A HISTORY OF THE DEFINITE INTEGRAL by BRUCE VICTOR KALLIO B.A., U n i v e r s i t y of B r i t i s h Columbia, 1961 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 reouired standard. THE UNIVERSITY OF BRITISH COLUMBIA August, 1966 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 avail able for reference and study. I further agree that permission.for extensive copying of t h i s thesis f o r scholarly purposes may he granted by the Head of my Department or by his representatives. I t 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 of The University of B r i t i s h Columbia Vancouver 8, Canada i i ABSTRACT T h e ' d e f i n i t e i n t e g r a l has an I n t e r e s t i n g h i s t o r y . I n t h i s t h e s i s we trace i t s development from the time of ancient Greece (500-200 B. C.) u n t i l the modern p e r i o d . We place s p e c i a l emphasis on the work done i h the nineteentn century and on the work of Lebesgue (1902) . The t h e s i s i s d i v i d e d i n t o f o u r p a r t s arranged roughly c h r o n o l o g i c a l l y . The f i r s t part t r a c e s the developments i n the period from the f i f t h century B. C. u n t i l the eighteenth century A. D. Secondary sources were used i n w r i t i n g t h i s h i s t o r y . The second part recounts the c o n t r i b u t i o n s of the nineteenth century. The o r i g i n a l works of 'Cauchy, D i r i c h l e t / Riemann, Darboux, and S t i e l t j e s are examined, the t h i r d part i s concerned w i t h the development of measures i n the l a t t e r part of the nineteenth century. This work leads to the Lebesgue i n t e g r a l . The f i n a l p a r t i s a b r i e f survey of modern idea s . i i i TABLE OP CONTENTS Page I A BRIEF HISTORY OF EARLY CONTRIBUTIONS 1 I I . DEVELOPMENTS IN INTEGRATION DURING THE 1 7 NINETEENTH CENTURY I I I THE DEVELOPMENT OF MEASURES - THE 29 LEBESGUE INTEGRAL IV A MODERN GLIMPSE 39 V BIBLIOGRAPHY *3 i v LIST OP ILLUSTRATIONS Page (1) Archimedes' h e u r i s t i c method f o r f i n d i n g area....4 (2) Use of the Method of Exhaustion. . . . .5 (3) C a v a l i e r i ' s use of i n d i v i s i b l e s . . . . . • 9 (4) Fermat's procedure, f o r f i n d i n g area 11 • under curve y = x.|j (.5) W a l l i s ' procedure'for f i n d i n g area .13 . . . . 2 under curve y = x V ACKNOWLEDGMENT I wish to express my thanks to Dr. Maurice Sion f o r suggesting the t o p i c , and f o r h i s great a s s i s t a n c e during the w r i t i n g of the t h e s i s , and to Dr. Robert C h r i s t i a n f o r h i s encouragement i n i t s p r e p a r a t i o n . The generous f i n a n c i a l support of the N a t i o n a l Research Co u n c i l and the U n i v e r s i t y of B r i t i s h Columbia i s ' g r a t e f u l l y acknowledged: CHAPTER ONE A B r i e f H i s t o r y of E a r l y C o n t r i b u t i o n s The Idea of the d e f i n i t e i n t e g r a l arose from the problems of c a l c u l a t i n g l e n g t h s , areas, and volumes of c u r v i -l i n e a r geometric f i g u r e s . These problems were f i r s t solved w i t h some success by the mathematicians of ancient Greece. Probably the e a r l i e s t attempt at a s o l u t i o n was one devised f o r c a l c u l a t i n g areas of c u r v i l i n e a r f i g u r e s . I t can be traced back to two Greek geometers, Antiphon (430 B. C.) and Bryson (450 B. C.) . They attempted to f i n d the area of a c i r -c l e by i n s c r i b i n g r e g u l a r polygons, and then s u c c e s s i v e l y doubling the number of s i d e s . By t h i s procedure they hoped to "'exhaust" the area of the c i r c l e , b e l i e v i n g that the polygon, would e v e n t u a l l y c o i n c i d e w i t h the c i r c l e . T h i s i m p l i e d that the circumference of the c i r c l e was not i n f i n i t e l y d i v i s i b l e , '"out must be made up of " i n d i v i s i b l e s " or " i n f i n i t e s i m a l s " . These ideas were vague and l e d to d i f f i c u l t i e s . I n f a c t , the ideas of i n f i n i t e s i m a l s and the I n f i n i t e caused so much d i f f i -c u l t y t h a t they were excluded from Greek geometry. Eudoxus of Cnidus (408-355 B. C.) i s g e n e r a l l y c r e d i t e d w i t h d e v i s i n g a method of f i n d i n g areas and volumes /Mich avoided these problems. This method, which l a t e r became mown as the Method of Exhaustion, was the Greek equivalent of i n t e g r a t i o n . I t used the b a s i c i d e a of approximating c u r v i l i n e a r f i g u r e s by r e c t i l i n e a r f i g u r e s but used only a f i n i t e number of these f i g u r e s . I t avoided the problems of the I n f i n i t e s i m a l and the i n f i n i t e hy the j u d i c i o u s use of a double r e d u c t i o ad absur--dum argument. The Method of Exhaustion was based on the f o l l o w i n g axiom, commonly c a l l e d the lemma, or p o s t u l a t e , of Archimedes. Two unequal magnitudes being set out, i f from the g r e ater there be subtracted a magnitude greater than i t s h a l f , and from that which i s l e f t a magnitude greater than i t s h a l f , and ' i f t h i s process be repeated c o n t i n u a l l y , there w i l l be l e f t some magnitude which w i l l be l e s s than the l e s s e r magnitude set out. ( [ 2 1 ] , P. 14) Using t h i s p r i n c i p l e , f o r example, one can conclude that a reg-u l a r i n s c r i b e d polygon can approximate - a c i r c l e so tha t the d i f f e r e n c e I n the areas can be made as small as one wishes. T h i s i s accomplished by successively, doubling the number of sides thereby decreasing the d i f f e r e n c e i n area by more than h a l f each time. The f o l l o w i n g example from E u c l i d ([21], pp. 374-375) i l l u s t r a t e s the procedure used i n the Method of Exhaustion. (This i s a condensed v e r s i o n ' o f the a c t u a l procedure.) Suppose one wished to.prove f o r two c i r c l e s ' 2 p that A 1 : A 2 = d^ : dg . where A 1 , Ag are areas of the c i r c l e and d-^ , dg , are t h e i r diameters. (The Greeks did not have numbers f o r geometrical q u a n t i t i e s because-of the problem of the Incommensurable but used proportions i n v o l v i n g f o u r geometrical quanti-t i e s , ) . One then used the double r e d u c t i o ad absurdum argument. Suppose f i r s t t hat 2 2 : Ag > d^ : dg . Then by the lemma there e x i s t s a polygon P, included i n A, and x 2 g -L such t h a t P^ : Ag > d^ : dg . Construct a s i m i l a r oolygbn P 0 i n A„ . Prom previous 2 2 2 " 2 r e s u l t s one knows that P.