1 1 2 3 5 ( ON THE INTEGRALS OF PERRON TYPE by CHENG-MING LEE B.Sc, Taiwan Normal University, Taiwan, China, 1963 M.Sc, Carleton University, Ottawa, Ontario, 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of MATHEMATICS We accept this thesis as conforming to the required standard The University of British Columbia April 1972 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by h i s representatives. It i s understood that copying or publication of t h i s thesis f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. Department The University of B r i t i s h Columbia Vancouver 8, Canada Date i i . Supervisor: Dr. P.S. Bullen ABSTRACT Perron's method of defining a process of integration i s through the use of major and minor functions. Many authors have adopted this method to define various integrals. In Chapter I, we give a very general abstract theory by f i r s t defining an abstract "derivate system" and then the cor-responding Perron integral. We show that this unifies a l l the integral theories of Perron type (of f i r s t order) known to us, in addition the abstract theories of Pfeffer [26] and of Romanovski [29] are contained in our theory as particular cases. Chapter II is devoted mainly to the study of Burkill's C^ P - integral. We know that the C P - integral is based on the theorem that i f M is n ° C - continuous in [a,b] , C DM(x) > 0 almost everywhere and C DM(x) > - °° n n — n nearly everywhere in [a,b] , then M is monotone increasing i n [a,b] . Burkill's original proof of this, [6] , contains an error and we give i t a new and correct proof. We also give a correct proof of Sargent's theorem that i f a function is CnP - integrable, then i t is CnD - integrable, [32] ; the original proof contains a gap. A scale of symmetric CP - integrals and a scale of approximately mean-continuous integrals are obtained in Chapter III and in Chapter IV, i i i . respectively. The f i r s t one is more general than Burkill's CP - scale, while the second one is more general than the GM - scale defined by E l l i s . Some other comparisons of various integrals are also given. iv. TABLE OF CONTENTS INTRODUCTION 1 CHAPTER I. THE GENERAL THEORY 4 1. SETTINGS 5 2. THE INTEGRAL * 10 3. CONVERGENCE THEOREMS 15 4. SOME GENERAL INTEGRALS AS PARTICULAR CASES 21 5. FURTHER PROPERTIES OF.THE INTEGRAL ON THE REAL LINE 25 CHAPTER I I . THE C P-INTEGRAL 34 n 1. THE CLASSICAL PERRON INTEGRAL 35 2. THE C P-INTEGRAL 37 n 3. THE C D-INTEGRAL AND THE C P-INTEGRAL 43 n n CHAPTER I I I . A SCALE OF SYMMETRIC CP-INTEGRALS AND THE MZ-INTEGRAL 50 . 1. THE SYMMETRIC de l a VALLEE POUSSIN DERIVATIVES 51 2. SOME PROPERTIES OF n-CONVEX FUNCTIONS 54 3. THE SC -DERIVATIVES AND THE SC -CONTINUITY 57 r r c •-4. THE SC P-INTEGRAL 63' n ,, 5. THE SC P-INTEGRAL AND THE P n -INTEGRAL 66 n 6. THE MZ-INTEGRAL AND THE SCP-INTEGRAL 71 CHAPTER IV. AN ACP-INTEGRAL AND A SCALE OF APPROXIMATELY MEAN-CONTINUOUS INTEGRALS " 77 1. ON THE MEAN-CONTINUOUS FUNCTIONS 77 2. A SCALE OF APPROXIMATELY MEAN-CONTINUOUS INTEGRALS 79 3. AN ACP-INTEGRAL AND AN AP2-INTEGRAL 88 REFERENCES 93 V. ACKNOWLEDGEMENTS I am deeply indebted to Professor P.S. Bullen for his encouragement, guidence and invaluable assistance during the preparation of this thesis. I also wish to thank Professor R.D. James for his constructive criticism of the draft form of this work, and Miss B. Kilbray for typing i t . The financial support of the National Research Council of Canada and the University of British Columbia i s gratefully acknowledged. 1. INTRODUCTION Various integrals defined for functions with domains in the real line have been generalized so as to apply to functions with domain in some abstract space. For example, the Lebesgue integral has been defined on a abstract measure space (see Saks [30]); the integral of Riemann type on the division space (see Henstock [11] and McShane [24]); the integral of Denjoy type on the Romanovski space (see Solomon [37]); the integral of Perron on certain topological spaces [26]. One of our purposes is to give a very general setting for Perron integrals. A so-called derivate system is defined in section 1, Chapter I, and then an integral theory of Perron type is obtained in the following sections. Doing this, we unify a l l the integral theories of Perron type, eg. the classical Perron integral, the CJ? - , SCP - , AP - integral of Burkill's [4] - [7], the MZ - integral defined by Marchinkiewicz and Zygmund [21], Kubota's AP - integral [19], and also the G-4 - integral defined by E l l i s i n [9] . For a good review of these integrals, we refer to James [14] and Jeffery [16]. In addition, we show that the P -and R - integral of Romanovski's in [29] and the integral of Pfeffer in [26] can also be obtained from our theory. In the theory of integrals of Perron type i t i s of interest to define more general concrete integrals. Thus the SCP - integral i s more general than the CP - integral; Kubota's AP - integral i s more general than Burkill's AP - integral, the GM - integral is more general than C P - integral. 2. We obtain via our general theory a scale of symmetric Cesaro-Perron integrals (SCP - scale) and a scale of approximately mean-continuous integrals (AMP - scale); this we do in Chapter III and IV, respectively. The SCP - scale is more general than the CP - scale of B u r k i l l , while the AMP - scale is more general than the mean-continuous scale due to E l l i s . The comparibility of these integrals i s then studied. We prove that the SCP - integral and the MZ - integral are in fact equivalent, and in section 5 of Chapter III the relation of the symmetric pn+"'" -integral (James [13]) and our SC n p ~ integral i s investigated. An ACP -2 integral and an AP - integral are defined and proved to be equivalent in section 3, Chapter IV. This generalizes the result [3] for n = 1 that the CnP - integral and the pn+"^ - integral are equivalent. Chapter II is devoted to the CI? - integral. A gap in Burkill's original paper [6] is f i l l e d , and so i s one in Sargent's paper [32]. We know that the theory of the CnP - integral is based on theorem 2.2 in [6]. However, the proof there is defective; see line 9 on page 546. We supply a proof of this theorem based on some concepts in [32].. Sargent has defined a CnD - integral and proved that i t is equivalent to the C^ P - integral. But there is a defect in her proof that a C P - primitive i s ACG (in n C^ - sense). We give a complete proof, which i s simpler than the one given recently and independently by Verblunsky in [38]. We close this introduction with some remarks about the notations used; A ^ B denotes the relative difference when A and B are sets; the symbol C to indicate inclusion, not necessarily proper; for real numbers a, b with a < b , we denote by [a,b] , ]a,b[ the closed and open interval, respectively; and [a,b[ , ]a,b] denote the half-open intervals by Theorem II. 3, we mean Theorem 3 in Chapter II, and similarly section 1.5, etc. If only Theorem 3 i s quoted, we mean Theorem 3 of the same chapter. 4. CHAPTER I. THE GENERAL THEORY Perron's method of defining a process of integration i s through the use of majorants and minorants (see Saks [30]). Many authors have adopted this method to define various integrals. As a typical example for our general theory in this chapter, we quote Burkill's definition of major functions for his SCP - integral in [7] . If f i s a function defined on [a,b] , a function M i s called a SCP - major function of f on [a,b] with base B (where B i s a subset of [a,b] with measure b - a and a , b £ B) i f (a) M i s C - continuous in B , and SC --continuous in ]a,b[ ; (b) SCpM(x) >_ f(x) almost everywhere in [a,b] ; (c) SCDM(x) > - °° except for a. countable set of points; (d) M(a) = 0 . For the definitions of C - and SC - continuity, and also SCDM(x) , see section III 3, below. F i r s t l y , we generalize the domain of the functions to an abstract space X with a distinguished family a of subsets of X . Secondly, we generalize the base B to the concept "base mapping B " , condition (a) to the "legitimate mapping", the. derivate SCP to the abstract "derivate operator", condition (c) to the so-called "inequality property". Then a "derivate system" i s defined and a naturally corresponding integral of Perron type arises. The f i n i t e additivity of the integral i s established and a convergence theorem similar to that of Lebesgue dominated convergence theorem is obtained. From this general theory, we obtain the integral theory of Romanovski [29] and of Pfeffer [26]. A differential property and a character ization of integrability i s obtained for the abstract integral in the case that domain of the function is the real line. §1. SETTINGS. Suppose X is a given set, a -a given collection of subsets of X and A C X . Define a by / o A = {A' | A' e a , A' CA} . 1.1 DERIVATE OPERATORS. Let A e a , B be a subset of a , V be a semi-vector space of (set) functions defined on 6 , where by a semi-vector space V we mean that F-j^ , F 2 e V implies cx^ F.^ + o^ F,, c£ V for a l l real numbers aj_»a2 — 0 A lower derivate operator on f is a mapping with domain 1/ * A such that for each v e V , and for each x e A , the image JZ(v,x) =Jlv(x) i s an extended real number, and satisfying the following axioms: (VI) for a l l x e A , £(0,x) = 0 ; (£2) for a l l x e A , w± , v 2 e V ,.£(v +v2,x) >. 'V(v ,x) + £(v 2 >x) whenever the addition on the right hand side makes sense; (£3) for a l l x . A , v e 1/ , a > 0 , £(av,x) = a£(v,x) ; (£4) for a l l x e A , £(-v,x) <_ - £(v,x) whenever both v and -v are in V . For each v e V , x e A , defining P(-v,x) = -£(v,x) , we c a l l V the upper derivate operator corresponding to £ . Letting V_ = {v | -v e \/} , we see that J/ x A is the domain of V . It is easy to see that V has properties (Dl) - (#4) of which the meaning is immediate. Furthermore, for v c V{) V_ , P(v,x) l£(v,x) , T>(csv,x) = a£(v,x) for a l l x e A and a < 0 If £(v,x) and t?(v,x) are equal, we say that v is V - d i f f e r -entiable at x , and the common value, denoted by P(v,x) or Vv(x) is called the V - derivative of v at x ; for example, clearly t?(0,x) = 0 for a l l x e A . 1.2. BASE MAPPINGS. Let A e a . A subset B of a^ w i l l be called a base in A i f A e 3 and for each A1 e 6 there exists a f i n i t e set of disjoint A. e 3 with A. A A' = <f> for each i and ijk. o» A' = A . By a base x mapping on a we mean a mapping B on a such that for each A e a , the 7. image 8(A) is a collection of bases i n A satisfying the following axioms. (81) e L , S 2 e 8(A) implies 3 ^ 3 2 e 8(A) . (82) 3 e 8(A) and A' e 3 imply 5 , = {A" | A" e 3 and A' ' «C A*} e B(A') (83) 3 i e B(A i) , e a for i = 1 , 2 and A.^ /•% k^ = <j> , A^ ^ A 2 e a implies that ^ « B = {A' ^ A' I A', e 3. for i = 1 , 2 , and 1 2 1 2 1 x l k^v k2 e a} e B(k±K/ A 2) . 1.3. LEGITIMATE MAPPINGS. Let F be an extended real-valued (set) function defined on Y , a collection of subsets of X . F is said to be superadditive on y i f F((e/A.) _> £ F(A.) for every f i n i t e collection {A.} of disjoint sets i 1 i 1 1 from Y for which [I A. e Y and the additions E F(A.) make sense. . x . 1 x i F i s defined to be subadditive i f and only i f -F is superadditive. If F is both superadditive and subadditive, we say that F i s additive. Given a base mapping 8 on a l e t M be a mapping such that for each A e a , 3 e 8(A) , the image M(A,3) is a semi-vector space of real-valued functions superadditive on 6 . If M satisfies further the following axioms, we say that M is a legitimate mapping on a with base mapping 8 (Ml) For any 3 X , 3 2 a 8(A) with ^ C B 2 , MCA.B^ ^ ) M(A,32) . 8. (M2) For any 3 e 8(A) and any A ' e 3 , M(A , 3) | A* = {M | B | M e M(A , 3)}^M ( A * , g ). (M3) For A. , A0 , B. , $_ as i n (B3) , i f M e M(A. ,3.) for i = 1 2 1 2 i l l then e yj &2 , $± « B2> , where M^CA1) = M ^ A p + M 2(Ap for any A ' = A ^ V J A^ i n 3-^ ® B 2 with A^ e 8 for i = 1 , 2 . (M4) K± = M2 on 3 and ^ e M(A , 3) implies that M2 £ M(A , 3) . (M5) M(A , 3) is closed under uniform sequence convergences in 3 • (i.e. i f F^ £ M(A , 3) for n = l , 2 , 3 , . . . , and F r F uniformly in 3 , then F e M(A ,3) •) 1.4. INEQUALITY PROPERTIES. By an inequality property on set functions we mean a property I satisfying the following axioms. (11) If F^ and F 2 are two set functions defined on a domain y and i f both F^ and F 2 satisfy the property I on y , then a^F^ + a 2 F 2 satisfies I on y whenever a - | ^ i + a2^2 m a ^ e s sense, where ct^ and are non-negative real numbers. (12) If a set function satisfies I on domains y , y^ respectively, i t does so on y ^ Y J and Y 1 ^ Y 2 (13) If F^ and F 2 are two set functions on Y with F^ _> F 2 and F 2 satisfies I on Y > then F^ satisfies I on Y • 9. If I i s an inequality property, we denote i t s dual property by I_ and by this we mean that F satisfies I_ i f and only i f -F satisfies I . We w i l l come across two kinds of inequality properties i n the examples considered later; one is defined by means of inequalities containing the lower derivates of functions; the other i s defined by means of inequalities containing the function values. 1.5. DERIVATE SYSTEMS. Let W be a fixed collection of subsets of X closed under countable set unions, (i.e. E^ e W for n = 1 , 2 , 3 , ... , imply that (J E e W) . For convenience, we say that a property P(x) is true n almost everywhere (a.e,) in A i f i t is true for a l l x in A except at most for points of a set in W Given a legitimate mapping M on a with a base mapping B , and an inequality properties I , suppose that for each A e a , 3 E B(A) , there exists a lower derivate operator 1?^ on M(A,3) • If the following axioms are satisfied, we say that (M , V_ , B , hi , I) is a derivate system on a . (PMl) For A± e o , A2 £ oA , M± e M(A± , 3 ±) , i = 1 , 2 , MJ - M2 on e^ y-v 3 2 , one has ^ (M.^ , x) = g (M£ ' x^ f o r e a c h x e A 2 . (PM2) If M e M(A , 3) with ^ ( M , x) > 0 almost everywhere in A and M satisfies the inequality property I , then M >_ 0 on 3 . 