@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 "McGregor, James Lewin"@en ; dcterms:issued "2012-03-12T16:43:38Z"@en, "1951"@en ; vivo:relatedDegree "Master of Arts - MA"@en ; ns0:degreeGrantor "University of British Columbia"@en ; dcterms:description "In the definition of the Perron integral of a function f (x) over a closed interval [a, b] a major function M(x) and a minor function m(x) are required to satisfy the conditions (i) M(x) and m(x) are continuous on [a, b] and M(a) = m(a) = 0 ; (ii) - ∞ ≠ Ḏ M(x) ≥ f (x) ≥ D [overscored] m(x) ≠ + ∞. It is shown that without restricting the generality of the integral one may impose the additional condition (iii) M(x) and m(x) are differentiable on [a, b]."@en ; edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/41334?expand=metadata"@en ; skos:note "•art fig AN INTEGRAL OF THE PERRON TYPE by James Lewin McGregor 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 t o the standard required from candidates f o r the degree of MASTER OF ART5?, Members of the Department of Mathematics THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1951 ABSTRACT In the d e f i n i t i o n of the Perron i n t e g r a l of a f u n c t i o n f (x) over a closed i n t e r v a l [a, b] a major function M(x) and a minor function m(x) are required to s a t i s f y the con-d i t i o n s ( i ) M(x) and m(x) are continuous on [a, b] and M(a) • m(a) ••» 0 ; ( i i ) - oo ^ D M(x) > f (x) > D m(x) ^ + °° . It i s shown that without r e s t r i c t i n g the generality of the integral one may impose the a d d i t i o n a l condition ( i i i ) M(x) and ra(x) are d i f f e r e n t i a b l e on [a, b] . i A c k n o w l e d g m e n t The writer wishes t o express h i s thanks to Dr. R.D. James of the Department of Mathematics at the University of B r i t i s h Columbia f o r h i s advice and guidance. His numerous c r i t i c i s m s and suggestions proved invaluable i n the preparation of t h i s t h e s i s . 1. Introduction 1.1. In t h i s paper an i n t e g r a l of Perron type c a l l e d the PN-integral i s defined. The major and minor functions used i n the d e f i n i t i o n of the PN-integral s a t i s f y more r e s t r i c t i v e conditions than those s a t i s f i e d by the corresponding functions used i n the d e f i n i t i o n of the Perron i n t e g r a l . It is shown that the PN-integral i s nevertheless equivalent to the Perron i n t e g r a l . 1.2 Notation and d e f i n i t i o n s . I f a, b are r e a l numbers with a < b , the open i n t e r v a l {x; a < x < b j w i l l be denoted by either (a, b) or (b, a) and the closed i n t e r v a l {x; a < x < b The l i m i t of the right hand side as h -> 0 , e x i s t s and is equal to Kg(x) f o r almost a l l x . Hence almost a l l points of E are density points of E . w i l l be denoted by either [a, bj or The Lebesgue measure of a measurable set E i s denoted by ( E | . I f E i s a measurable set of r e a l numbers, a number x i s c a l l e d a density point of E i f 2 I f E i s a measurable set of r e a l numbers the notation F C « E w i l l used to indicate that F i s a non-empty closed subset of E consisting e n t i r e l y of density points of E . 2. The r e s u l t s of t h i s section are reproduced with minor modi-f i c a t i o n s from a paper of Zahorski ([2]) . 