^ : p- = d^ : d p . Now ? 1 : A g > P ] : P 2 which i m p l i e s that A"2 < Pg . But t h i s i s impossible since the polygon Pg i s i n c luded i n A c . By a s i m i l a r argument 2 • 2 A. : A 0 < d n : d 0 leads to a c o n t r a d i c t i o n . i d x d .... 2 2 Hence the r e s u l t A^ : Ag =» d^ : dg i s proved. This Method of Exhaustion was used e x t e n s i v e l y by Eudoxus and h i s successors u n t i l the seventeenth century. The procedure had the advantages of being l o g i c a l l y c o r r e c t and i n -t u i t i v e l y c l e a r but had the disadvantages of being cumbersome to apply and d i f f i c u l t to deduce new r e s u l t s from. Archimedes (287-212 B. C ) , who' i s g e n e r a l l y consid-ered to be the g r e a t e s t mathematician of a n t i q u i t y , g r e a t l y ex-tended the work of f i n d i n g area and volumes of geometric f i g u r e s . He supplemented the Method of Exhaustion and devised an ingenious h e u r i s t i c method f o r f i n d i n g r e s u l t s before proving them formally. He was then able to a n t i c i p a t e many of the r e s u l t s of i n t e g r a l c a l c u l u s . The h e u r i s t i c method which Archimedes devised to get i n i t i a l r e s u l t s was based on the mechanical law of the l e v e r . The geometrical f i g u r e s I n question were v i s u a l i z e d as being "made up" of l i n e s or planes. The l i n e s or planes were then p i c t u r e d as being hung from one end of a l e v e r which was then balanced by a f i g u r e of known content and centre of g r a v i t y . Prom t h i s procedure the content of the unknown f i g u r e could be c a l c u l a t e d . The method i s i l l u s t r a t e d by the f o l l o w i n g example given by Archimedes ([20] P. 1.5-17). The problem was to show th a t i n the f o l l o w i n g diagram the parabolic segment ABC has area equal to Vj5 A ABC . I n the diagram D 'is the midpoint of chord AC , DBE and AKP are drawn p a r a l l e l to the a x i s of. the parabola, CF i s a tangent, CK = KH, CH i s v i s u a l i z e d as the l e v e r balanced at K , MO i s any Tine in A AFC p a r a l l e l to AKF and DBE . Archimedes proceeded as f o l l o w s : From the p r o p e r t i e s of the parabola and the c o n s t r u c t i o n s he showed that CK i s the median of A AFC and that ^ = ^ = = . He considered MO HK the f i r s t and l a s t term ^ = ^ and i n t e r p r e t e d t h i s mechani-c a l l y as meaning th a t l i n e segment OP" at H w i l l balance MO at N. • w i t h K being the fulcrum. This r e s u l t i s tru e f o r any position of MO i n A AFC . Since the geometric f i g u r e s are "made up" of l i n e s he concluded that p a r a b o l i c segment ABC a t H w i l l balance A AFC at i t s center of g r a v i t y . Since the cen-t e r of g r a v i t y of a t r i a n g l e i s 1/3 the distance along i t s median he concluded t h a t p a r a b o l i c segment ABC = 1/3 A AFC . By a previous r e s u l t Archimedes knew t h a t A AFC =.4 A ABC . Hence p a r a b o l i c segment ABC = 4/3 A ABC . Archimedes then r i g o r o u s l y proved, by the Method of Exhaustion, every re.sult suggested by the h e u r i s t i c procedure because he d i d not consider i t to be a v a l i d mathematical de-monstration. Many of h i s a p p l i c a t i o n s of the Method of Exhaust-i o n were quite ingenious. I n some problems, f o r example i n f i n d i n g the volume of a p a r a b o l o i d , he approximated the f i g u r e both from the i n s i d e and from the outside w i t h elementary f i g u r e s I n other problems h i s procedure was very s i m i l a r to th a t which we now use i n i n t e g r a l c a l c u l u s . For example, i n h i s o f f i c i a l proof t h a t the area;of the p a r a b o l i c segment ABC i s equal to 4/3 the area of A ABC he proceeded as f o l l o w s ([5]» PP- 5i-52) 3 F i g . 2 He approximated the area of the p a r a b o l i c segment ABC by suc-c e s s i v e l y forming t r i a n g l e s such as A AEB and A BDC . He t h j showed t h a t the area a f t e r the n step was A ABC ( 1 + J- + i -+ + ... -^n~1) • Rather than c o n s i d e r i n g a l i m i t and showing tha t the l i m i t i s equal to V 3 A ABC he completed the l a s t step by the double, r e d u e t i o ad absurdum argument. Using these methods Archimedes was able to f i n d areas, volumes, and centers of g r a v i t y of numerous geometric f i g u r e s . H i s r e s u l t s were a great i n c e n t i v e toward the f u r t h e r development of the s u b j e c t , e s p e c i a l l y i n the seventeenth century. During the two thousand year period from Archimedes u n t i l the s i x t e e n t h century i t appears that nothing s i g n i f i c a n t was done I n d e v i s i n g new methods and techniques f o r f i n d i n g area and volumes. However, two new i d e a s , u s e f u l i n the f u r t h e r development of i n t e g r a t i o n , were advanced during t h i s p e r i o d . One was the study of v a r i a t i o n . People began to study ideas such as v e l o c i t y , a c c e l e r a t i o n , d e n s i t y , and thermal content as p h y s i c a l quantities r a t h e r than as q u a l i t i e s . T h i s was the f i r s t step i n the development of the i d e a of a f u n c t i o n . The second i d e a , due to N i c o l e Oresme, (1323-1362) was the r e a l i z a -t