4 1 i 10. Note that by axiom (PMl) , we can always (without any ambiguity) write V(K , x) instead of P (M , x) . - —Ap §2. THE INTEGRAL. Given the set X , and a a collection of subsets of X , we let P = (M , V_ , 8 , W , I) be a derivate system on a ; i f we need 'other derivate systems on a we w i l l denote them by P 1 = (M 1 , V1 , 8 1 , W]_ , I 1) etc. 2.1. MAJOR AND MINOR FUNCTIONS. Let A e a , 0 e 8(A) , and f be an extended real-valued function defined and f i n i t e almost everywhere in A . A function M i s a P - major function of f on A with base 3 , written M e M^(A , 3) (Ml) M e M(A , 3) ; (M2) P(M , x) _> f(x) almost everywhere in A ; (M3) M satisfies I . A function m is a P - minor function of f on A with base 3 , written m e M.(A ,3) , i f - m e M ^ (A , 3) • We w i l l write M(A , 3) = (- M | M e M(A , 3)} • It is easy to see that m e M^(A , 3) and only i f 11. (ml) m E M(A , 3) ; (m2) V(m,x) f(x) almost everywhere in A ; (m3) m satisfies I_ . The following lemma is fundamental for our theory. LEMMA 1. For M e M (A , 6) , m e (A , 8) , M - m is superadditive and non-negative on B . In particular, M(A) >_ m(A) . Proof. It is t r i v i a l that M - m e M(A , B) , so that M - m is superadditive on B • As f is f i n i t e almost everywhere, i t follows from (M2) , (m2) and (Vl) that P(M-m , x) _> t?(M,x) - P(m,x) >_ 0 almost everywhere in A . Moreover, M - m satisfies I by (M3) , (m3) and (II) . Hence, M - m _> 0 on 3 by (VM2) , and the proof is completed. 2.2. THE DEFINITION OF THE INTEGRAL. If both M^(A , 3) and M^(A , 3) are not empty and iriff M(A) | M E M (A , B)> = sup {m(A) j m E (A , 3)} + + 0 0 , then we say that f is P - integrable on A with base 3 , and the 8 common value, denoted by (P) - / f , is called the P - integral of f A on A with base 3 • The set of a l l P - integrable functions on A with base 3 w i l l be denoted by P(A , 8) • The following lemma is an immediate consequence of lemma 1. LEMMA 2. f E P(A , 3) i f and only i f for each e > 0 there exist M £ M (A , 8),m E Mf(A , 3) with M(A) - m(A) < E . 12. LEMMA 3. Let $ 1 , 8 2 c 8(A) with ^ C ^ . If f e P(A , 3 2) , then f e P(A , B^) and two P - integrals are equal. In particular, i f 3 , 3' e 8(A) and f e P(A , 3) , f e P(A , 3') , then 3 3' (P) - / f = (P) - / f . A A Proof. This i s immediate from (Ml) , (VMl) and .(81) . Henceforward, we can often without any ambiguity leave the base unspecified. 2.3. ELEMENTARY PROPERTIES OF THE INTEGRAL. THEOREM 1. P(A , 3) is a vector space and the P - integral is linear on P(A , 3) • Proof. F i r s t , we prove that i f f e P(A , 3) then af e P(A , 3) for each real number a . For a = 0 , i t is t r i v i a l from (jPl) • Suppose that a > 0 . By (£3) and (V3) , i t follows that M e M , m e M => a M E M , . , a m e M . Hence af e P(A , 8) since aM(A) - am(A) af — a f can be made ar b i t r a r i l y small with M(A) - m(A) . The equality /af = a/f follows from the inequalities am(A) <_ /af _< a M(A) . For a < 0 , the proof i s similar. Secondly we prove that i f f^ e P(A , 3) for i = 1 , 2 then f 1 + f 2 e P(A , 3) and / ( f + f 2 ) = / f i + / f2 ' f o l l o w s f r o m (V2) and (T>2) , and the proof of the theorem i s completed. 13. THEOREM 2. If f e P(A , 3) and A' e $ . Then f e P(A' , $ ,) . Furthermore, i f A^ , A^ e 3 with A^ ^ = <j> and i ; = A , then / f - / ( + / Proof. If Me M (A , 6) , then M | 8 , e M(A* , 8.,) .by (M2) and (PMl) . Similar results hold for minor functions. By lemma 2, for each e > 0 , there are suitable major and minor functions M , m respectively with M(A) - m(A) < E . By lemma 1, M - m is superadditive and non-negative on 8 , so that M(A') - m(A') <_ M(A) - m(A) < £ . Thus, by lemma 2, f e P(A' , 8 A i ) • We now prove that J f = J f + / f » / f = inf{M(A) | M £ M (A , 8)} >. infiMCA-j) + M(A2) | M £ M (A , 8)} A >. inf{M 1(A 1) + M 2(A 2) | M± e M f(A ± , 3 A ) for 1 = 1 , 2 } i > in'f{M1(A1) | M1 E M f(A x , 8^)} + inf{M 2(A 2) | M£ £ M f(A 2 , 8 a )} = / f + / f , where the f i r s t inequality follows from the super-A l A2 additivity, while the second one follows from (M2) . Similarly, using minor functions, i t follows that / f .1 / f + / f > completing the proof. A A^ A2 THEOREM 3. If f E P(A ± , 8 ^ for i = 1 , 2 , where A± n A 2 = cjt and A ^ A e a , then f E P(A. ^ A , , M 80) and / f = / f + / f 1.4. Proof. This i s immediate from ( 8 3 ) , ( M 3 ) , (PMl) and Theorem 2. THEOREM 4. Let F e M(A , g) ^ M(A , g) and satisfy both I and I . If £>F(x) exists and i s f i n i t e almost everywhere in A , then VF e P(A , g) and / D F = F(A) . A Proof. It i s clear that F z Mpp(A , g) ^ Mpp^ » 3) » a n ^ t n e conclusion follows from lemma 2. We close this subsection by remarking that i f f = g almost everywhere in A and f e P(A , g) , then g e P(A , g) and the integrals of f i s equal to the integral of g . 2.4. PRIMITIVES. If f e P(A , 3) , then by Theorem 2, we see that f e P(A* , g^,) for each A' e g . Define F(A') = / f for each A' e g . F is called A' the primitive of f on A with base g . By Theorem 3, we know that F i s additive on g. , so that i t i s easy to obtain THEOREM 5. Let f e P(A , g) with primitive F , and M e Mf(A , g) , ra e M^(A', g) . Then M - F ,'F - m are both superadditive and non-negative on g . LEMMA 4. If f e P(A , g) with primitive F , then there exists a sequence {M^}CM^(A , g) and a sequence {m^ } C M^(A , g) such 15. that 0 ^ l^CA') - F(A') < - and 0 < F(A') - m^A') < - for each A' e 8 . Proof. This i s immediate from Theorem 5. THEOREM 6. If F is a primitive of f e P(A , 8) , then F e M(A , 8 ) A M(A , 8) . Proof. This i s immediate from lemma 4 and (M5) . The following general comparison theorem i s a direct consequence of the definition of the integral. THEOREM 7. Let P. = (M1 , V. , B. , N. , I.) be a derivate system on a 1 T 1 1 1 i —o for i = 1 , 2 . Suppose that M (A , 8) C M (A , 8) , , that each function satisfying 1^ satisfies 1^ , and that T?^ (M,x) -_<_ V^Qi,*.) for each M e M^A , 8) , then P^(A , 8) C P^(A , 8) and (P ) - / f = (P^ .) - / f for each f e P^A , 8) . §3. CONVERGENCE THEOREMS. With some further reasonable restrictions on the derivate system P = ( M , P , 8 , W , I ) , we w i l l now obtain some convergence theorems for our integral similar to those for the Lebesgue integral. Throughout this section, we assume that M(A , 8) satisfies the following additional axioms. 16. If f is a sequence of functions defined on a domain E , by f + f . we mean that f (x) -> f(x) as n -* 0 0 for each x e E and f (x) < f ., (x) n n — n+1 for each n and for each x e E . (M5') If {M }CZ M(A , 6) and Mn + M , then M e M ( A , 8 ) . n (PM3) For M £ M(A , 8) with M >_ 0 on 3 , fl(M,x) _> 0 for a l l x e A REMARK. It is clear that (PM3) is a very natural axiom, however axiom (M5') seems to be too much of a restriction. However, in the particular examples in later chapters, the "interval" functions in M(A , 8 ) are obtained from the "point" functions, so that the functions in M(A , 8 ) w i l l then be audi Live rather than only superadditive. If every function in M(A , 8) is additive, then (M5 1) follows from axiom (M5) . To see this, let M £ M(A , 8 ) for n = l , 2 , 3 , . . . , and M + M . We have to n > > > > n prove that M £ M(A , 8) . It is clear that M(A') > M (A') for a l l A' e 8 , — n and that M is additive on 8 • Thus, M - M^ i s non-negative and additive on 8 , so that M(A) - M (A) > M(A') - M (A') > 0 for each A' e 8 • n — n ~ Now, as M^(A) + M(A) , for each e > 0 there exists a positive integer n^ such that 0 <^ M(A) - M (A) < £ for any n >_ n . Hence 0 < M(A') - M (A') < e for each n > n A and for each A' £ 8 , i-e. — n — A M^ converges to M uniformly on 8 • That M e M(A , 8 ) then follows from axiom (M5) . 17. THEOREM 8. Suppose that f , f are functions defined and f i n i t e n almost everywhere in A and f e P(A , B) for n = l , 2 , 3 , . . . , n and f n ( x ) * f ( x ) almost everywhere i n . A . Then f e P(A , 8) and lim /f = / j. . ' n ' f n Proof. F i r s t , note that i f g(x) <_ h(x) almost everywhere in A and g , h e P(A , B) , then /g < /h . This follows directly from the definition of the integral. Now, let F^ be the primitive of f for n = l , 2 , 3 Then by Theorem 6, F e M(A , 8) />M(A , 8 ) • For each e > 0 , choose n ' * — M e M, (A , 8) such that 0 < M - F < e/2 n for n = 1 , 2 , 3 , ... , n t — n n n which is possible by lemma 4. In 8 , l e t N^ = M^ , and for n-1 _ n > 2 , N = M + J (M.-F.) . Then N e M(A , 8 ) and N > M . — n n V I l n n — n i=l Furthermore, N,_ = M , - + N - F n+1 n+1 n n > M . - + N - F t 1 since F < F , n — n+1 n n+1 n — n+1 > N since M ., > F ,, — n n+1 — n+1 Thus, N + N in 8 . By (M51) , N e M(A , B) • By (13) , N satisfies n I since M does and N > N > M . Moreover, by (V2) and (VM3) , i t n — n — n — — O easily follows that P(N,x) >^ f(x) almost everywhere i n A . Thus, we have proved that N e M^(A , 8 ) • Furthermore, since n-1 . n-1 N (A) < M (A) + I G/2 1 < F (A) + e/2 n + £ e/2 1 n — n , u- — n i=l i=l 18. n = F (A) + I E/2 1 for n = 2 , 3 , 4 , ... , n i=l one has inf {M(A) I M e IMA , B)} < N(A) < lim F (A) + e r — — n n As e is arbitrary, we see that inf {M(A) | M e M f(A , B ) } £ lim F^A) n Similarly, using minor functions, one can prove that M^(A , B ) is not empty and sup {m(A) | m e M (A , 8 ) } 1 lim F (A) n n Thus, by lemma 1, f e P(A , B ) and / f = lim F = lim / f , completing ' n n n the proof. THEOREM 9. Suppose that f , f are functions defined and f i n i t e almost n everywhere in A , and f e P(A , 8 ) for n = 1 , 2 , 3 , ... . Further, suppose that lim inf f (x) = f(x) almost everywhere i n A . If n inf {M(A) I M e M. _ . (A , g)} > -~ , then inf r n n 19, inf {M(A)|M e Mf (A, 3)} <_ lim inf : / f n A Proof. Let g (x) = inf f , ( x ) almost everywhere in A . Then n , K. k>n f i > 8 for each k > n and g (x) + f(x) almost everywhere in A k — n — n Hence, inf {M(A)|M e M (A, 3)} <_ inf {M(A)|M e M (A, 8)} for k _> n since 8 n fk because f. > g we have that Mr (A,8)ClM (A,8) . Thus, k — n f. g k °n inf {M(A)|M e M (A,8)} ± inf / f f c for n = 1,2,3 8 n k>n A Hence lim [inf{M(A)|M £ M (A,8)}] < lim inf f f n n n A Now, as g + f and inf {M(A)|M e M (A, 8')} > -°° , following n 83^ an argument similar to that in the proof of theorem 8, one proves easily that lim [inf {M(A)|M e M (A,8)}] > inf {M(A)|M £ Mr(A,8)> , g ~~ f n en and the proof i s hence completed. THEOREM 10. Suppose that h, g, f e P(A,8) and g(x) <_ f n(x) <_ h(x) almost everywhere in A for n = 1,2,3,... . If f is a function defined and f i n i t e almost everywhere in A with lim f (x) = f(x) almost every-n where in A , then f e P(A,8) and J f = lim J f n 20. Proof. Let < ) > = f - g , c j > = f - g , i j i = h - g . Then 0 <_ <j>n <_ ifi almost everywhere in A , so that 0 <_ inf c)> <_ sup <j> <_ almost n n every in A . Hence inf {M(A) |M e M. . , (A, 3)} > -°° and ' m i <p n n sup (m(A) Im e M , (A,3)} < +°° ~ ' —sup <p n n By Theorem 9 and i t s dual, we see that a H inf {M(A)|M E M\ (A,g)} < lim inf / (J) , n A b = sup {m(A)Im e M (A, g)} > lim sup / m — 9 ~ A N r n A as 9 N ( X ) <K X) almost everywhere in A . By lemma 1, we have a ^ b , and since in any case lim sup >_ lim inf , <J> z P(A, 8) and / 9 = lim / 9 N . n Now, f = 9 + g , so that by theorem 1, we see that f e P(A,8) and j f = lim J f , completing the proof, n 21. § 4 . SOME GENERAL INTEGRALS AS PARTICULAR CASES. We have mentioned in the introduction of this chapter that the integrals of Perron type defined by Romanovski [29] and by Pfeffer [26] can be obtained from our general theory. We now consider this point. 4.1 ROMANOVSKI'S INTEGRALS. In [29], Romanovski defines an abstract space, which i s called Romanovski space by Solomon i n [37] . These spaces of Romanovski contain, as special cases, the Euclidean spaces of any dimension. We now give the definition of a Romanovski space and show how the P - and R - integrals, defined by Romanovski on this space, can be obtained from our general theory. A t r i p l e (X,a,u ) is a Romanovski space i f X i s a second countable, locally compact metric space, u a non-negative countably addi-tive set function, f i n i t e on Borel sets with compact closure i n X and positive on non-empty open sets, a a distinguished family of subsets of X satisfying ten axioms. For a precise description of these axioms, we refer to [29], [37] . Let N = the family of a l l subsets of X with zero y-measure, B(A) = {a.} for each A e a • Then i t i s easy to see that B i s a base mapping according to the definition in section 1. Let F be a function defined on a. . Define A DF(x) = lim inf A eaA X E A ' 22. where B denote the closure of B in X . We say that F satisfies I on A i f DF(x) > - °° except perhaps for points of countably many boundaries of sets in a .We define F to be AC on A i f for each e > 0 there exists 6 > 0 such that £ |F(Ai)| < e whenever £ u(A.^) < 6 for any f i n i t e and disjoint {A.}<2 • F° r a nY subset E of A , let F E(A') = F(A') i f A' n E f $ , = 0 i f A'^ E = <|) . F i s said AC on E i f F„ is AC on A . Then F i s ACG on A E i f A i s a countable union of sets on each of which F i s AC Let F _(A') = F(A') i f F(A') < 0 , = 0 i f F(A') >_ 0 ; then F i s said to be ACG on A i f F is ACG on A It is obvious that I and ACG defined above are both inequality properties as defined in section 1. F i s said to be continuous from interior on A i f for each A' <_ A and e > 0 , there exists 6 > 0 such that A" C A' and y(A' - A") < <$ imply |F(A') - F(A") | < e . Let M(A,oA) = M(A) = {F|F is additive on and is continuous from interior on A} , and P = (M,D,B,N,Y), R = (M,D,B,W,ACG) . Then i t follows easily from lemmas on page 92 and page 95 in [29] that both P and R are derivate systems on a . The P-integral and R-integral are just those defined by Romanovski in [29] . By the theorem on page 77 [29], we see easily that both P- and R-integral have dif f e r e n t i a l proper-ties as given in thereom 11 in next section. Whether there i s a result similar to thereom 12 for the P-integral is an open question. The proof 23. of theorem 12 depends on the Zahorski function on the real line, so i f such a function could be constructed on an arbitrary Romanovski space, the question could be settled. 