@prefix vivo: . @prefix edm: . @prefix ns0: . @prefix dcterms: . @prefix skos: . vivo:departmentOrSchool "Science, Faculty of"@en, "Mathematics, Department of"@en ; edm:dataProvider "DSpace"@en ; ns0:degreeCampus "UBCV"@en ; dcterms:creator "Cuttle , Percy Mortimer"@en ; dcterms:issued "2012-02-14T17:43:10Z"@en, "1953"@en ; vivo:relatedDegree "Master of Arts - MA"@en ; ns0:degreeGrantor "University of British Columbia"@en ; dcterms:description "The Denjoy Perron Integral and the Perron Integral are known to be equivalent. It is shown that a Denjoy Perron Second Integral may be defined by extending the concept of Generalized Absolute Continuity. The Denjoy Perron Second Integral is shown to include the Perron Second Integral."@en ; edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/40696?expand=metadata"@en ; skos:note "A DENJOY-PERRON SECOND INTEGRAL by Percy Mortimer Cuttle A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department of MATHEMATICS We accept t h i s t h e s i s as conforming to the standard required from candidates for the degree of MASTER OF ARTS. Members of the Department of Mathematics THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1953 A b s t r a c t The Denjoy Perron I n t e g r a l and the Perron I n t e g r a l are known to be e q u i v a l e n t . I t i s shown t h a t a Denjoy Perron Second I n t e g r a l may be d e f i n e d by extending the concept of G e n e r a l i z e d Absolute C o n t i n u i t y . The Denjoy Perron Second I n t e g r a l i s shown to i n c l u d e the Per r o n Second I n t e g r a l . Acknowledgement The w r i t e r wishes t o express h i s thanks t o Dr. R.D. James of the Department of Mathematics at the U n i v e r s i t y of B r i t i s h Columbia f o r h i s guidance a a d v i c e , and f o r h i s suggestions and c r i t i c i s m s which proved i n v a l u a b l e i n t h e p r e p a r a t i o n of t h i s paper. I INTRODUCTION 1 . D e f i n i t i o n s 1 . The f u n c t i o n of a r e a l v a r i a b l e , F ( x ) , i s define d f o r a l l x . 1 . 2 The second o s c i l l a t i o n of a f u n c t i o n F(x) over an 2 i n t e r v a l I = ( a,6 ) denoted by 0 (F, I) i s the maxi of the numbers and where mum mn Sup a < t x < t 2 < <3 F ( t 2 ) - F(a) F ( t i ) - F(a) t 2 - a tjL - a m Sup a < t < t < /3 F(6 ) - F ( t 2 ) F(6 ) - F ( t x ) & - t. 3 - t i 1 . 3 The f u n c t i o n F(x) i s s a i d t o be a b s o l u t e l y continuous i n the second r e s t r i c t e d sense on a bounded set E, denoted 2 by AC , i f t o every € > 0 there corresponds an n > 0 x -such t h a t , f o r every f i n i t e sequence of non overlapping i n -t e r v a l s 1^. whose end p o i n t s belong to E, the i n e q u a l i t y ^ > l l k | < T\\ i m p l i e s t h a t l o 2 ( F , I R ) < £ 1 . 