2.1 Theorem of Lusin-Menchoff. Let M-^ be a measurable set of positive measure included i n a f i n i t e open i n t e r v a l (a, b) , a < b , and l e t be a non-empty closed set such that Then there i s a closed set such that Suppose further that the set (a, b) - has measure zero, and l e t p be an assigned p o s i t i v e number. Then the set can be so chosen that l ^ ( x , x+h)| ^ i _ 2 _ m _ p W whenever x e M and 0 <|hj< l/m , where m i s any positive integer s a t i s f y i n g m > m (x) = - i Q max{x-a, b-x) Proof: Let J denote the open set (a, b) - M2 . This set consists of a f i n i t e or countable sequence { N . Then i n every case oo, T = ^£-4 J n • n=l £ Numbers i n square brackets r e f e r to the bibliography at the end of the paper. / I f J„ i s a non-empty i n t e r v a l (a n,/^n), <*n < /3n> l e t e l , n - V2 (« n + /8B) , | J n | . e2k,n = a n + , k > 1, k i n t e g r a l , e2k+l,n - P -n - , k > 1, k i n t e g r a l . If b i s not an end point of J n denote the open i n t e r v a l s ( Gl,n> e3,n }> ( e 2 k + l > °2k+3,a ) b ? Jl,n> J2k+l,n respe c t i v e l y . I f b i s an end point of J n denote the open i n t e r v a l (©l^rijb) by J-^ n and l e t J 2k+1 n d e n o t e 3 3 1 empty set when k > l . I f a i s not an end point of J n denote the open in t e r v a l s (92,n> Q\\,nh (@2k+2,n> e2k,n> b ? J2,n> J2k+2,n re s p e c t i v e l y . If a i s an end point of J n denote the open i n t e r v a l (a, © i > n ) by J 2 n and l e t J 2 k n denote, an. empty set when k > 1 . Since Mg i s not empty the case J n = (a, b) does not a r i s e . F i n a l l y i f J n i s empty l e t Jk,n denote an empty set for k > 1 . Only a f i n i t e number of the i n t e r v a l s . .Jk'n have measure greater than one. I f 0 < l Jk,nl < 1 then there i s a unique posit i v e integer m such that m — v K ' n l m+1 and J. w i l l be said to be of the mth c l a s s . k,n Let p be a positive constant. I f J k n is of the mth class, and i f l e t F, denote a closed set with the properties k,n f U ^ , J k . n « l . and I f K,.nl> 1 o r i f \\Jk,n M l l ' 0 . l e t pk,n d e n o t e a n empty set. It w i l l be shown that the set oo oc M3 - M2 * 2 2 ^ F k n n=l k=l ' s a t i s f i e s the conditions of the theorem. By, i t s construction, consists e n t i r e l y of density points of M-j_, and To show that Mj i s closed l e t x be a l i m i t point of . I f x e M2 then x e Mj because M? C M3 • Suppose x e M2 . Then x is at a positive distance from because M 2 i s closed, and x i s not an end point of (a, b) because these end points are at a positive distance from M3 . It f o l lows that x i s an i n t e r i o r point of J and therefore 00 00 x e S \\ n=l k-1 K ' n where ,^k,n denotes the closure of Jk,n • Hence x e ~3s,r f o r some pair of integers s, r . Since the end points of \"Jg r are at a positive distance, from Mj the point x i s i n t e r i o r to J e _ • Therefore there i s a neighborhood of x t>,x which contains no points of M ^ which do not belong to F s > r 5 Hence x i s a l i m i t point of the closed set F_ _ and x e F S ) r C : M 3 . This proves that contains a l l i t s l i m i t points and i s therefore closed. It follows that To prove that K 2 r m G ( x ) « 1 max^x-a, b-xj-Let /\\(x, h) = / ^ ( x , x+h)| - / ^ ( x , x+h)j = /(Mj-M^fx, x+h)j . F i r s t consider the case when h i s p o s i t i v e . Suppose 0 < h < l/m . Then A (x, h) = I I \\ J K A - F k J + - M 3)(x 1, x + h)| where the summation i s over a l l i n t e r v a l s J, which are k,n contained i n (x, x +h) , and where . x-^ i s the left-hand end point of the i n t e r v a l J, which contains x + h if. such an exists, wh i l e x^ = x + h i f no such interval e x i s t s . I f ^ n ^ (x» x + h ) then Mk,n| < h < l/m , and therefore K , A - \\ n \\ < 2 - m \" P _ 1 K , A l • Hence / l(x, h) < 2\" m \" P \" 1 X! |Jk n M l | + |( M1 \" K 3)(x 1,x+h)j hTk ,-m-p-l|^ ( x^ x + h ) | + | ( M i _ M 3)(x 1,x +h)| < 2 < 2 -m-p-1 |h| + [(1^ - M^Hxj, x+h)( . 