h y p o t h e s i s A\|r € L J-j . B i i t by c o n d i t i o n ( b ) , we. have t h a t Aty $, L . T h i s 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 ( i n the sense o f norm) i n t o a n o t h e r 84-. bounded s e t . We know, t h a t a l i n e a r o p e r a t o r A i s c o n t i n u o u s i f f i t i s bounded. But. f o r a n o n l i n e a r o p e r a t o r , the n o t i o n s of c o n t i n u i t y and boundedness a r e - i n d e p e n d e n t o f one a n o t h e r . ; p 7.7 Example. C o n s i d e r t h e space 1 o f n u m e r i c a l sequences • 1 cp = [Q }^ C2,. . .} w i t h norm d e f i n e d by | II cp ii = { E *Jy* . 1 i = l 1 • , 2 L e t F be a f u n c t i o n a l i n 1 d e f i n e d by F(cp) = E C l ^ f - D - i . where the sum extends over t h o s e v a l u e s o f the i n d e x i , ' depending on cp , f o r w h i c h 1^ | >_ 1 . F o r each element 2 cp e 1 t h e r e i s a f i n i t e number o f such v a l u e s o f the i n d e x . The. f u n c t i o n F(cp) i s c o n t i n u o u s and i t i s bounded- and i n . f a c e q u a l t o z e r o on the sphere || cp || '<_ 1 and i t i s n o t bounded on any sphere w i t h r a d i u s l a r g e r t h a n one. The above example.'shows t h a t ' t h e boundedness 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 rom i t s con-t i n u i t y . 7.8 Theorem. sSuppose t h a t the o p e r a t o r A t r a n s f o r m s e v e r y f u n c t i o n i n L i n t o a f u n c t i o n i n L (p-^Pp 2l Then the o p e r a t o r A'. , i s bounded. 8 5 . Proof: We can assume w. l .o .g . that A6 = 0 . Since- A is continuous at 0 by Theorem 7.6, there exists an r > 0 s.t. P-i PT P O f |cp(s)| ±ds < r x implies f |Acp(s)| ^ds < 1 (l4) G G ~ p l p l P-, Suppose that u(s) e L and nr <_ [| u || <_ PT (n+l)r x , n is an integer. Let G^, , . . . , & n + 1 be a-?-, P-, part i t ion of G s.t. J [ u ( s ) f ds <_ r ( i = 1,2,.. . ,n+l) G. . x ,Po l / P o II „ I! P n 1 /Po So when | |Au(s) | | = {J jAu(a) | 2ds} 2 < [(-H-^ U-) we have by .(14) that G - r P o n + 1 P -f |Au(s)|,^ds < _ f |Au(s)| ^ds < n+1 1=1 G. 1 Q.E.D. 7.9 Definition.. Let T = (T,Y,,[i) be a a-f inite measure space. A function cp e Car (T) is said to be in Caratheodory p-class for. T denoted by cp e Carp(T) for 1 < p <_ » } i f • i t satisfies cp°x e L 1 (T) for x e L P (T) . Pi Po 7.10 Theorem. If A : L -> L i s an operator P^Pp h 1 defined on T by Au(s) = f[s ,u(s)] . where f e Car. (T) , then P T / P O f (s,u) |. < a(s) + b |u | x d P p where-•-b---is-a-p"ositive constant and -a(s)-e L (15) 86 Proof: By T heorem 7 . 8 , we can find a positive number b s Pp P-i • f | f (s ,u(s) | ^ds <_ b d whenever I* |u(s)| 1ds <_ 1 . T T P T / P O ' ' P-i / p • f | f ( s ,u ) - b |u | x d i f |f(s,u)|>b|u|. 1 Def ine cp( s ,u) = \ P-, /P L 0 . i f | f (s,.u) |<bfu| 1 . We have that |cp(s,u)| P 2 <_ | f ( s , u ) | P 2 - b P 2 | u | P l i f cp(s,u) ± 0 p l Consider an arbitrary function u(s) e L and l e t T + = {s e T : cp[s,u(s)] > 0} P1 Let f |u(s)| ds = n+f where n i s an integer and 0 _< € <_ 1 . The set T + can be partitioned into n+1 . sets P-, T 1 3 T 2 , . . . , T n + 1 s.t. J^|u ( s ) | xds < 1 i = 1 , 2 , . . . ,n+l . . Then M f [ s,u ( s)]j P 2 d s < (* J f [s,u( s ) ] | P 2 ds Po ' ' P T P O P O .b 2 J |u(s:)i 1ds <_ (n+l)b 2 - b 2 ( n+e) P < b 2 (16) Let (T^} be a sequence of sets of f in i te measure s.t. CO T l C T 2 C " " ' a n d T = U T k ' Since cp(s,u) i s continuous i=l w.r.t u at almost a l l s e T , we can define a sequence, ( u-^ts)}^^ of functions defined on almost a l l T s.t. u-k (s) = 0 when, s | T, and c p [ s , u v ( s ) ] .