4 The f u n c t i o n F(x) i s s a i d -to be g e n e r a l i z e d a b s o l u t e l y 2 continuous i n the second r e s t r i c t e d sense, denoted by AC G , on a bounded set E, i f F(x) i s continuous on E and i f E i s expressible as the sum of a sequence of bounded sets, on 2 each of which F(x) i s AC . x 1.5 The second r e s t r i c t e d o s c i l l a t i o n of a function F(x) over an i n t e r v a l I = ( a , Q ) , denoted by W(F, I ) , defined i f and only i f F'(a) and F f ( ^ ) exist and are f i n i t e , i s the maximum of the numbers n^ and n^ , where Sup n l a < t < /3 F ( t> I - F»(a) n 2 = Sup a < t < S F»-( 0 ) -F ( 6 ) - F(t) 6 - t 1.6 The upper (lower) r e s t r i c t e d second o s c i l l a t i o n of a function F(x) over an i n t e r v a l I = (a, (3 ) denoted by ¥ (F, I) , (W (F, I) ), defined i f and only i f F'(cx) and F* ( ft ) exist and are f i n i t e , i s the maximum (minimum) of the numbers n-j_ , (n^) and n^, (n 2) where n n (m) = S U P ( I N F ) ul> v i i l ' a < t < /3 Sup (Inf) n_, (n, ) = I' u ' 15 - t 1 . 7 The function F(x) i s said to be upper absolutely con-tinuous i n the second r e s t r i c t e d sense, i f F'(x) exists and 3 i s f i n i t e at every p o i n t of E, and i f t o every 6 > 0 the r e corresponds an TC\\ > 0 such t h a t f o r every f i n i t e se^-quence of non o v e r l a p p i n g i n t e r v a l s I k w i t h end p o i n t s b e l o n g i n g to E, the i n e q u a l i t y ^ Jl^l < T\\ i m p l i e s t h a t Z~W(F I k ) < € . 1.8\" The f u n c t i o n F(x) i s s a i d t o be lower a b s o l u t e l y continuous i n t h e second r e s t r i c t e d sense, denoted by 2 LAC , over a bounded set E i f F'(x) e x i s t s and i s f i n i t e at every p o i n t of E, and i f to each 6 > 0 th e r e corresponds an T\\> 0 such t h a t f o r every f i n i t e sequence of non o v e r l a p p i n g i n t e r v a l s 1^ whose end p o i n t s belong to E, the i n e q u a l i t y 5. jl^i < T\\. i m p l i e s t h a t I l ( F I k ) >- £ . 1.9 The upper and lower g e n e r a l i z e d extreme second de-ii r i v a t i v e s , denoted by A F(x) and 5 \" F ( x ) , r e s p e c t i v e l y , are d e f i n e d by the formulas [ l ] L\\ F(x) - h -> 0 h + k 0 r Lim i n f ' 2 b«F ( x ) = h -> o h + k k -» 0 F(x + h) - F(x) _ F(x) - F(x - k) h k F(x + h) - F(x) _ F(x) - F(x - k) h • k where h and k may tend t o zero i n any manner. // 1.10 I f ^ m ( x ) , (d) 8\"M(x) > - oo , A / m ( x ) < + o o , f o r a l l x i n (a,c) with the p o s s i b l e e x c e p t i o n of an enumerable s e t . 2.2 A f u n c t i o n f ( x ) d e f i n e d i n an i n t e r v a l (a,c) i s s a i d t o be i n t e g r a b l e over (a, b, c ) , a < b < c i f f o r every ( > 0 t h e r e e x i s t s a major f u n c t i o n M(x) and a minor fun c -t i o n m(x) such t h a t 0 < m(b) - M(b) < € . 2.3 I f f ( x ) i s i n t e g r a b l e over (a,x,c) then F(x) e x i s t s such t h a t F(x) i s the sup of a l l major f u n c t i o n s M(x) and F(x) i s the i n f of a l l minor f u n c t i o n s m(x) and 5 5 f i x ) dx = - F(x) . axe Furthermore (a) M(x) - F(x) i s convex i n (a, c) , (b) F(x) - m(x) i s convex i n (a, c) , (c) M(x) - m(x) i s convex i n (a, c) # 2 . 4 I f f ( x ) i s i n t e g r a b l e over (a, b, c ) , a < b < c then t h e r e e x i s t s a major f u n c t i o n M(x) and a minor f u n c t i o n m(x) such t h a t c - x £ x - a c - b ' b - a 0 < m(x) - M(x) < Max. f o r a l l x i n (a, c ) . 