6„ I f |(M, - M 3 ) ( X | , x + h)| » 0 , then A ( x , h) < 2-m-p-l| hj # I f |(Mj - M J H X J , x + h)| > 0 , l e t x^ denote the right-hand end point of the interv a l which contains x + h • Let x^ = slip |y; 3? e M 2 , y < x + h'j . Then x < x ^ < x 1 < x + h < x 2 , so x2 \" x l — x l ~ x3 1 b y c o n s t r u c t i o n o f t h e J k n» < (x + h) - x = h < l/m • Hence 1^ - M 3)(x 1, x + h)| < [(1^ - ^)(xlt x 2 ) | < 2 — P - ^ C x ! , x 2)| < 2° m-P- 1 |h| , so /\\(x, h) < 2- m-P\" 1|h| + 2- m-P- 1/h| = 2\"m-P |h| ; thus i n every case A ( x , h) < 2\" m\"Pjh| when G < h < l/m . Sim i l a r l y when h i s negative and 0 < |h| < l/m , i t can be shown that A(x, h) < 2- m-Pfh| . It follows that when 0 < |h| < l/m , I M 3 U , x + h)[ _ [ M j U , x+ h)| A(xf h) |h| \" ]h| \" |h| |Mx(x, x > h)/ 2 ^ > |h| - 7 . Given e > 0 , an integer N can be found such that 2 m P < e / 2 whenever m >N . And since x i s a density point of there i s a positive number 8 such that /Mjjx, x +h)/ — — - > 1 - e/2 whenever 0 < h < 8 . Hence i f m i s an integer and m>max\"(N, 1/8 } , then I V * , x +hj| i _ e Jh| whenever 0 < |hj < l/m . In particular, i f (a, b) - has measure zero, then |H,(x, x+h)| — ± - 1 whenever 0 < Jh < — ± — |h| so i f m i s an integer, m > m G(x) , then I V * ' X + h )l x 2-m-p |h| provided 0 < )hj < l/m . 2.2 Zahorski's Theorem: Let T be a bounded non-empty set of measure zero and of type G§ , and l e t |a, bj , a < b, be a f i n i t e closed i n t e r v a l containing T . Let e be a given posi-t i v e number. Then there is a function co(x) which is absolutely continuous on [a, b] and d i f f e r e n t ! able at every point of £a, bj , which has the properties (i ) ooT (x) = + «> i f x e T , ( i i ) 0 < co' (x)<+«> i f x e fa, b] - T , ( i i i ) u(a) = 0 , co(b) < e . Proof: Let J denote the open i n t e r v a l (a -1, b +1) , and l e t M • J - T = J . The sets J and GT are of type Fj. and hence t h e i r i n t e r s e c t i o n M i s of type F^ . . Let oo M = Y~\\ F' where {F k j - i s a sequence of closed s e t s . Let JJ, = (M| = ( j | and l e t t be a posit i v e number such that 0 < t < 1 . The sequence whose kth term i s the number converges to |M| = u- , Hence, f o r each posi t i v e integer k , there is an integer - n^ . such that (z! p{| > n | a ( l - t k ) , k - 1,' 2, 3, ... . (2) Let DI Q(X) be defined f o r x e J by the formula = min {x - (a - 1), (b + 1) - xj . and m Q(x) It w i l l f i r s t be shown that i t i s possible to determine a se-quence {^k} o r > closed sets such that 9. F k ° *k » k = x» 2> 3> ••• . (3) ^ O ^ C ' I , k - 2, 3, 4, ... , (4) [$ k| > j i ( l - t k ) , k - 1, 2,3, ... , (5) and i f x e $k-l , k ^2 , then, f o r every irfceger m which i s not l e s s than mo(x) , |$ k(x, x + h)l > 1 _ 2-m-k w h e n 0 < , h ( < l / m ( 6 ) oo and M = 2 Z $u . (7) k=l K Let ^ - F 1 # Then (3) and (5) hold f o r k - 1 . The set $1 i s closed, $1 O M f and J - ,M has measure zero. Hence by the theorem of 2.1., with p = 2 and ®± and M. i n place of M£ and % respectively, there i s a closed set P 2 such that $icr.p 2 o<» M , and i f x e $1, then f o r every integer m which i s not l e s s than m 0(x) , |P 2(x, x + h)| 0-m-2 , rt . k | . ./ -i—=— ' — • - > 1 - 2 whenever 0 < |h|< l/m . |h| Let $ 2 « P 2 + F 2 . Then (3), (4), (5) and (6) hold f o r k = 2 . The sets $3 , $4 , $ 5, ••• can now be determined one after another by the following procedure. Suppose the sets $1 , $2 , ... , $n » where n >^ 2 , have