i o n of a.connection between c e r t a i n geometrical p i c t u r e s and p h y s i c a l s i t u a t i o n s . Oresme devised the equivalent of a Car-t e s i a n coordinate system and represented v e l o c i t i e s by l i n e s on the coordinate system. He even I n t e r p r e t e d the area under the v e l o c i t y curve as repr e s e n t i n g the d i s t a n c e that the body t r a v e l l -ed. These Ideas were probably i n c e n t i v e s f o r the f u r t h e r d e vel-opment of i n t e g r a t i o n . The s i x t e e n t h century saw a r e v i v a l of i n t e r e s t i n the problems of quadratures, cubatures, and centers of g r a v i t y . T h i s renewed i n t e r e s t was caused mainly by the t r a n s l a t i o n of Archimedes* work i n t o L a t i n i n 1544. People f i r s t copied h i s formal method (The Method of Exhaustion) but soon they began to seek improvements and then to devise new methods f o r s o l v i n g the problems. The f i r s t suggested reform came from the Flemish engineer, Simon Stevins (1586) , and the I t a l i a n mathematician, Luca V a l e r i o (I.606) . They both attempted to avoid the double r e d u c t i o ad absurdum argument by a d i r e c t passage to the l i m i t . However, they s t i l l thought i n geometrical terms and did not have the a r i t h m e t i c ideas necessary to give p r e c i s e d e f i n i t i o n s . The unwieldiness of the Method of Exhaustion caused the mathematicians of the seventeenth century to drop the pro-cedure completely and to adopt the l e s s rigorous ideas of i n d i -v i s i b l e s or i n f i n i t e s i m a l s . I n f a c t , the period i n the seven-teenth century u n t i l the time of Newton and L e i b n i z .(1670) has been c a l l e d the Period of the I n d i v i s i b l e s ([12], P. 34l) . I n t e g r a t i o n became ass o c i a t e d w i t h the idea of summing these i n d i v i s i b l e s . The f i r s t to make extensive use of i n f i n i t e s i m a l s was Johann Kepler (1571-1650). He became i n t e r e s t e d i n l e n g t h , area, and volume problems wh i l e studying the laws of planetary motion. He was faced w i t h the problems of f i n d i n g the area of an e l l i p t i segment and the l e n g t h of an e l l i p t i c a r c . Kepler was a l s o i n -t e r e s t e d i n gauging the contents of wine casks. To solve these problems and others , Kepler v i s u a l i z e d t h a t geometric s o l i d s were made up of i n f i n i t e s i m a l s . Por example, a c i r c l e was made up of an i n f i n i t e number of t r i a n g l e s w i t h a common vertex and an i n f i n i t e l y small base, and a sphere was made up of an i n -f i n i t e number of i n f i n i t e l y small pyramids. To f i n d the content one merely added up the contents of the components. For example the area of a c i r c l e i s equal to the sum of the areas of the t r i a n g l e s and t h i s i s equal to one-half times the t o t a l sum of the bases (i.e. the circumference) times the r a d i u s . Using procedures such as t h i s , K e p ler was able to f i n d the contents of more than eighty new geometrical f i g u r e s . I t was undoubtedly Kepler's work th a t l e d Bonaventura Caval - i e r i ( 1 5 9 8 - 1 6 4 7 ) , an I t a l i a n J e s u i t mathematician, to develop h i s method of i n d i v i s i b l e s . H i s work was probably the most i n f l u e n t i a l one of t h i s p e r i o d . C a v a l i e r i was never too p r e c i s e as to what he meant by an i n d i v i s i b l e , but i t seems he v i s u a l i z e d p o i n t s as being i n d i v i s i b l e s of l i n e s , l i n e s as being i n d i v i s i b l e s of surfaces, and planes as being i n d i v i s i b l e s of volumes. To f i n d l e n g t h s , areas, or volumes, he added up the i n d i v i s i b l e s . To avoid the problem of the i n f i n i t e he always considered two geometric f i g u r e s and formed a correspondence between them. This approach i s i l l u s t r a t e d , by the s o - c a l l e d C a v a l i e r i ' s Theorem. , I f two s o l i d s have equal a l t i t u d e s , and i f se c t i o n s made by planes p a r a l l e l to the bases snd at equal distances'from them are always i n a given r a t i o , then the volumes of the s o l i d s are a l s o i n t h a t r a t i o . C a v a l i e r i ' s use of i n d i v i s i b l e s to prove p r o p o s i t i o n s can be i l l u s t r a t e d by the f o l l o w i n g simple example ( [ 5 ] , p. I l 8 ) . He was i n t e r e s t e d i n proving t h a t . p a r a l l e l o g r a m ACDP has area equal to double the area of A CAF or A CDF and proceeded as f o l l o w s : A r F i g . 3 I f EF: = CB and HE and BM are p a r a l l e l to CD then the l i n e s BM and HE are equal. Therefore, a l l the l i n e s of A CAF are equal to a l l the l i n e s of A CDF and the two t r i a n g l e s ' are t h e r e f o r e equal. A l s o the area of the pa r a l l e l o g r a m ACDF i s equal to twice the area of e i t h e r t r i a n g l e . By a s i m i l a r but more i n v o l v e d procedure, C a v a l l e r i was able to o b t a i n r e s u l t s which have been i n t e r p r e t e d ( [ 5 ] , p.120) r a m a m + 1 as being equivalent to the formula J x dx = • ^ , although • o . he thought of h i s work as p e r t a i n i n g only to geometrical con-s i d e r a t i o n s . H i s work was a g e n e r a l i z a t i o n of Kepler's as i t 10 went beyond the s p e c i f i c geometric problems. C a v a l i e r i ' s work on i n d i v i s i b l e s stimulated more math-ematicians to work on problems i n v o l v i n g areas and volumes. A l s some mathematicians, such as the Frenchman Roberval (1634) , developed ideas of i n d i v i s i b l e s independently. Thus there emerg ed, i n the period from 1630 to IbbO, a myriad of i n d i v i d u a l methods f o r s o l v i n g these problems. As S t r u i k ( [ 4 2 ] , p. 136) p o i n t s out, however, there evolved two d i s t