2 2 2 2 S e c t i o n 3 F u n c t i o n s AC G , UAC , LAC and AC x' x ' x x u Denjoy [ l j showed t h a t i f - oo < 5\"F(x) < A F(x) < + °° on a' set E then F' (x) e x i s t s and i s f i n i t e on E. T h i s r e s u l t i s used t o f i n d s u f f i c i e n t c o n d i t i o n s t h a t a 2 2 2 2 f u n c t i o n F(x) be AC G , UAC , LAC , or AC on X X x x a set E. Theorem 3 . 1 I f F(x) i s a continuous f u n c t i o n which f u l f i l s the c o n d i t i o n - ° o < S\"F(x) < Z\\ F(x) < at a l l p o i n t s of a set E, except perhaps those of an 2 enumerable subset, then F(x) i s ACG on E. ' x 6 Proof. For each pos i t i v e integer n, l e t A n denote the set of points x of E such that 0 < h, 0 < k, 0 < h + k < - implies -n < n + k F(x + h) - F(x) F(x) - F(x - k) h k and f o r every integer i l e t A n denote the common part of A n and of the i n t e r v a l i i + 1 n > n oo oo * - I Z n=l i=- oo n • I f I i s any i n t e r v a l (a, j3 ) whose end points belong i to one of the sets A^ , then f or a < x - k < x < x + h < f t where x e A^ we have 2 fF(x + h)' - F(,x) F(x) - F(x - k) + k L h k If we l e t k tend to zero In t h i s expression < n. - n < 2 h £(x_lhi_-_Flxi _ < n, Si m i l a r l y i f h tends to zero - n < h F , ( X , . . F(x) - F ( x - k) < n. Hence for a < t-^ < t 2 < 0 - n < ^ t - a F ( t 2 ) - F(a) t 2 - a - F'(a) < n. Hence - 5 ( t 2 - a) < I i t 2 l - Z J M . F t ( a ) < n .( t - a) , We o b t a i n 1 F ( t 2 ) - F ( a ) F ( t ! ) - F ( a ) n n t 2 - a t^Ta J < 2 ( t 2 \" a ) ^ ( t l - a ) , and hence F ( t 2 ) - F ( a ) F(t 1-) -F(a) t 2 - a t-j_ - a < § [(t2-a J + Ct-j^ -a )]< n( (B-a) S i m i l a r l y F ( B ) - F ( t 2 ) fi - t 0 F ( 8 ) - F ( t x ) a - t 1 < n(ft - o) , and hence 0 (F, I) < n ( l | . Then f o r any sequence of non o v e r l a p p i n g i n t e r v a l s b e l o n g i n g to one of the set s A 1 n 1 0 2 (F, I. ) < n I | l j . The r e f o r e F(x) i s AC on each o f the s e t s A n and hence 2 AC G v on E. Theorem 3 . 2 I f AF{x) < + °° and F T (x) e x i s t s and i s f i n i t e on a set E, t h e n E i s the sum of an enumerable sequence o f subsets E„ on each o f which F(x) i s UAC Proof For each p o s i t i v e i n t e g e r n l e t A n denote the set f o r which the i n e q u a l i t y 0 < h + k < — i m p l i e s 2 L FECX +>) - F(x) F(x) - F(x - k)1 . n . I E k J £ n and f o r h + k [ ' every i n t e g e r i l e t A^ denote the common par t of the set oo oo A n and the i n t e r v a l ( - , — - — ) . Then E = ^~ y A n i + 1 J. Then E = _ n=l i = - oo F o r any i n t e r v a l I = (a, 13 ) w i t h end p o i n t s b e l o n g i n g t o a set A we have f o r a l l a < x - k < x < x + h < ft n , — where x e A n h + k F ( x + h) - F(x) _ F(x) - F(x - k) h k < n L e t t i n g k tend t o zero i n t h i s e x p r e s s i o n we o b t a i n F(x + h) - F(x) _ F , ( x ) < n h _ and, l e t t i n g h tend t o zero F'(x) - F U » - k F U ' k ) < f k • • F o r a < t < (3 we have ^ - - , F ( t t ) - F ' ( a ) < f ( t - a ) , and F ' ( 0 ) - F < ^ ^ g ( t ) < n ( a . t ) ^ Hence W (F, I) < n | l | . T h e r e f o r e f o r any f i n i t e sequence of non o v e r l a p p i n g i n t e r v a l s w i t h end p o i n t s b e l o n g i n g t o a s e t £ W (F I k ) < n £|l I < 6 . For Z|l£ < | . n T h e r e f o r e F(x) i s UAC on each of t h e s e t s A n . Theorem 3 . 