already been determined so that (2) and (5) hold f o r 1 < k < n , and (2) and (6) hold for 2 < k < n . Since P n + 1 O M , and i f x e $ n, then f o r every integer m which i s not l e s s than m 0(x) , | p ( x x + h)| 1 n + 1 , ' -i >1 - 2-m-n-1 whenever G < |h| < l/m . |h| Let c p n + 1 = p n + 1 + F n + 1 . Then (3), (4), (5) and (6) hold f o r k - n + 1 . F i n a l l y (7) follows from (2), (3) and (4) . This proves the existence of the sequence It w i l l next be shown that f o r each number of the form a, p/2^, where p and § are integers with q. > 0 and p > 2 , a closed set $p/2°L can be determined so that fcj^Ofc^. O M whenever p/2 q < p T/2 q T • (8) For q «= 0 and p >1 l e t ^ l ^ * 5 ^p» where $p i s the pth set of the sequence {i>^| . Then (%) holds f o r q = 0 . For q >0 , the sets are determined recursively as follows. Suppose that the sets have been determined f o r q < n and p ^ 2q- , where n > 0 , i n such a way that (8) holds for q < n . Let ®if s f o r p 2 2 n . Then when p: > 2 , and Q****, are closed sets such that IF?* tr* ®j*. a • c • M . Let ®if+i denote a closed set such that Z**' Z\"' Xs\" By the theorem of 2.1., such a set exists. The r e l a t i o n (8) holds for q < n + 1 , because, i f q < n + 1 and p > 2^ , there is a po s i t i v e integer r such that F i n a l l y , f o r each r e a l number X > 1 , l e t % denote the closed set - i — X P / 2 C . (9) X p ' It follows from ( 8 ) and (9) that i f X i s of the form X « P/2 Q , where P, Q are integers with Q > 0 , B > 2 Q , then $\\ = . Suppose X.' > X > 1 . Then there are integers p, q such that q > 0, p > 2 , and X' > 2+1 > £ - > X . 2* 2<* Hence by (8) and (0) Therefore C.°$y whenever 1 < X < X' . (10) I f x e M then because of (7) the set £X; x e i s not empty, + oo i f x e T , in f {X - 1; x e i f x e M This function is defined on J and Let Z(x) = 0 < Z(x) < + oo i f x e M (12) Let c be a r e a l number and l e t E e denote the set [x; x e J, Z(x) > c] . I f 6 < 0 then E e = J an open set. I f c > 0 then f o r any x 0 e E c there i s a X 0 such that Z(x Q) > X 0 - 1 > c . Hence x belongs to the open set J - - ^ • But i f x Is any point of J - $k • then x e $x i s f a l s e f o r X < X 0 so Z(x) > X 0 - 1 > c • It follows that J - C E e and hence that E c i s open. Therefore £ x ; x e J , Z(x) > c} i s an open set for every r e a l number c . This proves that Z(x) i s measurable and lower semi-continuous on J . I f _0_ denotes -the ordinate set of Z(x) over J then oo |.nI < o-|«il + ZZ (k - 1 ) |«k - *k-il k=2 oo < ZI (k - - |$k_!l ) k=2 oo _ < Z2 (k - 1) n t k _ 1 , by (5) k=2 . £U±t # ( 1 3 ) ( 1 - t ) 2 Hence Z(x) i s summable over J . Let t be chosen so small that o < t < m i n ( i . • Then from (13) follows < e . (H) It w i l l be shown that the function w(x) = J X Z(t ) d t , x e J , (15) s a t i s f i e s a l l the conditions of the theorem. By the above de-f i n i t i o n , co(x) i s absolutely continuous on £a, b] , and w(a) = 0 . Also, since Z(x) is non-negative on J , and be-cause of (14) , 0 < oo(b) = j b Z(t)dt < < e . Since Z(x) i s lower semi-continuous on J , D co(x) > Z(x) , x s J . (16) In particular, i f x Q e T , then Z(x Q) = + oo y s 0 o)'(x 0) exists and has the value + -°° . Hence (x) = + °° for x e T 'By (12), the proof w i l l be complete i f i t can be shown that (x) exists and is equal to Z(x) at each point of M . Let x e M f let Y be any number greater than z(x) , and let n by the smallest integer greater than Y+ 2 . 0 < |h| < - i then mQ(x) 0 < co(x + h) - (o(x) . 