4.2 THE PFEFFER INTEGRAL. We recall Pfeffer's setting [26] and show how his integral of Perron type is obtained. Let X be a topological space and X~ = X ^ {°°} be the one-point compactification of X . For A C X , A denote the closure of A in X ; for A C X~ , A denote the closure of A in X~ . For each x e X~ , choose once and for a l l a l o c a l base of neighborhoods of x in X~ such that the cardinality of i s the smallest cardinality of local bases at x . Further, assume that for each x e X , U C X for each U e r x Let a be a pre-algebra of subsets of X such that ^ x C ° for each x e X . Also, assume that there i s a fixed integer p >_ 1 with the property that for each U e r there are disjoint sets U , U_ . . . 1»°°, 2,co, U from a for which (£/ U . = X . By X we shall denote P»°° i=l 1 , 0° n the system of a l l sets A e o such that A C U U . , where i-1 1 U i e ^ { r x ' X e X } f o r 1 = 1 , 2 , 3 n * Let G be a function defined on a , non-negative and additive on c , and f i n i t e on X 24. For each x e X , a certain family K of nets {Bu u e r,C} x is associated, where T is a cofinal subset of T . This mapping is assumed to satisfy six axioms, see [26]. For a function defined on , and for x e X , let //F(x,A) = inf {lim inf F(B )'I{B } e K. (o.)} , where a • a A A a K (a.) = {{B } e K | {B } cJ. o. } , and *F(x,A) = #(F/G)(x,A) . If A .A. /i. Ct X Ct i\ is fixed, we write #F(x) = #F(x,A) , *F(x) = *F(x,A) . Let M(A,o ) = M(A) = {F|F is superadditive on a. and there A A exists a countable set Z such that #F(x,A) > 0 for each r — x e 7, t J { « > } and #(-G)(x.A) > 0 for each x e Z „ } . F — r Let W = the family of a l l countable sets in X , and B(A) = {o-^ } . F is said to satisfy the inequality property I on A i f *F(x) > -«> for a l l point in A ~ Z . Let P = (M,*,B,W,T) . r Then i t follows from lemma 5.9 in [26] that P i s a derivate system on a . The P-integral i s just that defined by Pfeffer in [26]. Whether this P-integral has a diff e r e n t i a l property requires further investigation. 25. §5. FURTHER PROPERTIES OF THE INTEGRAL ON THE REAL LINE. Before studying some special cases, we are going to obtain a d i f f e r e n t i a l property of the integral.and a characterization of inte-grability and also a very general integration by parts formula for an abstract derivate system on the real l i n e . A different proof of the con-vergence theorem 10 is also given. Throughout this section, l e t X be the real line, a the family of a l l bounded half-open intervals like [a,b[ , N the family of a l l subsets of Lebesgue measure zero. For each function defined on and for each x e A , l e t D(F,x) = lim inf H d a > M ). t h e ordinary lower — r i _ i b-a xe[a,b] Ia,b[eo A derivate of F at x Let P = {M,P,B,N,I} be a derivate system on a satisfying (PM3) and the following additional axioms. (V5) Each Vti i s Lebesgue measurable. (P6) fl(M,x) >.D(M,x) . We also assume that for each 8 £ 8([a,b[) , the set B = L/ A is of measure b-a , where A denote the end points of the Ae6 interval A . Also, we assume that each M e*M([a,b[,8) i s additive on 8 • Then the interval function M on 8 is in one-to-one correspondence .'to the point function M on B as follows M (x) = M(Ia,x[) for each x e B ~ {a} , = 0 for x = a , M([x,y[) = M (y) - M (x) for each [x,y[ e 8 . 26. Should no ambiguities arise, we w i l l not distinguish the interval functions M on 3 and the point functions M on B Furthermore, we may c a l l B a base in {a,b] . Note that by the remark at the beginning-of section 3 (M5*) i s also satisfied. THEOREM 11. Suppose that f e P(A, g) with primitive F . Then t?F(x) exists and i s f i n i t e almost everywhere in A Proof. Let k,e be arbitrary given positive numbers. By lemma 2, there exist M e AL(A, 8) , m e MC(A,B) such that M(A) - m(A) < ke , f —x and also M(A) - F(A) < ke . Let E q be the set of points x for which at least one of the following inequalities _PM(x) >^ f (x) , ftn(x) <_f(x) f a i l s to hold. Then E q i s of measure zero. Observe that by theorem 6, F e M(A,8) so that M - F e M(A}8) . Hence by (V5) , P(M-F) is measurable, so that the set E ^ of points x in A on which P(M-F,x) >_ k i s measurable. We prove that u (E^) < e as follows, where u i s the Lebesgue measure on the real l i n e . Let R(A') = M(A') - F(A') for each A1 e 8 and R^(A1) = R(A') for A 1 e 8 , R,(A') = sup R(A") for A' e a. ~ 8 . Then i A II A A"eoAi Rj^ e M(A, 8) by (M4) , and R 1 is non-negative on aA • Therefore, D(R1,x) , and hence P(R1,x) by (P6) , exists and i s f i n i t e almost everywhere in A , and (1) - / P(R ,x)dx = (L) - / D(R ,x)dx <^ R^(A) = A A R(A) = M(A) - F(A) < ke , where (JL) denote that the integral concerned 27. is the Lebesgue integral. But (L) - / p(R l 5x) = (/.)-/ £(R ,x)dx > (L) - / P(R1,x)dx > ky (E, ) , A A 1 E. k k so that £ > v(Ej^) , which is what we want to prove. Now, for x i E, U E , fl(F,x) > P(M,x) - P(R,x) > f(x) - k . k o — — As k and e are arbitrary, i t follows that P(F,x) > f(x) almost every-where in A In a like manner, using minor functions, we can prove that PF(x) <_f(x) almost everywhere in A . Then i t follows that PF(x) exists and PF(x) = f(x) almost everywhere in A , completing the proof. COROLLARY 1. If f e P(A,3) , then f is measurable in A . COROLLARY 2. If f E P(A,g) , M e Mf(A,g) , m e Mf(A,8) , then pM(x) and Pm(x) exist and are f i n i t e almost everywhere in A Suppose that the above derivate system P satisfies in addition the following two axioms. Then we can obtain a characterization of integrability similar to that of McGregor in [22]. (Mfr) Each function M continuous in.. [a,b] belongs to M([a,b[,8) in the sense that the function M ([x,y[) = M(y) - M(x) for [x,y[£a r , r belongs to M([a,b[,$) L a, b L 28. (14) Let C £_ hi be closed under f i n i t e set unions. A function F satisfies the inequality property I i f and only i f PF(x) > -°° except perhaps for points of a set in C For convenience, we say that a property P(x) is true nearly everywhere (n.e.) in A i f P(x) is true for a l l x in A except at most for points of a set in C . Note that the property I defined in (14) i s a inequality property, but not every inequality property can be defined in this way. THEOREM 12. Let f be a function f i n i t e almost everywhere in A = [a,b[ Then f e P(A,3) i f and only i f for each e > 0 , there exist functions T,t such that (i) T e M(A,B) , t e M(A, 3) ; ( i i ) t?T(x) , pt(x) exist n.e. in A and are f i n i t e a.e. in A ; ( i i i ) + 0 0 j Pt(x) <_ f(x) ^PT(x) \ -°° n.e. in A : (iv) T(A) - t(A) < e . Proof. It i s clear that the conditions are sufficient. To see that the conditions are necessary, l e t f e P(A,8) . Then for each E > 0 , take M e W(A,g) , m e M(A,8) with M(A) - m(A) < e/2 , which i s possible by lemma 2. By corollary 2 to theorem 11, £>M(x) , and Pm(x) exist and are f i n i t e a.e. in A . Let E be the subset of A where at least one of M , m f a i l s to have a f i n i t e P-derivative. The set E i s of measure zero, so that there i s a set E^ of measure zero and of type G such that E C E- CA . 29. Let to be a point function defined on [a,b] with the following properties: (1) to is AC on [a,b] ; (2) u i s differentiable in the ordinary sense; (3) to ' (x) = +» for x e E 1 ; (4) 0 <_ CJ ' (x) < +» for x e [a,b] E 1 ; (5) a)(a) = 0 and w(b) < e/4 , That such a function exists i s well-known; see Zahorski [40] or McGregor [22]. As ixi is continuous in [a,b] the corresponding interval function on 0 ^ , also denoted by to , belongs to M(A, B ) by (M6) . Let T = M + CJ, t = m - a) . Then T e M(A,g) and t e M(A,B) . Let C be the set of points x on which P_M(x) > -» f a i l s to hold. For x e E 1 ~ C . £T(x) ^PMfx) + Du)(x)£+» , so that PT(x) = +=° ^ f (x) . For x e A .» [E U C] , PM(x) exists and i s f i n i t e , so that PT(x) exists and i s f i n i t e , and PT(x) = PM(x) + Dio(x) >_ PM(x) >_ f(x) . Similarly, Pt(x) exists n.e. in A and +°° =f= P(t,x) <_ f (x) a.e. in A . Furthermore, T(A) - t(A) = M(A) + OJ(A) - m(A) + OJ(A) < e . Thus, T,t satisfy a l l the required conditions, and hence the proof i s completed. COROLLARY. Let P = {M.P-pB.N, 1} be another such derivate system on o with P^MCx) = PM(x) n.e. in A whenever one of P.jM(x) , PM(x) exists n.e. in A . Then P(A,8) = P ^ A . B ) and two integrals of the same function are equal. 30. By theorem 12, we see that (A) we can use the ^derivatives instead of P-derivates in the d e f i n i -tion of major functions and minor functions; (B) the "almost everywhere" in (M2) and (m2) can be replaced by "nearly everywhere". Statement (B) is well-known for most of the particular integrals of Perron type while statement (A) is due to McGregor [22] for the classi c a l Perron integral. The proof here i s essential that of [22] . For a similar result for the P n-integral, see Bullen [3]. We w i l l use theorem 12 and i t s corollary to prove the equivalence of the SCP-integral and the MZ-integral in chapter III. If the derivate system does not satisfy some extra conditions, one can not get any reasonable integration by parts formula; but with some reasonable mild restrictions, which are unfortunately hard to check in particular cases, we obtain the following theorem. THEOREM 13. Let f eP([a,b[,B) and U be a bounded non-negative point function on [a,b], such that U'(x) exists and i s non-negative a.e. in [a,b], and such that the following inequalities make sense for each M e M^([a,b[,B) > m e JWp(['a,b[, g) . P(MU)(x) > M(x)U' (x) + U(x)PM(x) a.e. in Ia,b] U(mU) (x) < m(x)U' (x) + U(x)"Pm(x) a.e. in [a,b] P(MU) (x) > — C O n.e. in [a,b] P(mU) (x) < n.e. - in [a,b] 31. Then i f F i s the primitive of f , fU + FU1 e P([a,b[,3) , and / (fU + FU') = F([a,bI)U([a >bI) . Ia,M If, in addition, FU' E P([a,b[,8) then so i s fU and / fU = F([a,b[)U([a,b[) - / FU' . [a,b[ [a.bl Proof. Under the hypotheses, one can easily see that i f M e , then MU E M f u + F U? » and i f m e Mf then mU e M f u + p u, . Also, MU([a,b[) - mU([a,bf) is small when M([a,b[) - m([a,b[) i s small. Hence the required result follows easily. Now, we are in a position to give another proof of theorem 10 for the derivate system P on a satisfying the additional axioms C P M 3 ) , C P 5 ) , (P6) and (M6) , and also that a function F satisfies I whenever JJF(x) > -<» except perhaps for a countable set of points. LEMMA 5. If a function is I.-integrable, i t is P-integrable and two integrals are equal. Proof. By theorem 7, that i f a function is Perron integrable (see section II.1), i t i s P-integrable and two integrals are equal. It i s well-known that i f a function is Lebesgue integrable, i t i s Perron integrable and two integrals are equal. The conclusion then follows. LEMMA 6. Let f(x) ^ 0 almost everywhere in [a,b] . Then f i s I-integrable on [a,b] i f and only i f f is P-integrable on [a,b[ with a base B E B(Ia,b[) 32. Proof. The one implication i s given by lemma 5. To prove the other implication, l e t f be P-integrable on [a,b[ with a base B . Since |f| = f almost everywhere in [a,b] , i t follows that |f| is also P-integrable on [a,b] with base B . Clearly, the zero function 0 e M|f| ({a,b[,B) . Let M e M|f|([a,b[,B) . Then by lemma 1, M(= M - 0) is monotone increasing in B . Define M^(x) = M(x) for x e B , M (x) = sup M(t) for x e [a,b] ~ B . Then M^ i s mono-teBO[a,x] tone increasing in Ia,b] , so that M|(x) is L-integrable on [a,b] and hence so is VH since Mj(x) = ??M(x) almost everywhere in [a,b] by (PMl) and (£6) As f is measurable by corollary 1 to theorem 11, i t follows that f i s L-integrable on [a,b] since |f(x)| £ #M(x) almost everywhere i n [a,b] . The proof is hence completed. COROLLARY. Let ^ be P-integrable on [a,b[ with base B , and be L-integrable on [a,b] and f^ >^ almost everywhere in [a,b]. Then f ^ i s also L-integrable on [a,b] . THEOREM 10*. Let g,h,f (n = 1,2,3,...) be P-integrable on [a,b] with n a base B , and g(x) <_ f n ( x ) _^ h(x) almost everywhere in [a,b] for each n , and lim f (x) = f(x) almost everywhere in [a»b] . Then f n is P-integrable on [a,b] with base B and /f = lim /f ' ' n n 3 3 . Proof. Since 0 <_ f n(x) - g(x) £ h(x) - g(x) almost everywhere in [a,b] , both f - g and h - g are L-integrable on [a,b] by lemma 6. By Lebesgue dominated convergence theorem, we have lim (L) - J ( f -g) = (i-) -/ (f -g) • n Hence, by lemma 5, lim (P) - /( f -g) = (P) - /(f-g) . Now, g i s P-n integrable, so that f = (f-g) + g is also P-integrable and lim (P) - /f = (P) - /f , completing the proof, n We close this section by remarking that Kubota's abstract integral of Perron type [17], i s a particular case of the integral in this section. In fact, taking 8(A) = {o.} , W = {<}>} and the inequality property I to mean £F(x) > -°° , one gets Kubota's setting and his integral i f axiom QM) i s replaced by the equivalent axiom: ( P 4 ' ) P(v^+v 2) (x) = Dv^(x) + _P_v2(x) whenever the ordinary derivative Dv^(x) exists. Axioms ( I M ) and 0 P 4 ' ) are equivalent in the sense that one follows from the other by axioms (V2) , (_P3) , (V6) and the corresponding properties ( P 2 ) , ( P 3 ) and (V6) . Incidently, note that Kubota did not assume axiom (05) e x p l i c i t l y . However, in proving a result corresponding to our lemma 6, he did use implicitly (see the second last sentence in his proof of theorem 3 . 8 [17]) our corollary 1 to theorem 11, and axiom (£5) is essential i n the proof of this corollary. 