i n c t trends i n the work. C a v a l i e r i , T o r i c e l l i , and Barrow, (Newton's teacher) concentrated on a geometrical approach while Fermat, P a s c a l , Descartes, and W a l l i s used more of the new algebra and a l s o more of the new a n a l y t i c geometry which had been developed i n t h i s p e r i o d . Both groups were concerned w i t h the same b a s i c problem: P r a c t i c a l l y a l l authors i n the period from lc30 to 1660 confined themselves to. quest-. i o n s d e a l i n g w i t h a l g e b r a i c curves, espec-. i a l l y tnose w i t h the equations a y = b x and they found each i n h i s own way, formulas c m a equivalent to x" dx = ' • , f i r s t f o r p o s i -... ' J Q m-fl t i v e i n t e g e r s m , l a t e r f o r m negative i n t e g e r and f r a c t i o n a l . ( [ 4 2 ] , p. 13&) We w i l l consider i n d e t a i l two of the methods devised i n t h i s p e r i o d , f i r s t t h a t of P i e r r e Fermat, and then that of John W a l l i s . These methods were the most advanced of the period i n t hat the techniques used most c l o s e l y resemble the modern approach to the i n t e g r a l . 11 P i e r r e Fermat devised a precedure f o r c a l c u l a t i n g area under the curve f o r s p e c i a l curves. H i s ingenious procedure used a geometric s e r i e s and the new idea of a l i m i t . the curve of PP. 53-54) • Fermat devised t h i s procedure f o r f i n d i n g the area under xv' from 0 to b ( [ 5 ] , pp. 160-1615 [43] y = x "S .:..• F i g . 4. He f i r s t subdivided the i n t e r v a l from 0 to b , not i n t o a f i n i t e number of s u b i n t e r v a l s , but i n t o an i n f i n i t e number of i n t e r v a l s of unequal l e n g t h . He selecte d e < 1 and then p a r t i -tioned the i n t e r v a l by the po i n t s b, eb, e b, e^b, He formed the approximating sum and found i t formed an i n f i n i t e geometric progression. The formula f o r the sum was known at the time. • P P p S""= b q (b - eb) + (eb) (eb - 3%) + (e% ^ ( e b - e 3b) +.. = b§ .P 2p (b - eb) [1 + elf-- +' e-q+2 +. . . 1-e q S u b s t i t u t i n g e = E q he found: n p+q. _ q /1-E 4 . o"q~ (1-E) (1 + E + E S = b q (- T + q ) = — — t L-s. — ,q-l l.-B ( l - E ) (1 +.E + E2 + E P + q i ) To make the s i z e of the recta n g l e " i n f i n i t e l y s m a l l " he l e t e = 1 ( i n s i n u a t i n g a l i m i t as e approaches one ). The widths of the rect a n g l e s approach zero and E approaches one. He s u b s t i t u t e d E = 1 i n t o the sum and found i t to be a p±a r b £: a m equal to -~f-- b q . Hence h x H dx = b q P+q J-0- P+q As Boyer p o i n t s out ( [ 5 ] , P- l 6 l ) , Permat's demonstra-t i o n possesses many of the important c h a r a c t e r i s t i c s of the d e f i n i t e i n t e g r a l . There i s an equation of a curve, a p a r t i t i o n of the x - a x i s , a sum formed from the areas of approximating rec-t a n g l e s , and some idea of a l i m i t of the sum as the widths of the r e c t a n g l e s approach zero. Fermat, however, d i d not r e a l i z e the s i g n i f i c a n c e of the ope r a t i o n . He regarded the procedure as a method of s o l v i n g a p a r t i c u l a r geometrical/problem and had no thought of a g e n e r a l i z e d procedure. John W a l l i s ( l 6 l 6 - 1 7 0 3 ) was an E n g l i s h mathematician. He devised an i n t e g r a t i o n procedure which introduced a r i t h m e t i c i n t o the geometrical procedure and introduced the ide a of a l i m i t . W a l l i s ' procedure i s i l l u s t r a t e d by the f o l l o w i n g example taken from Hooper ([29], pp. 256-258) . I n t h i s example, W a l l i s was i n t e r e s t e d i n comparing the area under the curve 2 y = x between 0 and B w i t h the area i n the recta n g l e OBAC . ./ . F i g . 5 He began by subdividing the i n t e r v a l OB i n t o m + 1 equal parts and :formed approximating r e c t a n g l e s w i t h the heights selected so ••: •• o ' p p 2 that the t o t a l . a r e a would be p r o p o r t i o n a l to 0 + 1" + 2 '....+ m.. .......... 2 The area of rectan g l e OBAC i s p r o p o r t i o n a l to (m + l)m . Hence the r a t i o of the areas i s 0 + 1 p + 2 + — ' ' ' m . ' Sub-rri (m + 1) s t i t u t i n g values f o r m he found: . (1) a = 1 TTlt = 1 / 5 + ]-/6 ... (2) : m = 2 . . 4 HX 4=1/3+ 1A2 .. - . (3) m...5 ; ^ 9 : i - l / 3 + 1/16 He noted t h a t the greater the number of terms, the c l o s e r the r a t i o approximates 1/3 • I f t h i s i s continued to i n f i n i t y the d i f f e r e n c e " w i l l be about to vanish completely " ( [ 5 ] , p. 172). "Consequently the r a t i o f o r an i n f i n i t e number of terms i** 1/3 " a , ([5], p. 172). This r e s u l t i s equivalent to the formula x dx= ~ . o < W a l l i s was able by a s i m i l a r procedure to d e r i v e the formula x'dx = • f o r higher powers of i n t e g e r s and then he apparently a f f i r m e d the. r u l e f o r a l l powers, r a t i o n a l and i r -14. r a t i o n a l except n = -1. He was able to apply these r e s u l t s to problems of quadratures and cubatures. W a l l i s and Fermat came very close to our present idea of the d e f i n i t e i n t e g r a l . I n f a c t , according to Boyer ( [ 5 ] , P ...the b a s i s f o r the concept of the d e f i n i t e i n t e g r a l may be considered f a i r l y w e l l es-t a b l i s h e d i n the work of Fermat and W a l l i s . But, as he p o i n t s out • . . . i t was to become confused l a t e r by the i n t r o d u c t i o n of the conceptions of f l u x i o n s and d i f f e r e n t i a l s . These two c o n t r i b u t i o n s came from Newton and L e i b n i z . Newton and L e i b n i z are g e n e r a l l y considered to be the i n v e n t o r s of c a l c u l u s , as they devised algorithms f o r d i f f e r e n t i a t i o n and i n t e g r a t i o n , but t h e i r work marks a change i n the concept of the i n t e g r a l . Isaac Newton (1642-172?) was