3 I f 8 \" F(x) > - °° and F'(x) e x i s t s and i s f i n i t e ah a set E, then E i s the sum of an enumerable 2 sequence of subsets Sm on each of which F(x) i s LAC V . Proof S i m i l a r t o Theorem 3.2 Theorem 3.4 I f F'(x) e x i s t s and i s f i n i t e at a l l p o i n t s 2 of a set E, and i f F(x) i s * AC X on E, then F(x) i s 2 2 both UAC and LAC V on E. Moreover i f F(x) i s both 2 2 2 UAC X and LAC X on E then F(x) i s n e c e s s a r i l y AC X on E. Proof: I f F'(x) e x i s t s and. i s f i n i t e on E, we have f o r any i n t e r v a l I = (a , / 9 ) wit h end p o i n t s b e l o n g i n g t o E, F ( t ? ) - F ( a ) F ( t , ) - F ( a ) - F' t-^ - a f o r a < t± < t 2 < 13 F ( t 2 ) - F ( a ) F t t - ^ - F l a ) t 2 - a \" t-^ - a F ( t 2 ) - F ( a ) F ( t 1 ) - F ( a ) _ Hence- n x - n± < ^ _ a t l - a ' ^ n l \" *L • F ( 3 ) r F ( t 2 ) F ( B ) - F ( t x ) S i m i l a r l y n 2 - n 2 < 3 - t 2 8 - t± < n 2 - * 2 . Hence G 2 F I < W(F I) - ¥(F I ) . 2 2 T h e r e f o r e i f F (x) i s both UAC X and LAC X then F(x) 2 i s AC X on E. However from the d e f i n i t i o n s n l < m l • \\ < m2 , Si > - m l , £ 2 m 2 * T h e r e f o r e W(F,I) < 0 2 ( F , I ) , W(F,I) > - 0 2 ( F , I ) . T h e r e f o r e i f F(x) i s AC on a set E then F(x) i s 2 2 X both UAC X and LAC X on E. 4 . P r o p e r t i e s of Convex Fu n c t i o n s The r e s u l t s o f t h i s s e c t i o n are reproduced from the work of Hardy, L i t t l e w o o d and Polya [ 5 ^ 4 . 1 I f g(x) i s convex i n an i n t e r v a l (a,c) and bounded above i n some i n t e r v a l i n t e r i o r t o (a,c) then g(x) has the f o l l o w i n g p r o p e r t i e s : (a) g(x) i s continuous i n (a,c) ; (b) The . l e f t and r i g h t hand d e r i v a t i v e s g2(x) and g!(x) e x i s t vfor a l l x i n (a,c),* T (c) g M x ) < g j ( x ) f o r a l l x i n (a,c) ; (d) The d e r i v a t i v e g 1 ( x ) e x i s t s f o r a l l x i n (a,c) w i t h the p o s s i b l e e x c e p t i o n of an enumerable s e t . 5. P r o p e r t i e s o f major and minor f u n c t i o n s , and of the f u n c t i o n F(x) = - / f ( x ) dx. axe 'I Den j o y has shown t h a t A F(x) < + °° i m p l i e s t h a t Fix) < F{x) and t h a t 8»F(x) > - 0 0 i m p l i e s t h a t F ( x f < F ( x ) + [ l ] . However F(x) > F(x) except on an enumerable set and F(x) 2. F(x) except on an enumerable set [ 3 ] From these r e s u l t s i t f o l l o w s t h a t 1 . A m(x) < + 0 0 i m p l i e s m(x)= mCx) except on an enumer able s e t . 2. S , fM(x) > - oo i m p l i e s Mt>0« MCx) except on an enumerable, s e t . Theorem 5.1. Let F(x) = - \\ f ( x ) dx, M(x) axe any major f u n c t i o n , m(x) any minor f u n c t i o n . Then F' (x) , M'(x) and m' (x) e x i s t and are f i n i t e :on the i n t e r v a l (a,c) except perhaps on an enumerable s e t . Proof: M(x) = m(x) + g(x) where g(x) i s convex Th e r e f o r e M~(x) = m~(x) + g'(x) , • + + and M x = m ( x ) + g ' ( x ) , Therefore M (x) - M +(x) = m\"(x) - m +(x) , 0 = m\"(x) - m +(x) , and hence m~(x) = ni +(x) T h e r e f o r e ~m+(x) = m~(x) < m~(x) = m +(x) < m +(x) except perhaps on an enumerable s e t . Therefore m'(x) e x i s t s , except perhaps on an enumerable sub-s e t . -+ _+ Furthermore M (x) = m (x) + g'(x) ^ - M~(x) = m\"(x) + g 1 (x) , Therefore M +(x) - M~(x) = m +(x) - nf (x) = 0 . I t f o l l o w s t h a t M +(x) = M~(x) < JM'(X) = M +(x) < M +(x) except perhaps on an enumerable subset. T h e r e f o r e M' (x) e x i s t s , except perhaps on an enumerable subset. S i m i l a r l y s i n c e M(x) = F(x) + h(x) where h(x) i s convex F*(x) e x i s t s except perhaps on an enumerable subset. Theorem 5 . 