34. CHAPTER II. THE C P-INTEGRAL n The C^-integral was f i r s t defined by Burkill in [5], [6]. Since then, many authors have shown an interest in this integral; see for instance Bosanquet I I ] , James [13], Kubota [18], Sargent [31] - [34], and Skvorcov [35], 136]. We w i l l show how to obtain the C P-integral n from our general theory, and also state an integration by parts formula, which w i l l be used extensively in Chapter III. The theory of C^P-integral based on theorem 2(below). There is a defect in Burkill's original proof in [6] (see line 9, page 546). This de-fect was noted recently and independently by Verblunsky in [38]. We give a new and correct proof of this theorem.:'* For different proofs of stronger results, we refer to Sargent [31] and Verblunsky!39]. Sargent has defined a C^D-integral [32] equivalent to the C P-integral. However, there is a defect in her proof for theorem 4 n r (below) . This has also been given a correct proof recently and independently by Verblunsky in [38]. We supply another proof, which seems simpler and more direct, in the sense that we do not appeal to the deep de l a Vallee Poussin decomposition theorem used by Verblunsky. Throughout this chapter, as i n section 1.5, X is to be the real line, a the family of a l l bounded half-open intervals like [a,b[ , W the family of a l l subsets of Lebesgue measure zero. For each A E a > 35. let B(A) = io^} . It is easy to see that B is a base mapping. Legitimate mappings and derivate operators w i l l be defined later. Once a derivate operator V_ is defined, a function F w i l l be said to satisfy the property I i f and only i f jPF(x) > -°° except perhaps for a countable set of points. Note that for a l l the derivate operators used, the property I defined above i s an inequality property. As we have noted in section 1.5, corresponding point functions and the additive interval functions w i l l not be distinguished i f this causes no ambiguities. §1 THE CLASSICAL PERRON INTEGRAL. The classical Perron integral i s the C -P-integral of the next o section. We single i t out in this separate section because by doing so, we can make the induction arguments in next section clearer. For each A = [a,b[ , let M°(A) = {M|M is additive on and the corresponding point function i s continuous in [a,b]} . Let C^DMCx) or JJM(x) be the ordinary lower drivate of M at x . Then i t i s easy to show that P q = (M°,D,B,W,I) i s a derivate system satisfying a l l the additional axioms (PM3) , (£5), (£6), (M6), (74) in section 1.5, and also (M51) in section 1.3. The P Q-integral is just the classi c a l Perron integral; see [23], [30]. 36. From theorem 1.6, for P -integral, we see that the P -o o primitives are continuous. Moreover, i t is well-known that a P^-primitive is ACG (see Saks [30]), and conversely an ACG function is a P -primitive of i t s D-derivative. It i s also well-known that o in the definition of major functions, "DM(x) > -<» n.e." can be replaced by "i)M(x) > -°° everywhere" without affecting the generality of the resulting integral. An integration by parts formula for the P Q-integral reads as follows. THEOREM 1. Let f E P ([a,b[) and g be of bounded variation on [a,b] o Then fg e P ([a,b[) and JA fg = F(A)g(A) - / A F(t)dg(t) , where A = [a,b[ , and F is the P o-primitive of f and the integral in the right ha nd side is the Stieltjes integral. This theorem w i l l be used later. For the proof, we refer to Saks [30], McShane.[23[, or Gordon and Lasher [10], who provided a more direct proof from the definition of P -integrals. o The following notions w i l l be used and extended later. For an additive interval function F on o r , r > suppose that the correspon-Ia,bl ding point function F is P Q-integrable in a neighborhood of x e [a,b]. For h 4 0 , x + h in the neighborhood, write C,(F:x,x+h) = — . [ F , (x,h) where (x,h) = Ix,x+hJ i f h > 0 , = [x+h, x l i f h < 0 . Then F 37. is said to be C -continuous at x i f lim C (F;x,x+h) = F(x) ; and SC -continuous at x e ]a,b[ i f lim {C (F;x,x+h) - C (F;x,x-h)} = 0 , 1 h+0+ and SC^-continuous at a or b i f i t i s C^-continuous there. We end this section by remarking that S P q = (M,J3D,B ,W, I) is also J . v. r,™.,, \ -,J • r M(x+h)-M(x-h) a derivate system on a » where SDM(x) = lim mf rr , h+0+ h the symmetric lower derivate of M at x . This can be checked easily noting the recent result due to Mukhopadhyay 121], PROPOSITION. If j>DM(x) >_ 0 a.e. in [a,b] and SDM(x) > -co n.e. in [a,b] » then M is monotone increasing in [a,b] , where M i s a continuous function on [a,b] The SP o~integral i s more general than the P-integral, and might be more suitable for application to the trigonometric series (cf. [7] or section III.6). We may consider this SP Q-integral as the f i r s t of the SCP-scale of integrals defined below in chapter III. §2. THE C P-INTEGRAL, n We define a scale of derivate systems on a by induction as follows. For each Ia,bl , l e t M 1([a,b[,a r , r) = M*([a,b[) = {M|M i s ^-continuous in [a,b]} , and for each M e M*([a,b[) and C (M,x,x+h) - M(x) for each x e ]a,b] , le t C^MCx) = lim inf r-j^ • 38. the = ( M ^ J C ^ D . B S W J I ) is a derivate system on a follows easily from lemma on page 316 and lemma on page 319 [5]. Suppose that for n > 2 , the derivate system P =-,(MN~1,C ,0,3,^,1) has been defined. For each M e P ,([a,b[) n-1 n-1 n-i and for each x e [a,b] , h ^ 0 with x + h e [a,b] , let x+h C (M;x,x+h) = — (P ,) - f (x+h-t) n - 1M(t)dt . n , n n-i J h x Then M is said to be C -continuous at x i f lim C (M;x,x+h) = M(x) _ n h*o n Let M n([a,b[) = {M|M is C -continuous in [a,b]} > and for each M 1 n C (M;x,x+h) - M(x) define C DM(x) = lim inf — _n_ h^ 0 h / n + 1 Then i t can be shown that P = { M N » C D,B»N>I} is a derivate system on n n a • The P^-integral i s i n fact equivalent (see Bosanquet [1]) to the C J ? - integral of Bur k i l l in [5] , [6] . That the P^ defined above i s in fact a derivate system i s easy to check. We only prove the following theorem, of which the significance has been mentioned in the introduction. THEOREM 2. Let M be C -continuous in Ia,bJ and C DM(x) > 0 n n — a.e. in Ia,b] and C N DM(x) > -<» n.e. in [a,b] . Then M as a point function i s monotone increasing in [a,b] 39. To prove t h i s , we r e c a l l some notions introduced by Sargent i n {32] . For n > 1 , a function F i s said to be AC on a set E i f i t i s C P-integrable on an i n t e r v a l containing E , and i f f o r each e > 0 there e x i s t s a 6 > 0 such that I i n f {C (F;a ,x) - F(a )} > -e , n r r r a <x<b r r I i n f {F(b ) - C (F;b ,x)} > — E r a <x<b r n r r r f o r a l l f i n i t e sets of non-overlapping i n t e r v a l s {[a ,b ]} with end points i n E and such that J (b -a ) <_ 6 . The concept AC* i s defined i n a s i m i l a r way. If F i s both AC* and AC* , then F i s J n n said to be AC* . Applying the method i n the proof of theorem I i n [32], lemma IV i n [32] reads as follows. LEMMA 1. Let F be C -continuous i n [a,b] , and C DF(x) > n.e. n _n i n Ia,b] . Then [a,b] i s the union of a countable closed sets over each of which F i s AC* . n Generalizing the concept of AC functions (see Saks [30]), we say that a fun c t i o n F Is AC on a set E i f f o r each E > 0 there e x i s t s a 6 > 0 such that J (F(b ) - F(a )} > -e 40-. for a l l f i n i t e sets of non-overlapping intervals ([a^b^]} with end points in E and such that J ^ r - a r ^ <5 . AC i s defined in an obvious way. These notions AC and AC were f i r s t introduced by J. Ridder in [28]. Following parts of the argument used in the proof of lemma III in £32], we have LEMMA 2. Let F be C^P-integrable on [c,d] , and W= min{inf[C (F;c,x) - F(c)] , inf[F(d) - C (F;d,x)]} . — n n Then there exists a constant a independent of c,d such that F(d) - F(c) >_ -aW . Proof of THEOREM 2. By lemma 1 there exists a sequence {Efc} of closed sets with union [a,b] and on each of which M i s AC* . n By lemma 2, M i s AC on each E , k = 1,2,3,... , Let A be the set of points i n Ia,b] such that i f x e A , then there i s no interval containing x on which M i s monotone increasing. Then A i s closed and hence by the Baire category theorem, i f A i s not empty there i s an interval I&>m] and an integer k such that A f\]£,m[ i s not empty and A ^ [£,m] = E f c ^ [£>m] . As M is AC 41. on , M i s AC on A ^ [Ji,m] As M is monotone increasing on each of the intervals contiguous to A ^l£,m] w.r.t. l£,m] , by the C -continuity of M , i t follows that M is AC on [J£,m] Now, letting E > 0 be given, we prove that for each x E [£,m] x) > 0 , M(x.)-M(x) with C DM(x) there exists a sequence of points x. with x. -+ x n — n r x x M(t)-M(x) for which ± •—— > -e . Suppose to the contrary that <_ -e X for a l l t with t - x < 6 for any 6 > 0 . Then for 0 < h < 6 , one has / X + h ( x + h - t ) n - 1 ( M ( t ) - M(x))dt h x , , n x+h , 1-^nTT / (-+h-t) n- 1(-e(t-x))dt = - E , h x so that C DM(x) < - e , contradicting to C DM(x) > 0 , n — n — Let G be the set of points in l£,m] such that for x £ G , C^DMCx) >_ 0 . Then the measure of G is ra-Z . By the above asser-tion, for each x E G we can take a sequence of intervals ]x,x^[ with M(x±) - M(x) x J - x -> 0 for which > - E . This associates a V i t a l i i x^-x family of intervals with each point in G . Hence by the V i t a l i covering theorem, there i s a f i n i t e mutually exclusive set {]x_^,x_I[} of the family with J ( x | - x./) > (m-l) - n, n arbitrary, for which 42. T (M(x!) - M(x.)} > -e y (x! - x.) . . i i L l l l Let {]t^.,tj[} be the subintervals of f^m] complementary to the set {Ix.,xjj } . Then £ ( t | - t.) < n . Hence, as M i s AC in [ J2,m] , one has, for sufficiently small n » M(m) - M(£) x[{M(x!) - M(x.) } + \ {M(t!) -M(t.)} i 1 1 i J J >_-£ (^x_! - x.^ ) - £ ^_-e [ (m- £) - n - 1 ] As e i s arbitrary, one concludes that M(m) >_M(£) If & < c < d <_jn , i t can be shown in the same way that M(d) >_M(c) , so that M is monotone increasing in [ £,m] . This i s a contradiction since A (\ ] £,m[ is not empty. Thus, we conclude that A i s empty. Therefore, for each x e fa,b] , there i s an interval con-taining x such that M is monotone increasing in the interval. By Heine-Borel theorem, there is then a f i n i t e set of such intervals covering Ia,b] and i t then easily follows that M i s monotone increasing i n [a,b] , completing the proof. We remark that i t is well-known that M (A) (A) and CnDM(x) <_Cn+1DM(x) for each M e M^A) , x eA , for each n = 0,1,2, 3,... , where. CQD = D , Hence by theorem 1.7, we have the consistency of the P -scale (or C P-scale) starting from the c l a s s i c a l Perron integral n n (i.e. P o-integral i n §1). 43. The following Integration by parts formula w i l l be needed in the next chapter. For a proof, see Burkill {6J . THEOREM 3. For a <_b , l e t F(x) = (P ) - / f( t ) d t , and x Kl 52 5 n - l a a a where g is of bounded variation in Ia,bJ . Then fG e P n([a,b[) and 8 6 8 (P ) - / (fG )(t)dt = [FG ] - (P ) - / (FG )(t)dt , n ' n n n-± J n-1 where a < a < 8 < b , §3. THE C D-INTEGRAL AND THE C P-INTEGRAL. n n In .[32], Sargent has defined the C nD-integral by induction. The C QD-integral i s just the special Denjoy integral, which i s equivalent to the P -integral i n §1. For n > 1 , assuming that C .D-integral o — n-1 has been defined and i s equivalent to the P ,-integral (i.e. C ,P-n-1 n-1 Integral i n Burkill's notation), the C^D-integral i s then defined.as follows. A function f on [a,b] is C nD-integrable on [a,b] i f there \k is a function F C -continuous in [a,bl and AC G on la.bj such n n ff that C DF(x) =f(x) a.e. in [a,b] , That F is AC G on [a.b] n n means that there i s a sequence of sets with union Ia,b] such that 44. F i s AC (see §2) on each of the sets, n Sargent proved that the P^-integral i s more general than the C^D-integral; see theorem XI in [32]. In the proof of the converse, that the C^D-integral i s more general than (and hence of course equivalent to) the P -integral, theorem VIII [32], we noticed that there i s a defect since the n set (defined there) depends on the choice of e , so that the argu-ment breaks down. The purpose of this section i s to supply a correct proof. THEOREM 4. Let f be P-integrable on [a,b] with primitive F . Then * F i s AC G on [a,b] n Proof. Given E Q > 0 , by lemma 1.2., there exist a Pn-major function M and a P -minor function m such that M (b) - m (b) < e , and o n o o o o also M (b) - F(b) < e , F(b) - m (b) < e o o o o By lemma 1, there exists a sequence {E°} of closed sets such * —* o that M i s AC and m is AC on each E, , k = 1,2,3,... , where o n o n k U E° = [a,b] . k K For fixed k = 1,2,3,... , l e t ^ r » ^ r t ke t' i e contiguous o * o intervals of E, i n [a,b] . As M i s AC on E, , we have k o n k 45. (1) Y inf {C (M ;c x) - M (c )} > L , 1 n o r o r r c <x<d r r (2) Y inf {M (d ) - C (M-;d ,x)} > -° L , o r n o r r c <x<d r r and similarly, (3) I sup {Cn(mQ;cr,x) - m o(c r)} < +=° , (4) y sup {m (d ) - C (m ;d ,x)} < +=° , L o r n o r r Suppose that < x < d^ . Then C (F;c ,x) - F(c ) = C (M ;c ,x) - M (c ) n r r n o r o r x _ - - " - / ( x - t ) n X{M (t) - F(t)}dt + M (c ) - F(c ) x-c ) c r r > C (M ;c ,x) - M (c ) - {M (d ) - F(d )} + {M (c ) - F(c )} — n o r o r o r r o r r since M-F i s monotone increasing in [a,b] by theorem 1.5. It follows that I inf {C n(F;c r,x) - F(c r)} r c <x<d r r > y inf {C (M Jc ,x) - M (c )} - {M (b) - F(b)} > -° — u n o r o r o r by (1) and the fact M (b) - F(b) < e o o Similarly, using (2), (3), (4), we have I inf {F(d r) - C (F;d ,x)} > -« , \ sup {C (F;c x) - F(c )} < +» , r r r and 1 sup {F(dr> - C n(F;d r,x)} < +=° . Hence we have 46. (5) I sup |C n(F;c r,x) - F(Cy.') | < +» (6) I sup |F(d ) - C n(F;d r,x)| < +» r Now, we show that F i s AC on each E° . F i r s t , note that by lemma 2, we have that M is AC and m is AC on E.° , so that o — o k there exists a constant A such that 7 {M (x!) - M (x.)} > -A and V o l o i l 7 {m (x!) - m (x.)} < A for any f i n i t e set {[x.,x!]} of non-overlapping 7 O 1 O 1 1 1 intervals with end points in E^ . For such f i n i t e set {[x^,x^]} we have 0 < Y{M (x!) - M (x.)} - y{m (x!) - m (x.)