1 jX+h z ( t ) d t h >hj x = 1 / Z(t)dt + i r r A Z(t)dt l n l y % + 1 ( x , x+h) 'hl7(*n-%+l)(x, x+h) + J L / Z(t)dt . (17) N J (J-* n)(x, x+h) There i s a number V, such that Z(x) < < ^ , so that by (10), x e C-* . Thus x is a density point of $y+l and, i f 8 is any positive number, there is an integer N , N> rn^x) , such that (*, x+h)| > x _ s ^ G < | n | < l / . s # N Hence; |(* n - %+i)(x, x * h)| = l$n(x, x + h)| - |$ y +i(x, x+h)| < |h| - |h|(l - 8) - * M when 0 < |h| < l/N . Since Z(t) < n - 1 for t e $ n, i t follows that -RU Z(t)dt <_!_ (n - 1)8-|h| < n8 , when 0 < lh| < l/N . 14. Also, sincE Z(t) < Y f o r t e , -rir / 2<*>dt < * • (19) I\"\"1 J * F C I - « X , X+h) =7+1 OO Since J - $n - ZZ ($k+l - $k) + T' and since Z(t) < k f o r k=n t e - , i t follows that J L / Z(t)dt = J l - i . / Z(t)dt l h l J(J-» n)(x, x+h) k = n l h l J(*k +i\"*kM*, x+h) < S k l(^k-Hl-%)(x, x+h)| k=n |h| < fl k (1 - [ $ k ( x> x+h)l K ~ k=n lh| Let m be an integer greater than m^x) . Since n > V + 2 , x e ^k_2_ f o r every k > n , and by (6) x _ l $ k ( x , x+h)l < 2 ~ M - K W H E N G < \\h\\ < l/m . lhl Therefore, for 0 < |hl < l/m , _ i - / Z(t)dt < ZZ k 2 . , h I J(J-» n)(x, x+h) k = n < 2\" m £ k 2\" k . • k=l I f m -> oo then h •* 0 and l i m -rir- / Z(t)dt = 0 (20) h-*0 h / J(J-$n)(x, x+h) From (17), (18), (19) and (20) i t follows that D «(x) = l i m sup ^x+hl - < y + no . h-KJ h -15. Since t h i s holds f o r every p o s i t i v e number 8 , D c4x) < * , and since t h i s holds f o r every number ^ greater than Z(x) , D co(x) < Z(x) . (21) It follows from (16) and (21) that cof (x) e x i s t s and i s equal to Z(x) at any point x e M . This completes the proof of the theorem. 3. D e f i n i t i o n of the PN» i n t e g r a l . • . , < • ' . Throughout t h i s section, [a , b] w i l l denote a closed i n t e r v a l with a < b , and f(x) w i l l denote a function defined on [a, b] . 3.1. A pair of functions M(x), m(x) i s c a l l e d a pair adjoined to f(x) on [a, b] i f (i) M(x), m(x) are continuous on fa, b] and M(a) = m(a) = 0 ; ( i i ) the der i v a t i v e s M*(x), m* (x) e x i s t s and s a t i s f y the r e l a t i o n - oo ^ M» (x) > f (x) > m» (x) + + oo at every point of [a, b] . Functions s a t i s f y i n g the conditions imposeddon M(x)- and m(x) respe c t i v e l y are c a l l e d majors and minors for f ( x ) on [a, b} . 3.2. Lemma 1: I f M( x), m(x) i s a pair of adjoined to f ( x ) on [a, b] , then iVl(x) — m(x) i s monotone increasing on [a, bj . Proof: Because o f condition ( i i ) i n section 3.1., the difference M'(x) - m'(x) i s defined and non-negative f o r a l l x i n [a, b] . /Hence M(x) - m(x) has a non-negative derivative at each point of [a, b], and the lemma follows. 3 . 3 . I f M( x), m(x) is a pair adjoined to f(x) on [a, b] then M(b) * m(b) > M( a) - m(a) = 0 . The pair M ( x ) , m(x) is said to be e-ad.joined to f (x) on [a, b] i f M(b) - m(b) < e . The f u n c t i o n f (x) is said to be PN-integrable over [a, b] i f f o r every positive number e there i s a pair e-adjoined to f (x) on [a, b] . 3 . 4 Lemma 2: I f f(x) i s PN-integrable over [a, b] then f (x) i s PN-integrable over [a,^] f o r a l l such that a < £ < b . Proof: Let e be a positive number. By hypothesis there i s a pa i r M(x), m(x) e-adjoined to f(x) on [a, b] and hence on [a,£] . By lemma 1, M(f) - m(£) < M(b) - m(b) < e . Thus M ( x ) , m(x) i s a pai r e-adjoined to, f ( x ) on fa,^J , and t h i s proves the lemma. 