2 I f f o r a major f u n c t i o n M ( x ) , the i n e q u a l i t y 8 \" M(x) > - oo i s s a t i s f i e d on an i n t e r v a l (a, & ), then the i n t e r v a l (a , / 3 ) i s the sum of at most an enumerable i n f i n i t y of non o v e r l a p p i n g subsets on each o f which M(x) 2 i s LAC x Proof; From Theorem 5.1 M T (x) e x i s t s on (a,Q ) except perhaps on an enumerable s e t . Let E be the subset of (a , / 3 ) on which M ' (x) e x i s t s . Then (a, 3 ) = E + I where I i s an enumerable s e t . By 00 0 Theorem 3.3 E = 2 E , at most an enumerable i n f i n i t y of n=l 2 00 s e t s on each of which M(x) i s LAC X . Now I 0 = \"2. a n n=l where each i s a s i n g l e p o i n t . Hence M(x) i s t r i v i a l l y 2 00 00 LAC on each a . The r e f o r e ( a , 3 ) = ^ E n + ^ a n , n n=l n=l at most ah enumerable i n f i n i t y o f non o v e r l a p p i n g subsets 2 on each of which M(x) i s LAC Theorem 5.3 I f f o r a minor f u n c t i o n m(x) , the i n e q u a l i t y Z\\m(x) < + °° i s s a t i s f i e d on an i n t e r v a l (a, 6), then the i n t e r v a l (a,/3 ) i s the sum of at most an enumerable i n f i n i t y of non o v e r l a p p i n g subsets, on each of which m(x) 2 i s UAC X . Proof: S i m i l a r t o Theorem 5*2 Theorem 5.4 Let F(x) = ~f f ( x ) dx and l e t (a,A) axe be an i n t e r v a l i n t e r i o r t o the i n t e r v a l (a, c ) . Then F(x) 2 i s AC G on (a, 0 ). 13 oo Proof: By Theorem 5 .1 we can w r i t e (a,6 ) = E + X ' n=l where F ' ( x ) , M' (x) and m'(x) e x i s t and are f i n i t e on E and each a^ i s a s i n g l e p o i n t . 1 oo By Theorem 5.2 E = 5 . E n where M(x) i s n=l L.A.C. on each E . n Nov/ F(x) = M(x) - G(x) where G(x) i s convex. T h e r e f o r e f o r any sequence of non o v e r l a p p i n g i n t e r v a l s I k = ( a k , 3 K ) w i t h end p o i n t s b e l o n g i n g to one of the E n , and such t h a t £lj < n, we have f o r a, < t, < 3 K ' k k k n F ( t k ) - F ( a k ) a. F'(a k) M ( t k ) - M ( a k ) *k \" a k - M»(a k) G ( t k ) - G ( a k ) t k - a k - G'(aJi and n F ( 0 k)-F(t k) M'(3 K)-M ( l \\ ) - M ( t k ) \" 3 k - t,_ G(Q k)-G(t k) ^ - K Since G(x) i s convex G(tv) - G(au) ( n , 0\"' ,. k ) < ^ : °'**> < a . , B K ) Whence F ( t k ) - F ( a k ) \\ \" a k - F ' ( a k j ^ *M(t k)-M(a k) \\ \" a k - M'(ak) F ( f 3 J - F ( t k ) x K P K - t k M(6 K)-M(t k) G ' ( ^ k ) - G ' ( a k ) G'(3K)-G'(ak). Hence ^ W(F,I k) > £ W(M,Ik) - Z [ G ' ( ^ ) - G ' ( a k ) ] . k=l \" k=l Since g(x) i s convex k=l Z [ G * ( 6 ) - G T (a, )\"] < G » ( / 3 ) - G ' ( a ) . k=l 14. 