p r i m a r i l y i n t e r e s t e d I n the i d e a of the d e r i v a t i v e , which was a l s o being studied a t the time. He showed th a t the area under the curve could be c a l c u l a t -ed, not by a summation process as h i s predecessors had done, but by a process which depended on the Idea of d i f f e r e n t i a t i o n . For example, ( [ 5 ] , P- 191) he considered a curve w i t h a b s c i s s a x and o r d i n a t e y, w i t h area under the curve being given by . t n m+n z = ( ) ax n . . i f o represents the i n f i n i t e s i m a l i n c r e a s e v m+n' v 15. i n the a b s c i s s a then the augmented area w i l l he z + o«y =(=rr7r) m+n + a(x + d) n . I f , i n t h i s equation, one uses the binomial theorem, d i v i d e s through by Q > and then n e g l e c t s the terms i n -v o l v i n g Q (Newton was u n c e r t a i n of the j u s t i f i c a t i o n f o r t h i s procedure but was t h i n k i n g i n terms of a l i m i t concept), the m r e s u l t w i l l be y = ax n~ . Hence, I f the area i s z = (——) m+n m+n :a ax n the curve w i l l be y = ax n" . Conversely, i f the curve - n m + n i s y = ax n then the area w i l l be z - ( ) ax n . Thus . vm+n' to f i n d the area one could work backwards from the d e r i v a t i v e . Newton, consequently defined the i n t e g r a l , or f l u e n t , as he c a l l e d i t , as the in v e r s e of the f l u x i o n or d e r i v a t i v e and con-centrated on the methods f o r f i n d i n g d e r i v a t i v e s . L e i b n i z , (16^0-1716) working a t the same time as Newton, Was i n t e r e s t e d i n developing o p e r a t i o n a l r u l e s f o r sums and d i f f e r e n c e s of i n f i n i t e s i m a l s . He introduced the n o t a t i o n ijx and l a t e r ! x dx to represent the sum of a l l the values of the magnitudes x - or the i n t e g r a l of x , a name which was suggested by the B e r n o u l l i b r o t h e r s . However, i n d e v i s i n g r u l e s f o r the sum of the i n f i n i t e s i m a l s , L e i b n i z r e l i e d upon the f a c t that sums and d i f f e r e n c e s are i n v e r s e operations and he used the r u l e s f o r f i n d i n g d i f f e r e n c e s . For example, he derived the r u l e t h a t the d i f f e r e n c e (or d e r i v a t i v e ) of x n was nx11-"1' . Hence, the sum or i n t e g r a l of x n must be x n + 1 , : , n+1 1 6 . With the work of Newton i n t e g r a l had changed . . I t was no idea of a sum, hut was now viewed and L e i b n i z , the idea of the longer associated w i t h the as a secondary o p e r a t i o n . 17. CHAPTER TWO Developments i n I n t e g r a t i o n During the Nineteenth Century ' '. ' ' — Prom the time of.Newton and L e i b n i z u n t i l the beginning of the nineteenth century, i n t e g r a t i o n was viewed as the in v e r s e o p e r a t i o n to d i f f e r e n t i a t i o n . As we have noted, Newton had defined the i n t e g r a l as the Inverse of the f l u x i o n or d e r i v a t i v e , w h i l e L e i b n i z i n p r a c t i c e used the idea of an a n t i d e r i v a t i v e . I n the f u r t h e r development of the subject, Johann B e r n o u l l i and E u l e r a l s o stressed the i n t e g r a l as the in v e r s e of the d i f f e r -e n t i a l . Euler, i n f a c t , i n the p u b l i c a t i o n of h i s I n s t i t u t i o n e s c a l c u l i i n t e g r a l ! s of 17*58., defined i n t e g r a l c a l c u l u s as the method of f i n d i n g from a given r e l a t i o n of d i f f e r e n t i a l s , the q u a n t i t i e s themselves ( [ 3 2 ] , p. 664). He used the sum concept only as a means of approximating the value of the i n t e g r a l . .. The concept of a f u n c t i o n i n use at t h i s time was rath e r r e s t r i c t e d . I t u s u a l l y meant a quantity-, y r e l a t e d to a V a r i a b l e x by an equation i n v o l v i n g c e r t a i n constants, t o-. gether w i t h symbols to represent a r i t h m e t i c , t r i g o n o m e t r i c , 2 exponential or l o g a r i t h m i c operations. Por example, y = 3x , y = s i n x + 4x , y = a ... would be c l a s s i f i e d as f u n c t i o n s . Functions could a l s o be defined and represented g e o m e t r i c a l l y , but i t appears as i f the graph must be a smooth continuous curve before i t represented a true f u n c t i o n . A l s o i t was assumed th a t so:meho.w these true geometrical f u n c t i o n s could be represented by a s i n g l e a n a l y t i c expression, w h i l e a r b i t r a r y curves could IS. not be ( [ 3 5 ] , P • 3) . The work of J . B. F o u r i e r , published i n h i s famous book "The A n a l y t i c Theory of Heat" (1&07-1&22) ' forced a reexamination of these fundamental ide a s . F o u r i e r f i r s t showed th a t some discontinuous f u n c t i o n s could be represented by a s i n g l e a n a l y t i c expression, namely a tr i g o n o m e t r i c s e r i e s . For example, a f u n c t i o n equal to 1 from 0 t o a , and 0 from a to ir has a tr i g o n o m e t r i c expansion. Thus the requirement of having an a n a l y t i c expression did not d i s t i n g u i s h between a t r u e . f u n c t i o n and some a r b i t r a r y f u n c t i o n s . Moreover i t seemed no longer necessary to a s s o c i a t e the existence of a s i n g l e a n a l y t i c expression w i t h the d e f i n i t i o n of a f u n c t i o n because such expressions could apparently be determined a f t e r -wards. This work suggested a more general concept of a f u n c t i o n . I t a l s o forced a re-examination of the n o t i o n of • i n t e g r a l . I n the development of the t r i g o n o m e t r i c or F o u r i e r •series of a- discontinuous f u n c t i o n , the c o e f f i c i e n t s are de-f i n e d i n terms of the i n t e g r a l of discontinuous f u n c t i o n s . For example, i n expanding the f u n c t i o n f i n a t r i g o n o m e t r i c series, - - 2 f the c o e f f i e n t s a. are given by — i f ( x ) s i n i x dx or - j f ~ . . . . . . 