} < M (b) - m (b) < e — ^ O 1 O X ^ o x o i — o o o since M q - mQ is monotone increasing and non-negative. Combining the above inequalities, we have for any relevant set {[x_^,x^]} , - A < y{M (x!) - M (x.)} < A + e , u o i o l o - A -£ < y{m (x!) - m (x.)} < A , so that we have o L o 1 o 1 (7) Both M and m are BV on E° . o o k We prove further that (8) i f M e M^([a,b]) , m e M^([a,b]) , then both M and m are BV on K • 47. This in fact follows from M=m + ( M - m ) and m = M - (M - m) o o o o since M and m are BV on E? by (7), and as M - m and M - m o o k J * o o as both are monotone in [a,b], they are also BV on E° We have noticed that M is AC on E.° and m is AC on E.° o — k o k With the result (8), we prove further that (9) i f M e Mj([a,b]) , m e Mj([a,b]) , then M is AC and m is AC on E,C Jk ' To see this, let {]c ,d [} be the intervals contiguous to E° in [a,b]. r r K Define M^(x) = M(x) for x on E° , and on each Jc^d^f M^ i s defined such that the graph of M^ i s the linear segment joining the points (c r,M(c r)) and (d^MCd^)) . Then i t is easy to see that MA i s C -continuous in [a,b] and C DM. (x) > -<» n.e. in [a,b] . Hence by lemma 1 n * and lemma 2. M^ is (ACG) in [a,b] , that is [a,b] = U E^ , and MA k i s AC on each E, , where E, i s closed. Also, M. is BV in [a,b] — k k * x since M i s BV on E° by (8). Let G(x) = M^(x) - (L) / M^(t)dt . a Then G as a difference of an (ACG) function and an AC function is i t s e l f (ACG) . Furthermore, G'(x) = 0 a.e. in [a,b] . Hence, using the Baire's category theorem and the V i t a l i covering theorem, i t can be shown that G is monotone increasing in [a,b] and hence is also non-negative in x [a,b] . Therefore, M^(x) - ( L ) J M^(t)dt >^ 0 for each x in [a,b] , and a x! M*(xj_) - M A(x ±) L j 1 M;(t)dt for any [x.,xp C [a,b] x. l X As / M].(t)dt is AC on [a,b] , i t follows that M^ i s AC on [a,b] 48. As M = M. on E_° , i t follows that M i s AC on E° . Similar argu-* k — k ments hold for P^-minor functions, and (9) is hence proved. Now, we are in a position to prove that (10) F i s AC on E° . To do this, let e > 0 be given. Choose M e , m e with M(b) - m(b) < e/2 . Then for each f i n i t e set {[x^ ,x_^ ]} of nonoverlapping intervals in [a,b] , we have 0 <_ £{M(x.p - M( X i)} - £{m(xj) - m( X i)} £ M(b) - m(b) < e/2. 1 {m(x\) - m( X i)} < I( F ( X p - F(x.)} < j>{M(x*) - M(x.)} since M - m , M - F , F - m are a l l non-negative and monotone increasing in [a,b] . For such relevant {[x.,x!]} , i f x.,x! e E° , and i f I X X X K. £(x!^ - x^) i s sufficiently small, by (9), we have £{M(x.p - M(x±)} > -e/2 and £{m( Xp - m(xj.)} < e/2 . Combining a l l the above inequalities gives -e < I {F(xp - F( X i)} < e , so that F i s AC on E° , proving (10). 49. As the C n_jD-integral and the P n_^-integral are equivalent by * induction hypotheses, i t follows from (5), (6) and (10) that F is AC n on E° by theorem II in [32]. As k is arbitrary and U E° = [a,b] , i t follows that F is AC^G on [a,b] , completing the proof of theorem 4. We remark that the technique used in this chapter i s motivated by studying the paper [16], where the C^P-integral was investigaged i n great detail. 50. CHAPTER III A SCALE OF SYMMETRIC CP-INTEGRALS AND THE MZ-INTEGRAL Burk i l l has defined a SCP-integral i n [7], which i s more suitable for application to the trigonometric series than the CP-integral. Although this SCP-integral has been investigated by many people, no scale correspon-ding to the CP-scale has appeared in the literature. One of our purposes in this chapter i s to use the general theory developed in chapter I to give an SCP-scale of integrals. As a preliminary, we prove some lemmas concerning the de l a Vallee Poussin derivatives in section 1 and state two well-known theorems concerning n-convex functions in section 2. The results essential to the definition of our scale of integrals are proved in section 3. After developing the SC P-integral in section 4, section 5 i s devoted to i t s connection to the n James symmetric P -integral scale [13]. By the MZ-integral, we mean the integral defined by Marcinkiewicz and Zygmund in [21]. This MZ-integral solves the coefficient problem of the convergent trigonometric series. Burkill also used the SCP-integral to solve the same problem. However, i n his proof, he used an integration by parts formula, which remains unproved up to now. We prove in the last section that the MZ-integral and the SCP-integral are in fact equivalent. This implies that the SCP-integral does solve the coefficient problem. 51. §1. THE SYMMETRIC de l a VALLEE POUSSIN DERIVATIVES. Let F be a function defined on a bounded closed interval [a,b] , and x e ]a,b[ . If there are constants B .••,B_.,(r>0), depending on x but not on h such that 1 r h 2 k •> - {F(x +h) +F(x-h)} - J Q B 2 k - ^ T T =o(h 2 r) as h -»• 0 , then § ^ i s called the symmetric de l a Vallee Poussin (s.d.l.V.P.) derivative of order 2r of F at x , and we write &2r = D 2 r F ^ * I C i s c-'-ear t n a t i f I>2 r F( x) exists, so does D^FCx) for k = 0,1,2,...,r-l ,.and D 2 kF(x) = B 2 k • If D 2 kF(x) exists for 0 <_ k <_ m - 1 , (m >_ 1) , define 6 2 m ( x , h ) = 6 2 m(F;x,h) by ,2m 1 m-1 , 2k ( 2 ) W! 92m ( x' h ) = 2 { F ( x + h ) + F ( x " h ) } " I (2k)! D2k F ( x> > and l e t (3) D 2 mF(x) = lim sup &2 (x,h) , h-*0 m D_ F(x) = lim inf 6 „ (x,h) — 2 m , n 2 m h-K) Then a f i n i t e common value for D„ F(x) and D„ F(x) implies that D„ F(x) 2 m — 2 m 2 m exists and equals this common value. 52. In a similar way, the odd-ordered s.d.l.V.P. derivative is defined by replacing (1) by r 2k+l d') |{F(x +h) - F(x-h)} - J g 2 k + 1 - o ( h 2 r + 1 ) as h -> 0 . Similar changes can be made i n (2), (3). The following lemma is an extension and generalization of lemma 4, (i) i n [33]. For a partial converse in the non-symmetric case, see lemma 10 in [21]. LEMMA 1. Let H be a function and H'(x) = G(x) in a neighborhood of x . I f for some n , D G(x ) exists, then D MH(x ) exists and o n o n+1 o is equal to D G(x ) n o Proof. The proof i s by induction on n . To see that i t i s true for n = 1 , consider for sufficiently small h > 0 , 0 , (H;x ,h) = % i [ H ( x +h) + H (x-h)] - H(x )} . I o l o o o h In order to apply l'Hopital's rule, l e t 1 hZ f(h) = i[H(x o+h) + H(x o-h)] - H( X q) , g(h) = f j - • Then f (h) -»• 0 as h -> 0 since H i s clearly continuous in a neighbor-hood of x . Also, g(h) 0 as h -*• 0 . Furthermore, g'(h) = h =}= 0 , 53. H'(x +h) - H'(x -h) G(x +h) - G(x -h) and f ( h ) 0 -° ° 0 g*(h) 2h 2h which approaches to D..G(X ) as h -»- 0 . Hence 1 o lim 0„(H;x ,h) = D,G(x ) , which is what we want to prove. 2 o i o Now, suppose that the conclusion of the lemma is true for n < r , where r >_ 2 . Then.we prove that i t is also true for n = r as follows. Suppose r i s even, r = 2m say. As D„ G(x ) exists, so does D., G(x ) 2m o 2k o for 0 < k < m - 1 , and hence by the induction hypothese, D„, ,,H(x ) — — J r > 2k+l o exists and equals D G(x ) for 0 < k < m - 1 . Consider <£iC o < W H ' > V h ) " 7 2 m ^ ^[H(x o +h) - H(x o-h)]- I ^ I ) T D 2 k + 1H(x o)} . h k=0 Applying l'Hopital's rule, one gets lim 0- ,.(H;x ,h) = D„ G(x ) , which complete the proof for even r , _ 2m+i o 2m o h->0 A similar argument w i l l give the case for r odd. Note that, in particular, we can apply lemma 1 to the case that H i s the Lebesgue integral of a continuous function G in some interval. Following James [13], we say that a function F i s n-smooth at x i f D -F(x) exists and lim h© (F;x,h) =0 . By a similar argument h+0 n in the proof of lemma 1, one has LEMMA 2 . Let H be a function and H'(x) = G(x) i n a neighborhood of X Q Then H is (n+1)-smooth at x i f G i s n-smooth at x o - o 54. LEMMA 3. Let H be a function and H'(x) = G(x) in a neighborhood of x . Then for n > 1 , o — (4) D G(x ) > D , H(x ) > D t 1H(x ) > D G(x ) n o — n+i o n+1 o n o whenever 0 (G;x ,h) makes sense. n Proof. By lemma 1, i f 0 (G;x ,h) makes sense, so does 0 ,.(H;x ,h) n o n+1 o The inequalities (4) then follow from the inequalities (cf [12], p. 359) n. f'(h)- ,, f(h) . r f(h) .. . e f'(h) lim sup — T T T T > lim S UP —7TC > lim i n f —7TT > lim l n f i \ h+0 8 ( h ) - h -0 g ( h ) ~ h+0 8 ( h ) ~ fr*0 8 ( H ) for suitable choices of f and g §2. SOME PROPERTIES OF n-CONVEX FUNCTIONS. For the definition of n-convex functions, we refer to the papers mentioned below. The f i r s t result we want, due to James [13], [15] but is proved in a more complete form by Bullen [2], gives a set of conditions., which are sufficient for a function to be n-convex. The second result gives some important properties of an n-convex function. Before stating these, \.e r e c a l l some concepts. • A function F defined on [a,b] is said to satisfy the condition (C, ) in [a,b] i f 55. (a) F is continuous in [a,b] ; (b) ^2k^ exists, is f i n i t e and has no simple discontinuities in ]a,b[ for 0 <_ k. <_ r - 1 ; (c) F is 2r-smooth at a l l points i n ]a,b[ except perhaps for points of a countable set. Similarly, the condition (C_ is defined, so that the condition (C ) zr+1 n makes sense for a l l integer n >_ 2 A linear set i s called a scattered set i f i t contains no subset that i s dense-in-itself. Note that the union of two scattered sets i s also scattered [20]. then (1 <^ k <_ r) i s called the Peano derivative of order k of F at x , written a, = F,,.(x) , where a, ,a 0, a are constants k (k) 1 2 r depending on x only, not on h . It is clear that i f F ^ ( x ) exists, so does D^F(x) and two are equal. But the converse i s not true in general. *~ . . If i t is true that F(x+h) - F(x) = I k=l r If F posseses Peano derivative F (k) (x) 1 < k < r - 1 write r-1 Y_(F;x,h) = F(x+h) - F(x) - £ .(x) k=l v ' 56. Then define F, . ,(x) = lim sup y(F;x,h) t r ; s h-K)+ F, \ , > F , N j F / . are similarly defined, and -(r),+ ( r ) , - - ( r ) , - 3 F, N , , F, N are defined in a usual way. (r),+ ( r ) , - 3 then THEOREM 1. (cf.[2], theorem 16). Let F satisfy the condition (C n) in [a,b] and (i) D^F(x) >_ 0 almost everywhere in ]a,b[ ; ( i i ) D F(x) > -» for x e ] a , b [ ~ S , S a scattered set; n ( i i i ) lim sup h 0 (F;x,h) > 0 > lim inf h 0 (F;x,h) for x e S . x+0 n ~ ~ h+0 Then F i s n-convex in [a,b] THEOREM 2. ([2], theorem 7). Let F be n-convex in [a,b] . Then (r) (i) F exists and i s continuous in [a,b] for 1< r < n - 2 , /^ \ til where F (x) denote the ordinary r derivative of F at x ( i i ) both F,, , F, n are monotone increasing in [a,b] ; ... tic—X) ,— vn—i.) ,+ ( i - ) - ( F ( n _ 2 ) ) + ' a n d F ( n - D , - = < F < n " 2 ) j : j (iv) F^ n (x) exists a f a l l except a countable set of points. 57. §3. THE SC -DERIVATIVE AND THE SC -CONTINUITY, r r Let r >_ 1 , F be C^^P (i.e. P^ 0 f chapter II)-integrable on [a,b] , x e ]a,b[ , Cr(F;x,x+h) as defined in chapter II, and r+1 Ar(F;x,h) = — {Cr(F;x,x+h) - Cr(F;x,x-h)} , SC D F(x) = lim inf A (F;x,h) . • • — — h+0 r The notations SC^D , SC^D then have the obvious meanings. We c a l l SCrDF(x) , i f exists, the symmetric Cesaro derivative of order r of F at x , or simply SC^-derivative of F at x If lim {C (F;x,x+h) - C (F;x,x-h)} =0 , F is said to be h-*0+ r r SC -continuous at x . It is clear that F i s SC -continuous at x r r whenever i t is C -continuous at x , and SC DF exists and equals r ' r M C DF(x) whenever C DF(x) exists. But, neither of the converses is r r ' true. It i s also easy to check that SC^DF is measurable (cf. theorem 8 below), LEMMA 4. For r >_ 0 , l e t F be C^-continuous i n [a,b] . Then F has no simple discontinuities in ta»b] . In particular, every ^P-primitive of a function on [a,b] has no simple discontinuities i n [a,b] Proof. For r = 0 , the result i s immediate since the C -continuity o J i s just the ordinary continuity. For r >_ 1 , suppose that X q e ]a,b] , and lim F(x) = B . Then for each e > 0 , there exists 6 > 0 such that x-*x-o 58. or B - e < F(x) < B + e for x - 6 < x < x , o o B - e < F(x) < B + e for x - h < x < x , o — o where h is such that 0 < h < 6 . Hence (B-e)(x -x + h ) r _ 1 < F(x)(x-x + h ) r - 1 < (B+E)(x-x +h) r~ 1 o — o — o for x - h < x < x , which implies that o — o r x o B - E < — (C , P ) - / (x-x +h) r F(x)dx < B + e — , r r-1 J . o — h x -h o for 0 < h < <5 , so that lim C (F;x ,x -h) = B . h-K)+ r ° ° But F(x ) = lim C (F;x ,x -h) = lim C (F;x ,x-h) . Hence F(x ) = B ° h+0 r ° ° h->0+ r ° ° Similarly, i f x E [a,b[ , and lim F(x) = B' , then x-»x + o F ( X Q ) = B' . Hence F has no simple discontinuities in [a,b] The last statement of the lemma is now immediate since by theorem 1.6, every C P-primitive i s C -continuous. _ " 59. LEMMA-5. For n >_ 0 , let F be C P-integrable on [a,b] , and for n x e [a,b] , let x G n(x) = (CnP) - / F(t)dt , a x G k(x) = (CkP.) - / G k + 1 ( t ) d t , 0 < k < n - 1 , a G(x) = G (x) . o Then (i) G i s continuous in [a,b] ; ( i i ) i f F is SC . n-continuous at x , then D G(x) exists and n+1 n D G(x) = G (x) for 0 < k < [•"-] , and G is (n+2)-smooth at x , n— ik n-zk — — I and 0n+2(G;x,x+h) = A n + 1(F;x,h) ; ( i i i ) i f F i s C -continuous at x , then G, ,., N (x) exists and n+1 (n+1) Goo(x) = G k ^ f o r 0 — k — n + 1 ' w h e r e G n + i = F • Proof. (i) is immediate since G i s just a C oP-primitive. For ( i i ) and ( i i i ) , note that by integration by parts, n k C n + 1(F;x,x+h) = ( n^'{G(x+h) - G(x) - £ G^x)} , h k=l (5) n k C (F;x,x-h) = (n+1iii {G(x-h) - G(x) - I G, (x)} n+l (-h) n + 1 k=l k ' k 60. for h =j= 0 with x + h e [a,b] . Hence for n even, say n = 2m , (5e) C n + 1(F;x,x+h) - C n + 1(F;x,x-h) (2m+l)! m «-2k {G(x+h) + G(x-h) - 2 I - ^ r r y G (x)} ; ,2m+l » / • - v - " / - L ( 2k)! "2k1 h k=l and for n odd, say n = 2m + 1 , (5o) C n + 1(F;x,x+h) - C n + 1(F;x,x-h) „ m ,2k+l " tG(x+h> - G(x"h) " 2 X W W » h k-U For both cases, i f F i s SC .--continuous at x , then D G(x) exists n+1 n and D G(x) = G „, (x) for 0 < k < [—] , and G is (n+2)-smooth n—2K. n—2K — — 2 at x , where [^ ] = the greatest integer less that ~ + 1 . Further-more, 0n+2(G;x,h) = A n +^(F;x,h) , proving ( i i ) . ( i i i ) follows from the equality (5) . REMARK. If D G(x) = G 0 1 (x) for 0 < k < [£] , and G i s (n+2)-n -2k n -2k — — 2 smooth at x , then F i s SC ..