2- n r 1 T h e r e f o r e Y W(F,I k) > £ W(M,L ) - |G»(3) - G'(a}| . «t? ~ k=i ~ G(c) - G(6 ) However G*(^ ) < c - [3 a n ( * G t ( a ) > G < a ) - G< a> . T h e r e f o r e ~~ a - a G M f l ) - O . ( a ) < ° ( 0 ) - G( 3 ) . MSI^SUI . But M(a) = M(c). = F(a) = F ( c ) = 0 . T h e r e f o r e G(a) = G(c) = 0 and hence G ' ( 6 ) - G ' ( a ) < \" G ( Q J - , — c - D a - a whence - |>'( ft ) = G* fa j ] > ^ + . L J c- B a-a However G(x) = - [F(X) - M(x)] . T h e r e f o r e - [G'(/D ) - G ' ( a ) l > _ F( 6 )-M( ft ) _ F(a)-M(a) q - -ft a - a From Theorem 2.4 F(x) - M(x) < m(x) - M(x) < Max. ~ £ , £__ £ — — c-o * b-a f o r 0 < m(b) - M(b) < 6 , a < b < c . We choose m(x) and M(x) such that 0 < m(b) - M(b) < min (c - b) £ o , (b - a ) € c . The r e f o r e F( ft ) - M( ft ) . 6 _____ ^ „ o > F(a) - M(g) < € a - a ° 1 5 . n n I t follows that Y W (F,I, ) > Z W(M,I ) - 2 6 . . . k=l ~ K k=l K 2 Therefore F(x) i s LAC Y on each E . n 2 S i m i l a r l y E • 2 . sm w h e r e m ( x ) i s U A G X o n e a c h s m 2 and hence F(x) i s UAC„ on each S . Itl OO CX) oo Now (d>) = Z S E n S m + 21 a n n=l m=l n=l 2 2 and F(x) i s both LAC and UAC on each E n S m 2 x x and t r i v i a l l y AC on each of the points a.. x 2 \" Therefore F(x) i s AC G x on (a, 3 ) . 2 Section 6 . The d i f f e r e n t i a b i l i t y of the i n t e g r a l . This section i s reproduced with minor changes from a paper by James and Gage £2} . 6 . 1 Let f ( x ) be integrable over (a,b,c ) and l e t F(x) =. - $ f ( x ) dx . axe Then f o r almost a l l x i n (a, c) the function D nF(x) i s f i n i t e and equal to f ( x ) . This r e s u l t together with Denjoy's proof [l\\ that D\"F(x) = F\" (x) (the second ordinary approximate oa derivative of F(x) ) almost everywhere shows that i f f ( x ) 2 ( i s P integrable and F(x) = - ) f ( x ) dx then F(x) X2 axe i s AC G i n any i n t e r v a l (a, ft) i n t e r i o r to (a, c) and that 16. F\" (x) = f ( x ) f o r almost a l l x i n ( a , 6 ) . oa 2 S e c t i o n 7. P r o p e r t i e s o f AC G Y F u n c t i o n s . 2 Theorem 7.1 I f F ( x ) i s AC G x on a set E, then F ' ( x ) e x i s t s almost everywhere on E. Proof: L et I be an i n t e r v a l ( a , B ) with end p o i n t s ' a , 3 b e l o n g i n g to E and such t h a t a i s not i s o l a t e d on the r i g h t and 0 i s not i s o l a t e d on the l e f t . • Let | l | < nr\\ 2 Since F(x) i s AC G x on E we have f o r a < t ^ < t 2 < ^ F ( t 2 ) - F ( a ) Fit^ - F ( a ) Since a i s not i s o l a t e d on the r i g h t l e t tj_ approach a through p o i n t s o f E, and l e t t 2 approach (3 , then - I < 4% i ^ M - F + ( a , < _ | i = E _ . p % , < t . Hence F ^ l a ) and F + ( a ) are f i n i t e . S i m i l a r l y _ < < F - ( S ) - F < 8 ' - F ( A ) < F-(6) - F ' B > - < e . F_\"(0) and F~ ( B ) are f i n i t e . Let E^ denote the subset of E of p o i n t s which are not i s o l a t e d on both the l e f t and the r i g h t . Then f o r every p o i n t o f E-j_ the two D i n i d e r i v a t i v e s on the same non i s o l a t e d s i d e are f i n i t e . Hence F'(x) exists almost everywhere on E-^ [ V ] • Moreover the set of points isolated on both the l e f t and the right i s c l e a r l y enumerable. Therefore F* (x) e x i s t s almost everywhere on E . The D i n i derivatives are known to be measurable [43 when F(x) i s defined over a measurable set E; and