1 .. h ..... ... ^ j f ( x ) cos i x dx . These d e f i n i t e i n t e g r a l s could not be defined as the in v e r s e of a d e r i v a t i v e but they seemed to have some i n t e r p r e t a t i o n i n terms of area ([ 1 5 ], p. 196). Therefore they added impetus to the development of the i n t e g r a l i n terms of approximating sums. A. L. Cauchy (1S23) was the person who c l a r i f i e d these concepts. He suggested a more general d e f i n i t i o n of a f u n c t i o n and he restored i n t e g r a t i o n to a primary idea r a t h e r than a secondary op e r a t i o n . He f i r s t considered the concept of a f u n c t i o n . He began by d e f i n i n g an independent v a r i a b l e ([6], p. 17): When v a r i a b l e q u a n t i t i e s are r e l a t e d i n such a v yr: manner that given one of them one can conclude the value of a l l the others , the f i r s t q uantity i s c a l l e d an independent v a r i a b l e . The d e f i n i t i o n of f u n c t i o n followed d i r e c t l y : ... and the other q u a n t i t i e s , e x p r e s s i b l e by means of the independent v a r i a b l e , are c a l l e d f u n c t i o n s of t h i s v a r i a b l e . S i m i l a r l y , f u n c t i o n s of more than one v a r i a b l e were define d . Cauchy d i d not, however, t h i n k i n terms of the modern n o t i o n of f u n c t i o n because h i s l a t e r work suggested t h a t he thought of the v a r i a b l e s being r e l a t e d , not by any a r b i t r a r y r u l e , but by an equation. .' Cauchy next considered a s p e c i a l type of f u n c t i o n , which he named continuous and which he defined as f o l l o w s ([8], op. 19-20) : When the f u n c t i o n f ( x ) has unique and f i n i t e values f o r a l l x between two given l i m i t s , and the d i f f e r e n c e f ( x + i ) - f ( x ) i s an i n f i n -20. '. .' . i t e l y small q u a n t i t y , one says t h a t the func-t i o n f ( x ) i s a continuous f u n c t i o n of x ,o between the given l i m i t s . The stage was now set f o r Cauchy's d e f i n i t i o n of the I n t e g r a l ( 1 5 2 3 ) . He a r b i t r a r i l y r e s t r i c t e d himself by d e f i n i n g the i n t e g r a l only f o r continuous f u n c t i o n s , probably because con-tinuous f u n c t i o n s or those w i t h a f i n i t e number of d i s c o n t i n u i t i e s were the only' f u n c t i o n s which, at the time, were considered important. An o u t l i n e of h i s procedure (.£&], pp. 122-125) i s as f o l l o w s : L e t f ( x ) be a continuous f u n c t i o n of x d e f i n -ed between the two f i n i t e l i m i t s x=x and' x=X . o . -L e t x., x , x_, ... x = X be a p a r t i t i o n of o' x 2 n . ... [ x Q , X] and form the sum S = ( x 1 - x Q ) f ( X Q ) + ( x 2 - x i ) f ( x i ) ••• + (X - x n _ 1 ) f ( x n .) . Then the sum S approaches a d e f i n i t e l i m i t as the d i f f e r e n c e s (x^ - ) become i n f i n i t e l y s m all. This l i m i t which depends only on the f u n c t i o n f (x) and the values x Q and X i s c a l l e d the d e f i n i t e i n t e g r a l of fix) and i s •• >X , ' represented by the n o t a t i o n -j^f^x) dx . (The n o t a t i o n i s due to F o u r i e r . ) I t i s i n t e r e s t i n g to note that Cauchy's proof of the existence of the i n t e g r a l i s incomplete as he assumed uniform c o n t i n u i t y of the f u n c t i o n . Cauchy then proved the standard a l g e b r a i c p r o p e r t i e s of the i n t e g r a l . He a l s o apparently ( [ 5 ] , P« 2.6 0 ) gave the 2 1 . f i r s t r i g o r o u s demonstration of the fundamental theorem of c a l -cuius, jLe, i f f i s a continuous f u n c t i o n and F(x) = f ( x ) d x then F'(x) = f ( x ) . x o Cauchy next extended i n t e g r a t i o n to a c e r t a i n c l a s s of unbounded f u n c t i o n ([5], p. l4j5) . The f o l l o w i n g i s an o u t l i n e of the procedure: I f the f u n c t i o n f ( x ) becomes i n f i n i t e between x==x. . • o and. x=X at the.points ^ f i n i t e i n number) x^, x^... xm then the i n t e g r a l J* f ( x ) d x i s defined as: f (x)dx = l i m x ' e-o o x o X l ~ € ^ l .. x 2 " € ^ 2 X f ( x ) d x + J f ( x ) d x ...+ J f ( x ) d x x Q x 1 + e H l provided the l i m i t e x i s t s , where ^>•••^m' Ym and e are a r b i t r a r y p o s i t i v e constants. I f the l i m i t s of i n t e g r a t i o n are i n f i n i t e v t h e n . the i n t e g r a l f f x ) d x i s defined as : . • . l / e v • f f ( x ) d x = l i m [: f ( x ) d x + f ( x ) d x . + f ( x ) d x e-o l^-l/eu X i + € Y 1 xm + eYm provided the l i m i t e x i s t s where, u and y are a r b i t r a r y p o s i t i v e constants. I f I n the previous d e f i n i t i o n s a l l of the a r b i t r a r y constants are reduced to u n i t y one gets Cauchy's d e f i n i t i o n of the p r i n c i p a l value. Thus Cauchy•s work gave i n t e g r a t i o n i t s modern char-a c t e r . L a t e r developments-in the f i e l d were based on the found-a t i o n which he had provided. . The work of Lejeune D i r i c h l e t , a contemporary of Cauchy, on F o u r i e r s e r i e s motivated a f u r t h e r development•of the i n t e g r a l . D i r i c h l e t , i n 1&29, devised s u f f i c i e n t condi-t i o n s under which a f u n c t i o n could be represented by a conver-22. gent F o u r i e r s e r i e s . These conditions were ( [ 1 1 ], p. l 6 ) : ( 1 ) The f u n c t i o n has only a f i n i t e number of maxima .and minima. (2) The f u n c t i o n has only a f i n i t e number of discon-t i n u i t i e s . / • . . . - . / The second c o n d i t i o n was included because i t was only under t h i s c o n d i t i o n t h a t the i n t e g r a l s d e f i n i n g the c o e f f i c i e n t s were con-sidered . The next step i n the development seemed to be to a l t e r t h i s second;.condition by extending the i d e a of the i n t e g r a l . F i r s t D i r