-continuous at x . This i s clear since n+1 replacing G (x) by D 0 1 G(x) i n (5e) and (5o) one has that n—2K n—2K C n + 1(F;x,x+h) - C n + 1(F;x,x-h) = ^ h © n + 2 (G;x,h) . 61. LEMMA. 6. For n > 0 , let F be C P-integrable on [a,b] , and — n SC . -continuous in ]a,b[ , and G be defined as in lemma 5. If n+1 n If (a) SC n + 1DF(x) >_ 0 a.e. in [a,b] , (b) SC n + 1DF(x) > -*o for x £ ]a,b[ ~ S , S , a scattered set, then G i s (n+2)-convex in [a,b] Proof. This i s immediate since by lemma 5, ( i i ) , and lemma 4, G satisfies a l l the conditions in theorem 1 with n + 2 replacing n THEOREM 3. For n >^ 0 , let F be C^-integrable on [a,b] and SC .,-continuous in ]a,b[ . If n+1 (a) SC L 1DF(x) > 0 a.e. i n [a,b] , n+I — (b) SC n + 1DF(x) > -oo for x e ]a,b[ ~ S , S scattered, (c) F i s C ..-continuous in B [a,b] , n+i then F i s monotone increasing in B Proof. Let G be defined as in lemma 5. Then by lemma 6, G i s (n+2)-convex in [a,b] , so that by theorem 2, (iv), G^ n + 1^ and hence ^(n+^) exists at a l l except a countable set of points. By theorem 2, ( i i ) , G, - v i s monotone increasing where i t exists. Thus the condition (c) V.n+1; and lemma 5, ( i i i ) imply that F is monotone increasing in B 62. THEOREM 4. For n >_ 0 , let F be CMP-integrable on [a,b] , and x p ]a,b[ . If F is SC ..-continuous at x , then F i s SC , o n+1 o n+2 continuous at x , and o * SC ..DF(x ) > SC ,0DF(x ) > SC .0DF(x ) > SC ..DF(x ) . n+1 o — n+2 o — n+2 o — n+1 o Proof. Note f i r s t that F i s C^^P-integrable on [a,b] by the consistency of the CP-scale. Let, for x e [a,b] , 3n(x) = (CP) - / F(t)dt , 3fc(x) = (CfcP) - / G f c + 1(t)dt for 0 <_ k <_ n - 1 , x H n + l ( x ) " ( C n + l P ) " / F ( t ) d t > a x V x ) = ( Ck P ) " / Hk+l ( t ) d t f o r ° £ k £ n a Then H. (x) = G, (x) for 0 < k < n and H (x) = (L) - f G (t)dt k+i K — — o ' o By lemma (5), ( i i ) , G i s (n+2)-smooth at x , so that H i s (n+3)-o o o smooth at X q by lemma 2. Hence by the remark following lemma 5, F is SC ,„-continuous at x . The inequalities * follow from lemma 5 and n+2 o lemma 3, completing the proof. THEOREM 5. Let D e a sequence of SC^continuous functions in ]a,b[ , and each is C^-continuous in a set [a,b] with a,b e B and the measure of B i s b - a . Suppose that M^x) M(x) as k <» 63. uniformly in B . Then M is SCn-continuous in ]a,b[ and C^-continuous in B . Proof. Given s > 0 , choose k such that for a l l x e B , |M(x) - M^(x)| < -^ e . For each c E B , choose 6 > 0 such that |Cn(Mk;c,c+h) - M^Cc)! < -|e whenever |h| < 6 with x + h E [a,b] Then l-C (M;c,c+h) - C (M, ;c,c+h)| < -^ e , so that |c (M;c,c+h) - M(c) I < e 1 n n K . 3 n whenever |h| < 6 with x + h e [a,b] , proving that M is C^-continuous at c That M is SC^-continuous at each point c E ]a,b[ is proved in a similar way, only replacing ^ ( c ) > M(c) in the above argument by Cn(M^;c-h,c) and C^(M;c-h,c) , h now being restricted to c + h E [a,b] §4. THE SC P-INTEGRAL. n Let X be the real line a the family of a l l half-open intervals, W the family of a l l subsets of measure zero. For each positive integer n and each lower derivate operator SC D , I i s defined by n n SC^DFCx) > -» except perhaps for a scattered set of points. We are going to consider "point functions" instead of "interval functions", so that by a base B in [a,b[ , we mean that B C [a,b] and a,b e B and the measure of B is b - a . Throughout this section, we w i l l consider the base mapping to be the another extreme case B(A) = the family of a l l bases in A 64. For each interval [a,b[ and each base B in [a,b[ , l e t SM^ta.M.B) = {M|M is (^-continuous in B and SC -continuous in ]a,b[} n Define — — _ SC P = (SM.SC D,B,W,I ) . n n n Then by theorem 3, and theorem 5, i t i s easy to check that sC nP is a derivate system on a , which furthermore satisfies the additional axioms (V5), (P6) , (M6), (14) in section 1.5, and also (M5') . Therefore, we obtain a SC nP-integral for n = 1,2,3,... , a scale of symmetric CP-integrals. It follows from theorem 1.7 that this scale is more general than the scale of Burkill's CP-scale in chapter II since SM n^ Mn and SC DF(x) > C DF(x) . n — n As for the CP-scale, we have the consistency theorem for our scale. THEOREM 6. If f is SC^-integrable on [a,b[ with base B , then f i s also SC n +jP-integrable on [a,b [ with base B Proof. This i s immediate from theorem 4 and theorem I, 7. REMARKS. (1) Note that the definitions cf SC^-integral and Burkill's SCP-integral (see [7] or section 6 below) have different families C in (I4)-scattered sets and countable sets respectively. However, the two integrals are equivalent. For, letting M^ be a P Q-primitive of M , one has SCDM(x) = D2M^(x) Hence from the remark by James at the end of [15], the set of points x 65. where SCDM(x) = -<» is a G set and, i f at most countable, i t must be a scattered. (2) Burkill in [7] list e d an integration by parts formula for the SCP-integral and stated that the proof followed from that given for the CP-integral in [5]. This i s not true since the proof in [5] used essentially the following inequality CD(MG)(x) > M(x)G'(x) + CDM(x)G(x) , but we do not have a similar inequality for the SCD-derivate. For example, le t M(x) = x 2 for x > 0 , (-x) * for x < 0 , k for x = 0 , where k is any constant, and let G(x) = -x . Then SC D(MG)(0) = -» £ -k = M(0)G'(0) + (SC^MCO))G(0) . Thus, whether the formula for SCP-integral in [7] is true remains an open question. B u r k i l l in a recent letter to me agreed with this and said that the same point had been made to him by a young Russian mathematician some years ago. If such an integration by parts formula exists for the SC^P-integral, then one can use this to define the SC2P-integral instead of using the C^P-integral. Then a more general scale would be obtained by induction. Such a scale would be useful in application to the Cesaro summable trigonometric series. 66. , 1 1 §5. THE SC P-INTEGRAL AND THE P -INTEGRAL, n As we mentioned in the introduction of this chapter, in this section we are going to investigate the relation of the P^^-integral and the SC nP-integral. TiH~X By P -integral, we mean the modified symmetric one as i n [15] For convenience, we give the definition of i t s major functions here. Let f be a function defined almost everywhere in [a,b] , and let a^, i = 1,2,3,..., n+1 , be fixed points such that a = a^ < a^ < ... a < a ,- = b . A function Q i s called a J , -major function of f n n+1 x n+1 over (a.) i f l (a) Q satisfies the condition (C ,,) i n [a,b] (cf §2); n+1 (b) P^+-j_ Q( x) ^ .f(x) almost everywhere in [a,b] (c) I) +]Q(x) > -oo , x e ] a,b [ «. S , S a scattere< (d) Q(a.) =0 for i = 1,2,3,...,n+l . THEOREM 7. Let f be SC nP-integrable on [a,b[ with base B . Then f i s P n + 1-integrable over (a.,;c) , where a = a 1 < a _ < ... <a < a ,n I' 1 2 n n+1 b , c e [a,b] . Moreover, letting 67. x F n(x) = (SCnP) - / f(t) , x e B , a x F k ( x ) = ( C k P ) " / F k + l ( t ) d t ' X ? [ a ' b ] ' ° - k - n " 1 a F = F o , one has for a < c < a ,, , s — s+1 c n+1 * (-1)S / f ( t ) d n + 1 t = F(c) - I X(c;a )F(a.) , (a ±) n i=l where X(c;a^) = n (c-a.)/(a.-a.) i s a polynomial in c of degree at most n Proof. Let M be a SC^F-major function of f on [a,b[ with base B , and let t, t„ t . x 1 2 n-1 G(x) = (C P) - / (C.P) - / (C„P) - / ...(C .P) - / M(t )dt dt ....dt.dt. o ' l 4 l J n—1 ' n n n-± 2. ± a a a a Then by lemma 4 and lemma 5, G satisfies conditions (a), (b), (c) in the above definition. Hence i f we set n+1 Q(x) = G(x) - I X(x;a )G(a ) , i=l 1 then Q i s a J n +^-major function of f over ( a j ^ • Similarly, a SC P-minor function m yields a J , ..-minor function n J n+1 n+1 q(x) = g(x) - I X(x;a )g(a.) , i=l 1 -where g i s defined similar to G 68. For e > 0 , i f we choose M , m such that n+1 M(b) - m(b) < e/[l + £ (c;a.)](b-a) , then the corresponding Q , q have i=l 1 l'Q(c) - q(c)| < |G(c) - g(c)| + I |A(c;a.)| |G(a.) - g(a.)| < e . Hence, the P^ ^ - i n t e g r a b i l i t y of f follows. The equality * follows as above by using the property that F R can be uniformly approximated in B by a sequence of SC^P-major or minor functions. COROLLARY 1. F, - (x) exists for each x i n B and D ,F(x) exists (iU n-1 for each x e ]a,b[ . Furthermore, F, s = F on B , and D, F = F. (n) n k k on ]a,b[ for k = 0,1,2,...,n-l , where F , F^ are those in theorem 6. Proof. By theorem 1.6, F i s C -continuous i n B and SC -continuous in J n n n ]a,b[ , so that the required results follow from lemma 5. COROLLARY 2. There exists a function which i s P n + 1-integrable on [a,b] but not SC^P-integrable on [a,b] Proof. This i s similar to that of Cross in [8*] for n = 1 . In fact, i f n i s odd, l e t F(x) = x cos for x 4 0 , 0 for x = 0 ; i f n i s even, let F(x) = x sin ^ for x ={= 0 , 0 for x = 0 . 69. In either case, let f(x) = F^n+"^'(x) for x =f= 0 , 0 for x =f= 0 . Then D n +^F(x) = f(x) for a l l x , including x = 0 , and as shown by T1*T"1 James in [13] , f i s P -integrable over any interval containing 0 However f i s not SC^P-integrable over [o,b[ for any b > 0 . For otherwise, i t would follow from corollary 1 that F ^ ( 0 ) exists. But .not even F ^ ( 0 ) .exists. COROLLARY 3. Let f be periodic with period 2b , b > 0 . For n >_ 1 , l e t m = [^rp] . Then i f f is SC^P-integrable on [-2(m+l)b , 2(n-m)b[ with base B , one has O /" . f < t ) d n + l f c " < S Cn P ) " J . f ( t ) d t • (2b) (a i) l-b,b[ where (a ±) = (-2(m+l)b , -2mb , -2(m-l)b , -2b, 2b, 4b, 2(n-m)b) The proof, exactly similar to that of Cross in [8] for the unsymmetric case, i s omitted. 2 REMARKS. (i) Skvorcov [36] has pointed out that a function P -integrable 2 over two abutting intervals is not necessarily P -integrable over their n+1 union. We give an example to show that P -integral has the same property for n >^ 2 . Let F be as defined in corollary 2. Consider the function f defined by f(x) = F ( n + 1 ) ( x ) for x e ]0,-] , 0 for x e [—,0] , 70. where i = 2 i f n is odd and i = 1 i f n is even. Then (cf [13]) f is P n + 1-integrable over [- — ,0] with P n + 1-primitive G = 0 on IT [- — ,0] , and P n + 1-integrable over [0,—] with P n + 1 - p r i m i t i v e F on ir IT • 2 [0,-] . For n = 1 , i t is well-known (cf [8**]) that f is not P -TT integrable over [-—,—] . We show that i t is also the case for n > 2 TT TT — Suppose, to the contrary, that f is P n + 1-integrable over t-^1,—] with Pn+"'"-primitive H . We show that f i r s t H / 1 N (0) and IT ir K-i) ,-then H.,.,. ,(0) exists. Note that on [-^ -,0] , H-G is a polynomial v.iy,+ TT of degree n at most ([13]), and so is H-F on [0,—] . Hence both TT (H-G).. _(0) and (H-F) . (0) exist. As G . (0) exists, we see \ i / , \1/ ,+ (1/ ,— that H. . (0) exists. To see that H,..* ,(0) exists, note f i r s t that v , — ( i j > + M-H is (n+l)-convex on [- —,—] (cf. [13]) for any J , -major function TT IT n+1 M of f on [- —,—] , so that D nH(0) exists since H = M - (M-H) . TT TT n-1 In particular, DJI(0) exists, where i = 2 i f n is odd and i = 1 i f n is even. If i t i s i = 1 , then H...... ,(0) exists since H,.... _(0) \l/,+ \ i / , — exists. If i t is i = 2 , then H i s smooth at 0 , so that h Q ) + ^ exists since (0) exists. Thus, we have proved that both ( H - F ) ^ + and H,.... .(0) exist. Then i t follows that F , - v ,(0) exists, a v.l^,+ v.1;,+ contradiction, and our proof is hence completed. ( i i ) Unlike that for p n + 1 - i n t e g r a l , note that our S c n p~ integral has the "additive" property by theorem 1.3. ( i i i ) Necessary and sufficient conditions for a function p n + 1 -n+1 integrable over two abutting intervals to be P -integrable over their union are under consideration. Note also that the comparison to Taylor's AP-integral might be interesting (cf. [8**]). 71. §6. THE MZ-INTEGRAL AND THE SCP-INTEGRAL. Throughout this section, X , a, B, hi, w i l l be the same as in section 4, and once a derivate operator V is chosen, I w i l l be defined by J?F(x) > -co except for a countable set of points. We show how to obtain using our general theory the MZ-integral of Marchinkiewicz and Zygmund [21] and the SCP-integral of Burkill [7], and then prove that they are in fact equivalent. For each P Q-integrable function M (see section II.1) on [a,b] , and for each x e ]a,b[ , let „ \ i . . 1 -i • • r I-** M(x-fu) - M(x-u) , I5sM(x) = lim inf — lim i n f J — du , h->0+ e-»0+ e 2 U and also r> \ u • c 1 i • • c M(x+u) - M(a) , BsM(a) = lim inf r- lim inf J du h->Of e->0+ e U „ N , . . _ 1 , . . j- M(b) - M(x-u) , BsM(b) = lim inf — lim inf J — * - du . h^0+ e-*0+ e U These are called the lower Borel derivates. We have THEOREM 8. _BsM i s measurable. Proof. F i r s t , note that the function <f)(M;x,h,e) = j M ^ x + U ^ ^ M ^ x u^ d u e is continuous in x . For, by the second mean value theorem (see [29], [1 6 ] ) , there exists T with e < T < h such that 72. 9 (M;x+Ax,h,E) -9 (M;x ,h,£) , T+Ax e+Ax T e+Ax = -r-{/ M(x+u)du - / M(x+u)du - / M(x-u)du + / M(x-u)du} T e T-Ax e ^ h+Ax T+Ax h T-Ax + — {/ M(x+u)du - / M(x+u)du - / M(x-u)du + / M(x-u)du} h T h-Ax T Note that T depends on Ax . However, as the P^-primitive as a point function i s continuous in the closed interval concerned, i t is uniformly continuous there. Hence each integral in the right hand side of the above equality tends to zero with Ax . Hence 9(M;x+Ax,h,e) -> 9(M;x,h,e) as Ax -y 0 , proving the continuity of 9 in x Now, l e t $(M;x,h) = lim inf 9(M;x,h,e) . Then $ is measurable in x since 9 is continuous in e . Furthermore, $(M;x,h) i s continuous in h since by simple calculations, $(M;x,h+Ah) - f h + A h M(x+u) - M(x^ul d u ^ 0 a s A h ^ 0 . h 2 U Hence j?s M(x) = lim inf — $(M;x,h) is measurable in x , completing h+0+ h the proof. THEOREM 9. Let B be a base in Ia,b[ and M be a function defined on [a,b] such that M i s C--continuous in B and SC -continuous h in ]a,b[ , and furthermore lim j .