= max cp("s,u] -k<u<k 8 7 . p i So " ^ ( s ) € L a n d w e s e t a ( s ) = sup cp(s ,u) = l i m c p [ s , u k ( s ) ] -oo<U<co k-»oo 16) and F a t o n ' s lemma i m p l i e s t h a t Po P^- Po f I a( s) I ^ds < sup f |cp(s,u, ( s ) |. ^ds < b d " T ~ k T K Po a ( s ) e L S i n c e a ( s ) = sup cp(s ,u) _> sup { | f ( s , u ) | - b | u [ 1 2 ] , -03<U<00 - £ » < U<oa P-1/P0 we have t h a t | f ( s , u ) | <_ a ( s ) + b [ u | * f o r s e T , u e ( - » , o o ) - . 7.11 Remark. L e t T = ( T , _.,(_) be a f i n i t e a t o m - f r e e measure s p a c e . I t f o l l o w s f r o m t h e above theorem t h a t cp € C a r P ( T ) 1 < p < 00 i f f | c p ( s , u ) | <_ a(s') + b | u | P f o r some a e L 1 ( T ) . , 7.12 D e f i n i t i o n . F o r s e S , t e T and u , e (-00,00) } t h e n o n l i n e a r i n t e g r a l o p e r a t o r A d e f i n e d b y Acp(s) = f K [ s , t , c p ( t ) ] d t (17) T i s c a l l e d P . S. U r y s o n ' . s o p e r a t o r and i t t a k e s m e a s u r a b l e f u n c t i o n s t o m e a s u r a b l e f u n c t i o n s where. S ,T a r e Lebesgue n m e a s u r a b l e s u b s e t s o f R and K : SxRxT -* R i s a r e a l v a l u e d f u n c t i o n w h i c h i s m e a s u r a b l e on SxT f o r each f i x e d v a l u e o f 88 i t s second argument and c o n t i n u o u s o n . R f o r a l m o s t a l l a r g u - : m e n t s . i n SxT . r 1 7•13 Remark. L e t C(S) denote the c l a s s o f ' c o n t i n u o u s , f u n c t i o n s d e f i n e d on S , t h e n an i m p o r t a n t s u b c l a s s o f (17) i s t h e c l a s s o f U r y s o n ' s o p e r a t o r s whose r ange i s i n , C ( S ) ' where S i s compact . T h i s s u b c l a s s i n c l u d e d t h e . c a s e ' i n w h i c h the k e r n e l cp . i s i n d e p e n d e n t o f i t s f i r s t argument so t h a t t h e o p e r a t o r A r e d u c e s t o a r e a l - v a l u e d f u n c t i o n a l F d e f i n e d b y F ( x ) = J c p ( x ( t ) , t ) d t _____(18). T ' ' • ' I n t h i s s e c t i o n , we c h a r a c t e r i z e f o r a l l c r - f i n i t e measure space T = ( T , E , | j ) and a l l compact H a u s d o r f f s p a c e s , 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 A : L P ( T ) — C(S) , 1 <_ p < co , w h i c h have t h e f o r m (17) and a l s o i n p a r t i c u l a r , we c h a r a c t e r i z e , f u n c t i o n a l s on L P ( T ) o f t h e f o r m g i v e n i n ( l 8 ) . T h i s l a t e r c h a r a c t e r i z a t i o n e x t e n d s o u r e a r l i e r r e s u l t s o f A . D. M a r t i n and V . J . M i z e l [ l ] and V . J M i z e l and K . S u n d a r e s a n [2] g i v e n • I n p r e v i o u s s e c t i o n s , c o n c e r n i n g f u n c t i o n a l s o f t h e f o r m F ( x ) = | cp (x ( t))dn ( t ) (19) T , d e f i n e d e s s e n t i a l l y on n o n a t o m i c a - f i n i t e measure s p a c e s . L e t T = ( T , E , p ) be a f i n i t e measure s p a c e . 7 . 