i c h l e t h imself attempted to do t h i s by extending the i n t e g r a l to f u n c t i o n s whose set; e of d i s c o n t i n u i t i e s has a f i n i t e number of accumulation p o i n t s . An example of t h i s type of f u n c t i o n i c " " ~ ' f o r t , n e o n l y accumulation p o i n t i s 0. T h e ; i n t e g r a l was defined as f o l l o w s ([35], p• 10) : • . The accumulation p o i n t s of e w i l l d i v i d e the i n t e r v a l [a, b,] i n t o a f i n i t e number : of p a r t i a l i n t e r v a l s . L e t [ a , £>] be one of them. The i n t e r v a l [ a + h, S - k] w i l l c o n t a i n only a f i n i t e number of p o i n t s of e and one can consider the Cauchy i n t e g r a l B-k j f(,x)dx provided i t e x i s t s . Then ra+h " ' • • -3 ,.B-k; f ( x ) d x = 11m f ( x ) d x provided the J a §^8 Ja+h 23. l i m i t e x i s t s . The i n t e g r a l over [a,b] i s then j u s t the sum of the i n t e g r a l s over the . i n t e r v a l s . i This i n t e g r a l apparently was not e x t e n s i v e l y used, p a r t l y because the o r i g i n a l paper was never p u b l i s h e d / b u t mainly because i t was superceded by the i n t e g r a l of Riemann. G. B. Riemann (l&54) was a l s o i n t e r e s t e d i n extending the c o n d i t i o n s of D i r i c h l e t . . I n f a c t , he was i n t e r e s t e d i n f i n d i n g not only s u f f i c i e n t but necessary c o n d i t i o n s under which the r e p r e s e n t a t i o n can occur. This l e d him qui t e n a t u r a l l y to an i n v e s t i g a t i o n of the meaning of the symbol | f ( x ) d x . The r e s u l t of the i n v e s t i g a t i o n was the famous Riemann i n t e g r a l . rb Riemann began by co n s i d e r i n g what f ( x ) d x meant J a f i r s t f o r bounded f u n c t i o n s . U n l i k e Cauchy he made no other assumptions about the f u n c t i o n s . An outline of h i s procedure ( [ 4 6 ] , p. 239) i s as f o l l o w s : L e t x n' x„, ... x ' be an i n c r e a s i n g seauence 1' 2" . n - l 0 of values i n (a,b) and l e t 5 ^ = x^ - a , 5 ^ = x 2 ~ xl* '"'* ' ' ' ^n = k ~ x n - l ' ^ o r m the sum S = 6 x f (a + € 1 6 a ) + 6 2 f ( x 1 + € 2 52^ + ••• + 6^f(x„ , + e *_) where the e. are p o s i t i v e n > n - l n n' i proper f r a c t i o n s . The value of the sum S depends upon the choice of the i n t e r v a l s \ and the numbers e. . I f t h i s sum has the property t h a t i t approaches a f i n i t e number A as the 8^ approach zero, no matter how (1) I t i s mentioned i n [35] p . 10 24. 6.. and .€. are chosen, the value A i s the d e f i n i t e i n t e g r a l f ( x ) d x . I f the sum S a r>b does not have t h i s property then f ( x ) d x ; has no meaning. -b a Riemann a l s o defined f ( x ) d x f o r f u n c t i o n s f a which have a s i n g u l a r i t y at a po i n t c , . a <_ c <_ b ([46], p .24C). - r c - a i -b Form the i n t e g r a l s S ^ f ( x ) d x + I f ( x ) d x a c+a p ', f o r a^, a^, a r b i t r a r y p o s i t i v e constants. • Now l e t and a g approach zero independ-e n t l y . I f S approaches a l i m i t then t h i s l i m i t i s defined as f f ( x ) d x . a Under what c o n d i t i o n s imposed on the f u n c t i o n f w i l l the i n t e g r a l e x i s t ? This i s the next question that Riemann answered. He showed that a necessary and s u f f i c i e n t c o n d i t i o n f o r the i n t e g r a l of a f u n c t i o n to e x i s t i s that given a > o then the sum of the lengths of i n t e r v a l s where the os-c i l l a t i o n of the f u n c t i o n i s greater than that a can be made as small as one would l i k e . T his statement suggested the idea of a; measure of a set and may have, been a stimulus i n the development of that, concept ( [ 4 ] , p. 249). Riemann's work thus introduced the property of i n t e g r a b i l i t y of a f u n c t i o n and widened the c l a s s of i n t e g r a b l e f u n c t i o n s to i n c l u d e many discontinuous ones. 25. G. M. Darboux [10] (1875) while attempting to make Riemann!s work on integration more precise, suggested a d i f f e r e n t approach to the i n t e g r a l through the use of upper and lower sums. Darboux began by defining p r e c i s e l y the supremum and infiaium m^ of a bounded function f on an i n t e r v a l [a,b]. He then subdivided the Interval (a,b) by the points x^, x 2,.... x„ , and formed the new sums: • n - l • M ; . T M1 61 + M2 62 + Mn; 6n m m m 61 + m 26 2 + m n6 n Where 5± = x ^ ^ ^ 6Q - - a , 6 n = b - x ^ M. = sup f(x) m. = i n f f(x) ' 1 x. ,< x < x. x. x < x. i - l — — i i - 1 — — i Then he proved what became known as the Darboux Theorem ([10],p. 6 5 ) : I f 6i £ 6 f o r a l l I , then there exists f i n i t e numbers M , , m . such that lim M = M . , • . 6-0 l i m m = m^-. . . _ ab • • - 6*0 •• . • He did not c a l l these l i m i t s the upper and lower i n t e g r a l s nor did he use the notation Jf(x)dx and Jf(x)dx . These c o n t r i -butions apparently came from Jordan i n 1&92 ([2&], p. 464) .. • '. ............ pb - • . In considering the i n t e g r a l f(x)dx Darboux started a with the c h a r a c t e r i s t i c sum. ! £ = 6 1f(a+9 16 1) +. 8 2f(x 1+© 26 2) . .