^(x+H,)— M ( x HZju exists ( f i n i t e e->0+ e . or infinite) for a l l x except perhaps for a countable set of points, 73. h =}= 0 with x + h e [a,b] . If _BsM(x) >_ 0 almost everywhere in ]a,b[ and ~JBsM(x) > _°° except for a countable set of points, then M i s monotone increasing in B Proof. Let be the P o-primitive of M . Then by lemma 30 in [21], D^M^Cx) >_ BsM(x) > JBsM(x) except for a countable set of points, where _ M1(x+h) + M (x-h) - 2M(x) DJ1 (x) = lim sup - . Hence, as a point function, 1 1 h+0+ h M^ is convex in [a,b] by theorem 2, and so M is monotone in B since M^'(x) = M(x) for x in B by the C^-continuity of M in B , completing the proof. Now, we are in a position to define the MZ-integral as well as the SCP-integral. For each base B in [a,b[ let SM([a,b[,B) = {M|M is C^-continuous in B and SC^-continuous in ]a,b[} , (i.e. the — 1 SM of section 4) and SMR([a,b[,B) = {M|M e SM([a,b[,B) and lim / M ( x + U ) ~*("*~^- du e->0+ e exists (f i n i t e or infinite) except perhaps for a countable set of points} Define SCP = (SM, SC^D, B, W,T) , SCPR = (SMR, SC^.B.W,!) , MZ = (SMR, Bg,B,W,T) . It i s easy to see that both SCP and SCPR are derivate systems on a That MZ i s also a derivate system on a follows easily from theorem 8 and theorem 9. Thus, we can define the SCP-, SCPR- and MZ-integrals. The SCP-integral i s just that of Burkill's in [7], while the MZ-integral i s just that of Marcinkiewicz and Zygmund in [21] except that the latter was defined by using Lebesgue integrals instead of the P Q-integral. 74. REMARK. It is easy to see that a l l the derivate systems SCP , SCPR and MZ satisfy the extra axioms in section I.4> except that SMR([a,b[,B) may not contain a l l the functions continuous in [a,b] . However, the function to used in the proof of theorem 1.12 belongs to SMR([a,b[>B) so that a l l the results in section 1.4 are applicable to the SCP-, SCP^-, and MZ-integral. To see that to e SMR([a,b[,B) , we need only show that lim f — ^ i ^ L - i i l du exists (finite or infinite) except perhaps e-0+ e r. . _> t0 f i n i t e , so that w(x+u) — w(x—u) w'(x) = +00, so that — 2 ^ — l s positive for small u ; in both „ , «. , . f h a(x+u) - ^ (x-u) , . t cases, we see that lim j — du exists. £-0+ £ 2 U Now, we establish two lemmas, which w i l l be used to prove the main result of this section (i.e. theorem 10 below). LEMMA 7. Let M e SMR([a,b[,B) . Then BsM(x) exists i f and only i f SC^DM(x) exists.* Proof. Let M, be the P -primitive of M . Then i t i s easy to see 1 o r J that SC1DM(x) = D M (x) and SC^DM(x) = D~2M (x) and so the conclusion follows from lemma 28 in [21], LEMMA 8. Let M e SM([a,b[,B) and SC;LDM(x) exist n.e. i n [a,b] . Then M e SMR([a,b[,B) . 75. t Proof.. Let SC.DM(x ) exist an d let ^(t) = / {M(x +u) - M(x -u)}du 1 o 1 o o o For 0 < k < h , h M(x +u) - M(x -u) h . / — * = ° — d u - / ^ u k 2 U k 2 U _ l n.(k.) . r h ] f e M d l l i " 2 1 h k + J. 2 Q u ' k u by integration by parts. By the SC^-continuity of M , ~* 0 as k + 0+ . For SC.DM(x ) f i n i t e , is bounded for small 1 o 2. ( ) u u ; for SC^DM(x) = -H» or -» , 2 ^ s °^ constant sign for small u u . In a l l cases, one sees that h . . h M ( X Q + U ) - M ( X Q - U ) lim / ^ 2 du exists, so that lim / - du k+0+ k u k+0+ k U exists ( f i n i t e or i n f i n i t e ) , completing the proof. THEOREM 10. The SCP- , SCP - and MZ-integral are a l l equivalent. K Proof. By lemma 7, one sees that the corollary to theorem 1.12 applies to the derivate systems SCPT1(=P) and MZ(=Pn) , so that the SCP -integral and the MZ-integral are equivalent. To see that they are also equivalent to the SCP-integral, note that by theorem 1.7, the SCP-integral i s more general than the SCP D-integral. It remains to show that the MZ-integral i s more general than the SCP-integral. To do this, l e t f be a SCP-integrable function. Applying theorem 1.12 to the derivate system SCP , one obtains for f an appropriate : _1 •• / 76. SCP-major function T and an appropriate SCP-minor function t . Then by lemma 8 and lemma 7, one sees that T, t are respectively relevant MZ-major and minor functions for f , so that i t is MZ-integrable, completing the proof. We have remarked that the integration by parts formula for SCP-integral stated by Burkill in [7] remains unproved. Hence his proof of theorem 5.2 in [7] (- the SCP-integral solves the coefficient problem for the convergent trigonometric series) breaks down. However, this theorem remains true by our theorem 10 since i t has been proved in [21] that the MZ-integral solves the coefficient problem. We remark that the proof in [21], without using integration by parts but using formal 2 multiplication of series (also see James' P -integral), applies to the SCP-integral too. CHAPTER IV AN ACP-INTEGRAL AND A SCALE"OF APPROXIMATELY MEAN-CONTINUOUS INTEGRALS. Many authors have generalized the continuous classical Perron integral to integrals that are approximately continuous; see, for example, Bu r k i l l [A], Kubota [19]. It would be nice i f one can generalize the Burkill's C^P-integral in the same way. We are only able to do so for n = 1 . One of our purpose in this chapter is to obtain such an ACP-integral, and then using a method due to Bullen in [3] to obtain an 2 AP -integral, and prove that they are equivalent in some suitable sense. E l l i s [9] has defined a scale of mean-continuous integrals, of which the definition i s simpler in the sense that the approximate derivative is used for a l l orders of this scale. With the same idea, we w i l l obtain a scale of approximately mean-continuous integrals, which i s more general than and seems more natural than (§1 below) the scale of E l l i s . §1. ON THE MEAN-CONTINUOUS FUNCTIONS. We prove that the mean continuity scale of E l l i s i s just Burkill's scale of Cesaro continuity (theorem 1 below), which gives a motivation for a more natural approximately mean-continuous integral (section 2). The GM-integral scale [9] starts from a function integrable in the general Denjoy sense (see Saks [30]). E l l i s called such a function 78. 1 x+h F M -continuous i f — / F(t)dt -»• F(x) as h -> 0 for each x . By 1 h J x theorem 1 below, this i s just a C^-continuous function, and hence i s special Denjoy integrable. This is why we say that the E l l i s integral seems somewhat unnatural in the sense that i t starts from the general Denjoy integrable functions. We recall that the M^-continuity in [9] was defined in the same way as the C^-continuity (cf section II.2) except that the ^ n _ j _ ~ integral was used instead of the C^P-integral. THEOREM 1. A function is M -continuous in an interval i f and only n i f i t i s C -continuous in the interval. n This has been proved by Sargent in [33], page 120. However, we give another proof here. Proof. Note that the GM^ ^-integral i s more general than the C^ jP-integral, so that a C -continuous function is M -continuous. To prove " n n the converse, let F be M -continuous in [a,b] . Then F i s GM .-' n n-1 integrable on [a,b] , and x+h n — (GM .) - / (x+h-t) n - iF(t)dt = F(x) + o(l) . n n-1 J h x as h -> 0 . Let x Fn (x) = (GM ) - / F(t)dt , n-1 n-1 J a x F k(x) = (GM^ ) - / F k + 1 ( t ) d t for 0 <_ k <_ n - 2 a 7 9 . Then, using the integration by parts formula for the GM^-integral, one gets that -7— 0 7 (GM^) / (x+h-t) " F(t)dt = F o ( x f h ) - F o(x) - J ^ F^) . X K—X Hence as h -> 0 , n F (x+h) = F (x) + V F, (x) + o(h ) , where F = F . o o , u. K n k=l th It then follows that F i s the n Peano derivative of F . A s o F i s continuous, i t follows from lemma 11.1 of James [13], that F o i s C -continuous in [a,b] , completing the proof. §2. A SCALE OF APPROXIMATELY CONTINUOUS INTEGRALS. This scale w i l l be defined inductively. In a manner analogous to the definition of the Cn-mean, the M -mean of a function F i s defined " n to be b M (F;a,b) = — 2 — • / ( b - t ) n _ 1 F ( t ) d t n (b-a) n a for any positive integer n , where the integral in the definition of M^ -mean is the general Denjoy integral, and the integral involved in the definition of M -mean for n > 2 i s the AM ..P-integral defined below, n — n-1 & The function F i s said to be AM -continuous at x i f n o 80. applim Mn(F;x,x+h) = F ( X Q ) , where "applim" means"approximate l i m i t " (cf Saks [30]). A function F i s (ACG) on a set E i f E can be covered by a countable sequence of closed sets on each of which F is AC (see section II.2). Note that (ACG) i s an inequality property as defined i n section 1.1. Let X, a, N be as in section 1.4, and for each A e a , let B(A) = {a A ' For each positive integer n , and for each A e a l e t AMn(A) = AMn(A,a^) = {M|M is AMn-continuous in A} , and for each M , and each x , let .™,/ s , . • ^ M(x+h) - M(x) ADM(x) = applim inf —* J— h-»0_ x+heA Let AM P = ( A U N , AD, 8, N, (ACG)) . Then we are going to show that n — AMnP i s a derivate system on a , so correspondingly we have an AMnP-integral for n = 1,2,3,... , thus obtaining a scale of approximately mean continuous integrals. The integral in the definition of M^ -mean for n. >_ 2 is i n the sense of AM^ ^P-integral. Thus, in defining the AM^P-integral for n > 1 , we assume that AM ,P-integral has been defined with some — n-1 properties, where the AM oP-integral i s taken to be the general Denjoy integral (see remark at the end of this section). 81. We remark that we might define in a similar way another scale of integrals starting from the AP-integral of Burkill [4]. However, doing this, we are unable to prove the consistency of the scale. That AM^ P i s in fact a derivate system on a follows easily from theorem 2 and theorem 3 below. THEOREM 2. For n = 1,2,3 i f F is AMn-continuous in [a,b] and (ACG) in [a,b] with ADF(x) >_ 0 almost everywhere in [a,b] , then F is monotone increasing in [a,b] Proof. This follows from the usual proof of monotonicity (cf. the proof of theorem II.2) by applying the Baire category theorem, the V i t a l i covering theorem and the following lemma. LEMMA 1. For n = 1,2,3,..., let F be AM r-continuous in [a,b] and monotone increasing in ]a,b[ , then F is monotone increasing and continuous in [a,b] Proof. F i r s t , we prove that F(a) <_ F(x) for each x e ]a,b] . Suppose to the contrary that F(a) > F(x ) for some X q e ]a,b] . Let e = F(a) - F(x ) . Then e > 0 and F(a) - F(t) > F(a) - F(x ) > e/2 for o — o a l l t e ]a,x o] since F is monotone increasing in ]a,b[ . Hence for each x e ]a,x o] , x M (F;a,x) = — (AM ,P) / (x-t) n - J-F(t) dt n . sn n-i J (x-a) a x 1 < — (AM P) - / (x-t) n - J"(F(a) - e/2)dt = F(a) - e/2 , , x-a) a 82. so that applim M (F;a,x) <_ F(a) - e/2 < F(a) , a contradiction of the x->a+ fact that F i s AM -continuous at a , n Similarly, one can prove that F(b) >_F(x) for each x e [a,bl , and hence F i s monotone increasing i n Ia,bJ To show that F is continuous in [a,b] , suppose to the contrary that F is not continuous at X q for some X q e [a,b] . Note that F(x +) and F(x -) exist (only one of them exists i f o o X q = a or b) since F is monotone in [a,b] . Again, by the monotonicity, either F ( X q - ) < F ( X q ) or F ( X Q ) K F ( X D + ) • Suppose that F(x ) < F(x +) , and let T = F(x l) - F(x ) > 0 . Then by a similar o o o o calculation to the one above, we have applim M N (F;x o,x) > F ( X q ) + T/2 > F ( X Q ) , x+x0+ a contradiction. If ^ C X Q ) > F C X O - ) ,.a similar argument can be given. Thus F i s continuous at each point of [a,b] Note that in the above arguments, we use the •'.;•>>=•?' property that i f f., f. are both AM .P-integrable with f, < f„ , 1 2 n—1 1 — 2 then J f ^ <_ jf^ ' Thus, the proof of this lemma is completed by the following theorem, which we prove by induction. 83. THEOREM (<^n) Let f 1 , f 2 be AM nP-integrable on Ia,b[ and f £ f 2 almost everywhere i n [a,b] . Then b b (AM P) - / f(t)dt < (AM P) - / f(t)dt . n = 0,1,2,3,... . a a Proof. The assertion i s true for n = 0 , since the AM oP-integral i s just the general Denjoy integral. Suppose that the assertion i s true for n = k - l , k > _ l . Then the assertion is also true for n = k by the definition of AM P-integral, and the proof is then completed by induction. THEOREM 3. For n = 1,2,3,... , let continuous functions such that F, ->• F k F i s AM -continuous, n Proof. We only prove i t for n = 1 . a similar proof can be given. {F, } be a sequence of AM -k ^ n uniformly i n Ia,b] . Then For n > 1 , using Theorem (>_ ,n-l) , Let c e Ia,b] , and given e > 0 , choose k so that |F k(x) - F(x)| < -je for a l l x in Ia,b] . Then by theorem (£,0) , we have iM^F ^ C j C+h) - M1CF;c,c+h) | < - J E i f h > 0 with c + h e [a,b] . As F^ is AM^-continuous at c , the set E^ of points x for which |M^(FkJC,x) - F]j^c) I > "3 s has zero density at c . For each x ^ E 1 and x near c , we have 84. |M1(F;c,x) - F(c) | <_ ^ ( F j c . x ) - M 1(F k;c,x)| + ^ ( F ^ c ^ ) - \Cc) | < e . As E is arbitrary, we see that F i s AM^-continuous at c , completing the proof. The general properties of our AM nP-integral follow from the general theory in Chapter I. In addition we prove the consistency of this scale. THEOREM 4. If n > 1 then an AM ..P-integrable function i s also — n-1 AM^P-integrable and two integrals are equal. Proof. For n = 1 , let f be AM QP-integrable with primitive F Then F being continuous i s AM^-continuous. It i s then easy to see that F i s both an AM^P-major and -minor function of f and the proof is then completed. Now, suppose that i t i s true for n = k , k >_ 1 . We prove that i t i s true for n = k + 1 . To do this, by theorem 1.7, i t suffices to show that i f F is AM^-continuous, then F i s AM^+^-continuous. As F Is AM^-continuous, i t i s AM^ ^P-integrable and hence i t is P-integrable and two integrals are equal by induction hypotheses. The AM^^-continuity of F then follows by applying the integration by parts formula, which we w i l l prove below, (Theorem 5). 