1 4 Lemma. L e t F b e a r e a l - v a l u e d f u n c t i o n a l o n L P ( T ) <, 1 <_ p < co , w h i c h s a t i s f i e s . ( i ) F ( x + y ) - F ( x ) - F ( y ) = C_, = c o n s t a n t , w h e n e v e r x , y = 0 a . e . ( i i ) F i s u n i f o r m l y c o n t i n u o u s r e l a t i v e t o L ° ° n o r m o n e a c h b o u n d e d s u b s e t o f . L C ° ( T ) . ( i i i ) F i s c o n t i n u o u s r e l a t i v e t o L P n o r m , i f . p < » and i s c o n t i n u o u s w . r . t . b o u n d e d a . e . c o n v e r g e n c e i f p = co , t h e n f o r e v e r y r e a l ' n u m b e r h , t h e s e t f u n c t i o n d e f i n e d b y v ^ ( E ) = F ( h x E ) , f o r E € S i s a yi-continuous m e a s u r e . P r o o f : B y t a k i n g - F ^ = F + C^, w h i c h i s a f u n c t i o n a l o f t h e same t y p e as F w i t h C_, = 0 , c o n d i t i o n ( i ) r e d u c e s t o t h e ' 1 c a s e w h e r e 0-^ = 0 i . e . F ( x + y ) = F ( x ) + F ' (y) w h e n e v e r x y = 0 a . e . n Now v h((p) = F(hxi,) = 0 . L e t E n = U E. , a n d E ± nE j . = cp f o r ± 4= J t h e n E n.s? E i m p l i e s hx £ - hx E a n d hence . ' W = F ( h X E n ) - P(hx E) = v h ( E ) H e n c e i s a c o n t i n u o u s m e a s u r e o n T . 90. Now we s t a t e t h e f o l l o w i n g theorems b y V . J . M i z e l [8] t h a t p r o v e t h e i n t e g r a 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 t r a n s -f o r m a t i o n s on L ^ - s p a c e s , 1 <_ p <_ co . 7.15 Theorem. L e t T = _(T,E_,|_i) be a f i n i t e measure space and l e t F be a r e a l - v a l u e d f u n c t i o n a l cn L P ( T ) , 1 <_. p <_ <» > t h a t s a t i s f i e s ( i ) F (x+y) - F ( x ) - F ( y ) = c o n s t a n t = C^,. whenever x y = 0 a . e , ( i i ) F i s u n i f o r m l y c o n t i n u o u s r e l a t i v e t o L°° norm on each bounded s u b s e t o f L ° ° (T ) . ( i i i ) F i s c o n t i n u o u s r e l a t i v e t o L p - n o r m , i f p < co and i s c o n t i n u o u s w . r . t . bounded a . e . c o n v e r g e n c e i f p = CO . Then t h e r e e x i s t s a f u n c t i o n cp e C a r p ( T ) s . t . • r ( * ) F ( x ) = - C „ + J cpox dp f o r x e L P ( T ) T where cp can be^ t a k e n t o s a t i s f y (a) cp(0,*) = 0 a . e . and i s u n i q u e up t o s e t s o f the f o r m RxN w i t h . N a n u l l s e t i n T . • ' _ | C o n v e r s e l y , V cp c C a r p ( T ) s a t i s f y i n g - (a) and f o r e v e r y Cp e R , (*) d e f i n e s a f u n c t i o n a l s a t i s f y i n g ( i ) , ( i i ) , a n d _ _ ( i i i ) — — • " " ' - . — 91. The above results.extend to a-finite measure spaces and the proof for p = » i s as i t is and for p < <» i t i s '•' va l id i f the phrase "bounded subset of l / ° ( T ) " is replaced by-bounded subset of L°°(T) which is supported by a set of f in i te measure. 7.17 Theorem. Let T = (T,T,,|a) be a f in i te measure space and ie t A ' L P (T) C(S) 1 <_'p~ <_ » be a transformation where S is a compact•Hausdorff space. Suppose A satisfies the conditions (ia) A(x-t-y) = A(x) -i- A(y) whenever x.y = 0 . a.e. ( i la) A' i s uniformly continuous relative to L°° norm on', . each bounded subset of LC°('T). . ( i l i a ) A is. continuous relative to L p norm i f p < » and • i s continuous w. r . t . bounded a.e. convergence i f p = cs . Then there exists a transformation cp : S - Car p(T) s.t . (**) A(x)(s) = f cpox. d|_' T The transformation cp can be taken to satisfy (a) cp(-s)oO = 0 a.e. V s e S , in which case cp i s unique for each s up to- sets- of the form RxN . with N a nu l l set in T . Moreover, cp "has the following additional properties: (b) The mapping -s - cp( s ) °x e L 1 (T) "Is weakly continuous V x e L P (T) . 