3 .5 Lemma 3 : I f f(x) i s PN-integrable over [a, b] then the function F(x) which i s the i n f of a l l the majors M( x) is also the sup of a l l the minors m(x) f o r f (x) on [a, b] . Proof; Let m^x) be any minor f o r f (x) on [a, b] . I f M(x) i s any major f o r f(x) on [a, b] then by lemma 1 M( x) - m Q(x) > 0 ', a < x < b , and hence F(x) - m 0(x) > 0 , a < x < b . Since t h i s i s true f o r every\" miner m Q(x) , i t follows that i f 1 7 . G(x) i s the sup of a l l the minors f o r f (x) on [a, bj then Fix) - G(x) > 0 , a < x < b . If e i s any positive number, there i s a pair % ( x ) , m}_(x), e-adjoined to f (x) on [a, bj • Therefore F(x) - G(x) < % ( x ) - m^x) < % ( b ) - m^b) < e , a < x < b , and hence F(x) - G(x) < 0 , a < x < b . It follows that F(x) = G(x) , a < x < b , and the lemma i s proved, 3.6 I f f(x) i s PN-integrable over [a, b j , the fu n c t i o n Fix) defined i n lemma 3 i s c a l l e d an i n d e f i n i t e PN-integral of f(x) on [a, bj , and the number F(b) i s c a l l e d the PN-integral o f f(x) over [a, b] . The function F(x) is denoted by F(x) = (PN)J X f ( t ) d t . 4, Equivalence of the PN-integral and the Perron i n t e g r a l . In t h i s section i t is shown that the PN-integral is equi-valent to the Perron i n t e g r a l . The d e f i n i t i o n of the Perron i n -t e g r a l i s given below, and those of i t s properties which~will be needed are stated without proofs. (For the proofs see Saks [ l ] ) . 4.1. Let f (x) be a function defined on a closed i n t e r v a l [a, b] where a < b . A pair of functions / D f (x) > D t ( x ) t + 0 0 i s v a l i d . Functions s a t i s f y i n g the conditions imposed on (p(x) and f i x ) respectively are c a l l e d Perron ma.iors and Perron minors f o r f(x) on [a, bj . The Perron pair q>(x), f i x ) i s said to be e-adjoined T k e to f (x) on [a, b] i f «j*(b) - t(b) < e . ^ Function f (x) i s defined to be Perron integrable over £a, bj i f f o r every posit i v e number e there i s a Perron pair e-adjoined to f(x) on [a, b] . I f f(x) i s Perron integrable over {a, b] there i s a function *7 (x) which i s the i n f of a l l the Perron majors and the sup of a l l the Perron minors f o r f (x) on [a, b] • The function \"7 (x) is c a l l e d an i n d e f i n i t e Perron in t e g r a l of f (x) on [a, bj and i s denoted by 7 ( x ) = (P) J X f ( t ) d t . The Perron i n t e g r a l of f(x) over [a, b] i s defined to be the number *7(b) . The function \"7 (x) has the properties ( i ) i f (b) - f { b ) ) + 2oj(b) < £ + 2L = e . 2 4 I f x e T then since D cp(x) > -;o=>: , D M( x) ;> D cp(x) * co» (x) = + 00 . Hence M'(x) exists and s a t i s f i e s M»Cx) = + <*> > f ( x ) If x e [a, b] - T then cp T(x) e x i s t s and i s f i n i t e , so M'( x) exists and i s f i n i t e and s a t i s f i e s M'(x) = cp'(x) = D (x) > f ( x ) \" . -Hence f o r a l l x i n fa, bj , M(x) has a deri v a t i v e s a t i s f y i n g . 00 ^ M' (x) > f(x) . A sim i l a r argument shews that f o r a l l x i n fa, bj , m(x) has a derivative s a t i s f y i n g f (x) > m» (x) + + eo , Hence M(x), m(x) i s a pair e-adjoined to f ( x ) on fa, bj . Since \"e i s any positive number i t follows that f (x) is PN-integrable over f a , bj . This shows that the condition is necessary and completes the proof of the theorem. Bibliography [l] Saks, S., Theory of the Integral, Hafner, New York, 1937« 1 welchen die Math. J., 43, 12} Zahorski, Z., Uber die Menge der Punkte i nAbleitung unendlich i s t , Tohoku * 321 - 330, 1941, "@en ; edm:hasType "Thesis/Dissertation"@en ; edm:isShownAt "10.14288/1.0080630"@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 "An integral of the Perron type"@en ; dcterms:type "Text"@en ; ns0:identifierURI "http://hdl.handle.net/2429/41334"@en .