85. THEOREM 5. Let F(x) = (AM P) - / f(t)dt , and a G (x) = / / / ... / g(t )dt dt ... dt_dt. , for x e [a,b] n J J J J n n n-1 2 1 a a a a where g i s continuous and of bounded variation. Then fG i s AM P-n n integrable over [a,b] and B 8 8 / (fG n)(t)dt = [FG] - J FG n_ 1Ct)dt a a a for a < a < 8 < b . Proof. We only prove i t for n = 1 . The general case can then be proved by induction. Without loss of generality, we suppose that g (and hence G^) is non-negative in Ia,b] . Let M be an AM^P-major function of f on Ia,b{ , we are going to show that MG^ i s an AM^P-major function of Fg + fG^ . To do this, we have to show that MG^ i s AM^-continuous, (ACG) i n Ia,bJ and ADCMG^ >_ Fg + fG 1 almost everywhere in Ja,b] That MG^ has the last two properties i s t r i v i a l . We prove that MG^ i s AM^-continuous as follows. It i s clear that MG^ i s AM oP-integrable. Using the integral tion by parts formula for the AM P-integral, we have . x+h. x+h x+h M1(MG1;x,x+h) = ± / MG1 = £ l O y ^ C t ) ! F x g J , t where F (t) = J M(u)du . Hence x x+h M1(MG1;x,x+h) = £ Fx(x+h)G](x+h) - | J F x g . x As G^ i s continuous and M i s AM^-continuous, one has applim f F (x+h)GXx+h) = M(x)G, (x) . h-K) h X 1 1 ± x+h F g i s continuous so that — J F g -* F (x)g(x) = 0 as h -> 0 X ft. X X X Hence applim M1(MGn;x,x+h) = M(x)G 1(x) + 0 = M(x)G n(x) , proving h+0 that MG^ i s AM^-continuous at x A s i m i l a r argument f o r an AM P-minor function proves that 3 3 Fg + f G 1 i s AMjP-integrable and / (Fg+fGj) = FGj Now, by theorem 1.6, F i s AM^-continuous i n [a,bj , so that F i s general Denjoy integrable, and hence so i s Fg . Hence Fg i s AM-jP-integrable, so that f G 1 = (Fg + f G ^ - Fg i s AM.jP-int e g r a b l e , by theorem 1.1. "Furthermore, 3 3 3 / fG^ = FGjj - / Fg , completing the proof, a a a 87. THEOREM 6. The AM nP-integral is more general than the ©^-integral of E l l i s i n [9]. Proof. It i s true for n = 0 , from the definition. Suppose that i t i s true for n = k , k > 0 . Then we prove that i t i s true for n = k + 1 . To this end, let f be GM^^-integrable. Then there exists a M^^-continuous (ACG) function F such that ADF = f almost everywhere. Thus by the induction hypotheses, F is AM^ P-integrable and hence AM^^-continuous. Hence, i t is easy to see that F serves as both AM^^P-major and -minor function of f , and hence f is AM^-jP-integrable and (AM^P) - Jf = (GM^) - / f , completing the proof. We end this section by the following remarks. REMARKS. (i) Instead of starting from the general Denjoy integral, we can start from the AP-integral, where AP is a derivate system defined by AP = ($°, AD, B, W, (ACG)) , where M° is the legitimate mapping defined i n section II.1, i.e. M° (A) = {M|M is continuous in A} . However, one can prove that i n fact this AP-integral is equiva-lent to the general Denjoy integral. Cii) For n = 1,2,3,,.. , l e t M^A) = {M|M is M^continuous in A} , and l e t M P = {W1, AD, 8, N. (ACG)} . Then MP is a ' n — ' n derivate system. The M^P-integral can be proved to be equivalent to E l l i s GM n~integral, which was defined by a descriptive method of Denjoy's. 88. §3. AN ACP-INTEGRAL AND AN AP ^ -INTEGRAL. For each, general Denjoy integrable function F , l e t M (F;x,x+h) - M(x) ACD F(x) = applim inf — h+0 h / 2 and l e t I be the inequality property defined by ACD F(x) > for a l l x . Let ACP = (AM1, ACD, 8, W, T) , where AM 1 , B, W are defined as in section 2. Then, i t can be checked that ACP is a derivate system on a . This ACP-integral is just a special case of Ridder's C P ^ - i n t e g r a l in {27]. We w i l l prove that the ACP-inte-2 gral i s equivalent to an AP -integral defined below in the sense of theorem 6 below. 2 Before defining the AP -integral, we prove a lemma. Let F be function such that ADF(x) (= applim F ( x + h ) —F(x). exists. i i * - A T\ s \ -,. . c FCx+h) - F(x) - hADF(x) , and define AD- F(x) = applim inf — L , and h-H) h similarly for AD2 F(x) . LEMMA 2. Let F be continuous such that ADF(x) exists for each x in [a,b] and AD2F(x) >_ 0 in [a,b] . Then F is convex in [a,b] 1 2 Proof. Let Gn(x) = F( x) + x for x in [a,b] , n = 1,2,3,... Then AD„G(x) = AD_F(x) + — > 0 . We prove that G is convex in / n z n n fa,b] , so that F , the limit of G^ , i s also convex in [a,b] , and the proof w i l l then complete . 89. To show that G i s convex in [a,b] , suppose to the contrary n that G i s not convex in [a,b] . Then there exists an interval n [a,B] C [a,b] such that the function (B-x)G (a) - (x-a)G (B) H(x) = G ( x ) ~ " n n B-a is sometimes positive in [a,B] . As H(a) = H(B) = 0 , the continuous function H assumes a positive maximum in ]a,Bt at X q say. Then we have H ( X q ) >_ H(X) for each x e [a»B] and A D H ( X Q ) =0 , so that A D 2 H ( x Q ) <_ 0 , which contradicts the fact that A D „ H ( X ) = A D . G (x ) > 0 2 o L n o Now, using the modified approach to the P n-integrals used 2 in [3], we define an AP -integral. Let f be a function defined on 2 [a,b] . Then a function M continuous in [a,b] is called an AP -major function of f on [a,b] i f (a) ADM(x) exists and i s f i n i t e for each x in [a,b] ; (b) AD^ ,M(x) >^ f (x) almost everywhere in [a,b] ; (c) AD^ ,M(x) > - o o for each x in [a,b] ; (d) ADM(a) = 0 = M(a) . 2 If -m i s an AP -major function of -f on [a,b] , then m i s called 2 an AP -minor function of f on [a,b] 90. 2 2 LEMMA 3. Let M be an AP -major function and m an AP -minor function of f on [a,b] . Then M-m is non-negative and convex on [a,b] Proof. Let G = M-m . Then G is continuous in [a,b] , ADG(x) exists and i s f i n i t e for each x in [a,b] , AD^GCx) >_ 0 for x e [a,b] ~ E , where E is of measure zero, and AD^GCx) > -» for each x e [a,b] Let E n be a G. set of measure zero with E C E. C [a,b] , I o x and let u> be the function used in the proof of theorem 1.12 with E/4 replacing e/b-a , and write x * (x) = (L) / <o(t)dt ; a then if»'(x) = (JJ(X) , 1 S continuous, ADiJ/ (x) = ip'(x) £ £ £ £ exists and is f i n i t e for each x , AD^if^Cx) = o)'(x) >_ 0 , AD^^g (x) = +50 for each x £ E , and 0 <_ ^ e ( x ) JS e For each E = ~ , write = i|/ , and define G = G + h n £ n n Then by lemma 2, G^ i s convex in [a,b] , so that the lim i t function G i s convex in [a,b] . That M-m i s non-negative follows from the convexity and the conditions M(a) - m(a) = 0 = ADM(a) - ADm(a) , completing the proof. 91. 2 2 In ease that f has both AP -major functions M , and AP -minor functions m on [a,b] , and sup m(b) = inf M(b) f +j° ,. we 2 m M say that f i s AP -integrable on [a,b] , and the common value, denoted 2 b 2 by (AP ) - / f(t)dt , i s called the AP -integral of f on [a,b] a 2 It follows from lemma 3, that i f f i s AP -integrable on [a,b] , so is on each [c,d] C. [a,b] THEOREM 6. f is ACP-integrable on [a,b] i f and only i f f i s 2 2 X AP -integrable on [a,b] . Furthermore, i f F(x) = (AP ) - / f(t)dt , x a then ADF(x) exists and ADF(x) = (ACP) - / f(t)dt , x u F(x) = (P) - / (ACP) - / f(t)dt du . a a Proof. We w i l l only prove the f i r s t assertion, since the proof of the last one being similar to that i n section III.5. (i) Suppose that f i s ACP-integrable on [a,b] . Let M be an ACP-major function of f on [a,b] , and x G(x) = (P) - / M(t)dt a Then G i s continuous on [a,b] with G(a) = 0 and ADG(x) = M(x) , 2 ADG(a) = M(a) = 0 , AD_2G(x) = ACDM(x) , so that G i s an AP -major function of f on [a,b] . A similar result holds for minor functions, 2 and the AP -integrability of f follows. 92. ( i i ) Suppose that f i s AP -integrable on [a,b] . Let G 2 be an AP -major function of f on [a,b] . Then ADG(x) exists and i s f i n i t e on [a,b] , so that ADG(x) is Denjoy integrable with G as a primitive. Furthermore, ADG(a) = 0 , ADG is AC^-continuous in [a,b] , and ACD (ADG) (x) = AD2G(x) , so that ADG is an ACP-major function of f on [a,b] . A similar argument for the minor functions completes the proof. 93. REFERENCES L.S. Bosanquet, A property of Cesaro-Perron integrals, Proc. Edinburgh Math. Soc. (2) 6 (1940), 160-165. P.S. Bullen, A criterion for n-convexity, Pacific Jour, of Math. 36 (1971), 81-98. , The P n-integral, to appear in Journal of the Australian Math Soc. J.C. B u r k i l l , The approximately continuous integral, Math. Zeit. 34 (1932), 270-278. , The Cesaro-Perron integral, Proc. London Math. Soc. (2) 34 (1932), 314-322. , The Cesaro-Perron scale of integration, Proc. London Math. Soc. (2) 39 (1935), 541-552. , Integrals and trigonometric series, Proc. London Math. Soc. (3) I (1951), 46-57. G. E. Cross, The relation between two definite integrals, Proc. Amer. Math. Soc. 11 (1960), 578-579. , The relation between two symmetric integrals, Proc. Amer. Math. Soc. 14 (1963), 185-190. , On the generality of the AP-integral, Canada J. Math., Vol, XXIII, No. 3 (1971), 557-561. H. W. E l l i s , Mean-continuous integrals, Jour. Canad. Math. Soc. 1 (1949), 113-124. L. Gordon and S. Lasher, An elementary proof of integration by parts of Perron integral, Proc. Amer. Math. Soc. 18 (1967), 394-398. R. Henstock, Generalized integrals of vector-valued functions, Proc. London Math. Soc. (3) 19 (1969), 509-536. E.W. Hobson, The Theory of Functions of a Real Variable, 3rd Ed., Cambridge (1927). R.D. James, Generalized n t b primitives, Trans. Amer. Math. Soc. 76 (1954), 149-176. 94, , Integrals and summable trigonometric series, Bul l . Amer. Math. Soc. 61 (1955), 1-15. , Summable trigonometric series, Pacific Jour, of Math. 6 (1956), 99-110. R.L. Jeffery, Non-absolutely convergent integrals, Proc. of Second Canada Math. Congress, Vancouver (1949). Y. Kubota, A generalized derivative and integrals of Perron type, Proc. Japan Acad. 41 (1965), 443-448. , On a characterization of the CP-integral, Jour. London Math. Soc. 43 (1968), 607-611. , An integral of Denjoy type II, Proc. Japan Acad. 42 (1966), 737-742. K. Kuratowski, Topology, New York, Acad. Press (1966). J. Marcinkiewicz - and A. Zygmund, On the differ e n t i a b i l i t y of functions and summability of trigonometric series, Fund. Math. 27 (1937), 38-69. J.C. McGregor, An integral of Perron type, U.B.C. thesis (1951), unpublished. E.J. McShane, Integration, Princeton (1944). , A Riemann integral that includes Lebesgue-Stieltjes, Bochner, and stochastic integral, Mem. Amer. Math. Soc. No. 88 (1969). S.N. Mukhopadhyay, On Schwarz dif f e r e n t i a b i l i t y IV, Acta Math. Acad. Sci., Hung., 17 (1966), 129-136. W.F. Pfeffer, An integral in topological spaces I, Jour. Math, and Mech. 18 (1969), 953-972. J. Ridder, Cesaro-Perron integration, CR. Soc. Sci. Varsovie 29 (1937), 126-152. II , Uber den Perronschen Integralbegr-if f und seine Beziehung zu den R-, L- und D-Integration, Math. Zeit. 34 (1931), 234-269. P. Romanovski, Integrale de Denjoy dans des espaces abstraits, Sbornik (N.S.) 9 (1941), 67-120. S. Saks, Theory of the Integral, Warsaw (1937). 95. [31] W.L.C. Sargent, On sufficient conditions for C P-integrable functions to be monotone, Q. Jour. Math. Oxford Ser. 12 (TJ941), 148-153. [32] , A descriptive definition of Cesiro-Perron integrals, Proc. London Math. Soc. (2) 47 (1941), 212-247. [33] , On the generalized derivatives and Ces^ro-Denjoy integrals, Proc. London Math. Soc. (2) 52 (1951), 365-376. [34] , Some properties of (1 -continuous functions, Jour. London Math. Soc. 26 (1951), 116-121. [35] V.A. Skvoicov, Some properties of CP-integrals, Mat. Sk. (N.S.) 60 (102) (1963), 304-324 (Math. Reviews 27, #264). 2 [36] , Concerning definition of P -and SCP-integrals, Vestnik Moskov.- Univ. Ser. I. Mat. Mech. 21 (1966), No. 6, 12-19 (Math. Reviews 34, #7765). [37] D.W. Solomon, Denjoy integration in abstract spaces, Mem. Amer. Math. Soc. No. 85 (1969). [38] S. Verblunsky, On a descriptive definition of Cesaro-Perron integrals, Jour. London Math. Soc. -(2) 3 (1971), 326-333. [39] , On the Peano derivatives, Proc. London Math. Soc. (3) 22 (1971), 313-324. u [40] Z. Zahorski, Uber die Menge der Punkte in welchen die ableitung unendlich i s t , Tohbku Math. Jour. 48 (1941), 321-330.
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On the integrals of Perron type Lee, Cheng-Ming 1972
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Title | On the integrals of Perron type |
Creator |
Lee, Cheng-Ming |
Publisher | University of British Columbia |
Date Issued | 1972 |
Description | Perron's method of defining a process of integration is through the use of major and minor functions. Many authors have adopted this method to define various integrals. In Chapter I, we give a very general abstract theory by first defining an abstract "derivate system" and then the corresponding Perron integral. We show that this unifies all the integral theories of Perron type (of first order) known to us, in addition the abstract theories of Pfeffer [26] and of Romanovski [29] are contained in our theory as particular cases. Chapter II is devoted mainly to the study of Burkill's C[sub n]P - integral. We know that the C[sub n]P - integral is based on the theorem that if M is C[sub n] - continuous in [a,b] , C[sub n] DM(x) ≥ 0 almost everywhere and C[sub n] DM(x) > - ∞ nearly everywhere in [a,b] , then M is monotone increasing in [a,b] . Burkill's original proof of this, [6] , contains an error and we give it a new and correct proof. We also give a correct proof of Sargent's theorem that if a function is C[sub n]P - integrable, then it is C[sub n]D - integrable, [32] ; the original proof contains a gap. A scale of symmetric CP - integrals and a scale of approximately mean-continuous integrals are obtained in Chapter III and in Chapter IV, respectively. The first one is more general than Burkill's CP - scale, while the second one is more general than the GM - scale defined by Ellis. Some other comparisons of various integrals are also given. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-03-22 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080482 |
URI | http://hdl.handle.net/2429/32730 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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