i f i n addition the Di n i derivatives are VB over a set C E then they are approximately derivable almost everywhere on E [ V ] . We use these r e s u l t s to prove , , 2 Theorem 7»2 I f F(x) i s AC on a bounded measurable set E, then F\" (x) exi s t s almost everywhere on E. 7 oa Proof: F'(x) ex i s t s almost everywhere on E (Theorem 7»U» Let be the set on which F'(x) i s defined . E = E, + E„ where E i s of measure zero. 1 o o Since E i s measurable so i s E^ and |E| = | E-J Let I = (a,6 ) be any i n t e r v a l with end points belonging to We have - 0 2(F,I) < F ( ^ ^ ( < X ) -F'(a) < 0 2(F,I) , - 0 2(F,I) < F'(B) F ( j f ] \" F ( a ) < 0 2(F,I) . n - a ~ 7 Hence - 2 0 2(F,I) < F'(6 ) - F»(a) < 2 0 2(F,I) , 2 and hence F»(x) i s AC on the set E. i f F(x) i s AC 1 x on E-^ . Moreover F'(x) being AC on E-j_ i s necessarily VB on E1 [4] . Now on F'(x) c o i n c i d e s w i t h the D i n i d e r i v a t i v e s . T herefore F'(x) i s approximately d e r i v a b l e almost every-where on and hence almost everywhere on E. S e c t i o n 8. D e f i n i t i o n of the I n t e g r a l . 2 D e f i n i t i o n &.1 f ( x ) i s s a i d t o be D x i n t e g r a b l e over an i n t e r v a l (a,c) i f there e x i s t s a f u n c t i o n F(x) such t h a t 2 1. F(x) i s AC G on ( a , c ) . x ' 2. F\" ( x ) = f ( x ) almost everywhere. :oa 2 The f u n c t i o n - F(x) i s c a l l e d the i n d e f i n i t e D x i n t e g r a l o f f ( x ) . 2 D e f i n i t i o n 8.2 The d e f i n i t e D x i n t e g r a l of f ( x ) over 2 a < b < c, i s g i v e n by the formula the i n t e r v a l ( a , b , c ) , denoted by Dv f f ( x ) dx , a,b,c J? j f ( x ) dx = F(a) + £-=-^ F(c) - F ( b ) . x a,b,c C a c \" a 2 Theorem 8.3 I f f ( x ) i s P i n t e g r a b l e i n the i n t e r v a l • 2 (a,b,c) then f ( x ) i s D i n t e g r a b l e i n any i n t e r v a l i n t e r i o r t o ( a , c ) . Proof: By Theorem 6.1 i f F(x) = - P x / f ( x ) dx then axe F o a = f ^ a l m o s t everywhere an ( a , c ) , and from Theorem 5 . 4 F(x) i s AC G x on any i n t e r v a l (a,/3 ) i n t e r i o r t o ( a , c ) , 2 hence f ( x ) i s i n t e g r a b l e i n the D x sense. B i b l i o g r a p h y Denjoy, A., Lecons sur l e C a l c u l des C o e f f i c i e n t s C a u t h i e r - V i l l a r s , Paris, 1 9 4 1 . James, R.D. and W.H. Gage, T r a n s a c t i o n s of the Royal S o c i e t y of Canada, t h i r d s e r i e s , s e c t i o n 111, Volume XV, Ottawa, 1946. McShane, E . J . I n t e g r a t i o n . P r i n c e t o n , 1944 . Saks, S. Theory of the I n t e g r a l . Warsaw, 1937. Hardy, G.H., L i t t l e w o o d , J.E. and Polya, G. I n e q u a l i t i e s . Cambridge, 1934. "@en ; edm:hasType "Thesis/Dissertation"@en ; edm:isShownAt "10.14288/1.0080658"@en ; dcterms:language "eng"@en ; ns0:degreeDiscipline "Mathematics"@en ; edm:provider "Vancouver : University of British Columbia Library"@en ; dcterms:publisher "University of British Columbia"@en ; dcterms: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."@en ; ns0:scholarLevel "Graduate"@en ; dcterms:title "A Denjoy-Perron second integral"@en ; dcterms:type "Text"@en ; ns0:identifierURI "http://hdl.handle.net/2429/40696"@en .