92. ( c ) The mapping x — "cp(s)<>x i s u n i f o r m l y c o n t i n u o u s r e l a t i v e t o Lc° norm'on each bounded s u b s e t o f LC°(.T) , u n i f o r m l y i n ^s. (d) The mapping x -• cp (s ) »x . i s weakly continuous'-on L p'(T)' u n i f o r m l y i n s , i f p < <» '/'If x n -• x "boundedly . a. e. then l i m P (cp(s)ox )dp -» 0 u n i f o r m l y i n . s and n !-i(E)-0 E ' i f p = o . C o n v e r s e l y e v e r y t r a n s f o r m a t i o n cp : s -»'Carp(T) 1 <_ p <_ OD } s a t i s f y i n g ( a ) , ( b ) , '(c) and '(d) determines, by means of (**) a t r a n s f o r m a t i o n A : L P ( T ) - C ( S ) s a t i s f y i n g ( i a ) , ( i i a ) and ( i i i a ) . . The ab.pve r e s u l t a l s o extends t o c r - f i n i t e measure spaces. F o r 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 i s added. i (e) I f x -* x boundedly a.e., then f o r any sequence E. ^ <J> , J (cp(s)ox )du - 0 u n i f o r m l y i n s and 'n . E. i F o r p < co , I t i s v a l i d i f the p h r a s e "bounded subset I o f . L°°(T) " is , , r e p l a c e d by "a s e t o f f i n i t e measure". The p r o o f i n Theorem 7 . 1 5 . u t i l i z e s the Lemma 7 . 1 4 and t h e r e p r e s e n t a t i o n •(*) i s then e s t a b l i s h e d by use o f the V i t a l i convergence. The c o n v e r s e u t i l i z e s the Lemma 7 - 4 by N e m y t s k i i ' i n [ 1 2 ] and Banach-Saks Theorem. The l a s t theorem, 7 - 1 7 , u t i l i z e s Theorem 7 - 1 5 and V i t a l i - H a h n - S a k s Theorem on con v e r g e n c e . o f measures. 9^ i • BIBLIOGRAPHY [1] A. D. M a r t i n and V. J. M i z e l , A rchive f o r R a t i o n a l Mechanics and A n a l y s i s , V o l . 15, No. 5, pp. 353-367-[2] V. J. M i z e l and K. Sundaresan, Archive f o r R a t i o n a l Mechanics and A n a l y s i s , V o l . 30, No. 2. pp. 102-126r [3] N. Friedman and M. Katz, : " A d d i t i v e f u n c t i o n a l s on LP • spaces", Canad. J . Math. 18 (1966), pp. 1264-1271.: [4] 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 " , ; Archive 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. Schwartz, L i n e a r Operators, Par t I , New York, I n t e r s c i e n c e . [6] E. Hewit and K. Stromberg, Real and Ab s t r a c t A n a l y s i s , Springer V e r l a g , New York, 1965-[7] M.. M. Day, The spaces L p 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 , Representation of n o n l i n e a r transformations on L p spaces, Bull... Amer. Math. S o c , V o l . 75, No. 1, (1969), PP- 164-168. [9] D. H.. Heyers, A note 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, (1938), pp. 76-80. [lO] J . V. Wehausen, Transformation 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 the theory 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. Burlak, Macmillan, New York, 1962. pp. 20-32. [12] V. V..Nemytskii, Existence and uniqueness theorems f o r no n l i n e a r i n t e g r a l equations, Mat. Sb. 4l, No. 3, (1934). [13] A. C. Zaanen, " I n t e g r a t i o n " , North Holland Pub. Coy. Amsterdam, 1967. 

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