+ ^ ( ^ + 9 ^ ) He then formed h i s upper and lower sums, M Arid m , discussed • 2 b . above, and showed that a necessary and s u f f i c i e n t c o n d i t i o n that the sum £ has a l i m i t as 6-0 (6. < 6) i s that the l i m i t s M , 1 — ab and m^ of M and m are equal. A. s i m i l a r procedure to t h i s i s used i n many modern t e x t s to de f i n e the Riemann i n t e g r a l . Darboux was a l s o able to give a completely v a l i d proof t h a t a continuous f u n c t i o n was i n t e g r a b l e . I t i s d i f f i c u l t to say f o r c e r t a i n but t h i s seems to be one of the e a r l i e s t proofs of t h i s r e s u l t . (Heine [ 2 3 ] ( 1 S 7 2 ) had considered the ide a of uniform c o n t i n u i t y and had shown that a continuous f u n c t i o n on [a,b] i s uniformly continuous. T h i s could have l e d to e a r l i e r proofs of t h i s r e s u l t ) . During the l a t t e r p a r t of the nineteenth century ap-pa r e n t l y many i n t e g r a l s were devised f o r unbounded f u n c t i o n s ( [ - + 4 ] , p. 2 3 6 ) • These i n t e g r a l s were extensions of the Riemann i n t e g r a l . They d i d not achieve l a s t i n g importance but have some h i s t o r i c a l i n t e r e s t . We w i l l consider one example to i l l u s t r a t e the type of procedure. A. Harnack ( l S & 4 ) [ l b ] devised an i n t e g r a l f o r unbound-ed f u n c t i o n s whose set of s i n g u l a r i t i e s can be enclosed i n a f i n -i t e number of i n t e r v a l s w i t h t o t a l l e n g t h as small as one would wish. (By modern terminology the set of s i n g u l a r i t i e s has-content zero.) An o u t l i n e of h i s procedure ( [ l b ] , p. 2 2 0 ) i s as f o l l o w s : Let f be the f u n c t i o n and enclose the s i n g u l a r i t y p o i n t s i n a set E c o n s i s t i n g of a f i n i t e number of i n t e r v a l s of t o t a l length e .. L e t f, be e q u a l to 0 i n E and to f everywhere e l s e and suppose f-, (x)dx e x i s t s . I f t h i s i n t e g r a l approaches a f i n i t e l i m i t as e approaches 0 t h i s l i m i t i s said to be the i n t e g r a l of f from a to b . . I n lSg4 T, J . - S t i e l t j e s [40] introduced a completely new i d e a i n t o the h i s t o r y of i n t e g r a t i o n , an i d e a which was u n r e l a t e d to other developments i n the f i e l d . While working on questions i n v o l v i n g the d i s t r i b u t i o n ' o f mass along a l i n e , S t i e l t j e s suggested the i d e a of an .int e g r a l ' i n v o l v i n g two func-t i o n s .'..•'-';'. He began by c o n s i d e r i n g a monotone i n c r e a s i n g f u n c t i o n , cp defined on the p o s i t i v e x - a x i s w i t h cp(o) = o . The func-t i o n could be v i s u a l i z e d as r e p r e s e n t i n g a d i s t r i b u t i o n of mass w i t h the p o i n t s of d i s c o n t i n u i t y r e p r e s e n t i n g the p o i n t s of condensation of mass. With t h i s i n t e r p r e t a t i o n an i n c r e a s i n g f u n c t i o n represents a p h y s i c a l example of a measure. S t i e l t j e s then considered the moment about the o r i g i n of such a d i s t r i b u t i o n i n [a,b] and proceeded as f o l l o w s ([40], p. o7l) : ' L e t a=x::; b=x_ , and place between x and o n " o x the n - l values x < x, < x 0 . . . < x„ . n o 1 2 n Next p i c k n numbers e, , e_, ... . • 1 2 n sucn that ' <_ £ x^ . Then form the sum e 1[cp(x 1)-cp(x 0) ]+e 2[cp(x 2) -cp(x1) ]. . .+en[cp(xri) 7cp(x n_ 1)], The l i m i t of the sum (as max ( x ^ - x ^ ) approaches 0) i s by d e f i n i t i o n the moment of the d i s t r i b u t i o n about the o r i g i n . S t i e l t j e s then g e n e r a l i z e d t h i s procedure by consider-i n g the sum f(e L)[ Cp(x 1) -cp(x Q) ]+f( e 2) [cp(x 2) - c o(x 1) ]+..f ( e n ) [ ^ M f o ) ! where f i s any continuous f u n c t i o n . This sum w i l l have a l i m i t as max (x. - x. ,) aporoaches 0. This l i m i t i s designat-ed by " f(x)dcp(x) and i s now c a l l e d the S t i e l t j ' e s i n t e g r a l of f w i t h respect to cp . S t i e l t j e s d i d not extend t h i s i n t e g r a l beyond the case where f I s continuous and <p i s monotone. The.only property he proved was the f o l l o w i n g : .b . b f(x)dcp(x)=f(b)cp(b)-f(a )cp(a) - <p(x)df(x) . t a 29. CHAPTER THREE The Development of Measures - The Lebesgue Integral As.we have seen, the concept of the i n t e g r a l i s c l o s e -l y r e l a t e d to the; concept of area. Area, along w i t h l e n g t h and volume, were h i s t o r i c a l l y amongst the f i r s t examples of the general i d e a of measure. These examples a l l have the character-i s t i c property of being non-negative and a d d i t i v e . From a n t i q u i t y u n t i l the nineteenth century, these measures were c a l c u l a t e d only f o r very r e g u l a r geometric sets such as the set of p o i n t s under a continuous curve. The pro-cedure, as we have noted i n the case.of area, was to'approximate the sets from the i n s i d e and/or the outside by a ' f i n i t e number of simple f i g u r e s . For example, Archimedes, i n c a l c u l a t i n g the volume of a paraboloid used approximations by r e c t i l i n e a r s o l i d s , both from the i n s i d e and from the o u t s i d e . The advances i n a n a l y s i s i n the nineteenth century seemed to motivate a more i n t e n s i v e study of measures.: As.we have. seen Riemann's c o n d i t i o n f o r the i n t e g r a b i l i t y of a func-t i o n ( t h a t the sum of the lengths of the i n t e r v a l s on which the . o s c i l l a t i o n > a can be made as small as we l i k e ) suggests the id e a of a measure f o r c e r t a i n new subsets ( [ 4 ] , p. 249). With the development of set. theory, many more sets were considered. The problem then presented (according to. some sources ([13]., p 150)) was how to a s s o c i a t e a measure not only w i t h the r e g u l a r sets but a l s o w i t h a r b i t r a r y subsets. 30. The f i r s t methods introduced by S t o l z , ([4l], p..151), /Harnack, ([17], P- 2 4 l j , and Cantor, ( [ 7 ] , pp. 473-^75), (15S4-1&&5), a l l used the same b a s i c i d e a . A set E i n , f o r example, was covered by a f i n i t e number of i n t e r v a l s . The measure, m(E), was defined as the l i m i t of the sum of the lengths as the longest of the i n t e r v a l s approached zero. This measure, however, was un-s a t i s f a c t o r y because i t d i d not have the a d d i t i v e property even; f o r commonly used s e t s . For example, i f A i s the set of r a t i o n a l s i n [0 ,1 ] , A* i s the complement of A , rti i s the measure, then m(A') = 1 , m(A) = 1, m(AUA') - 1 . Hence m(AUA») k- *(A) + m(A') . Probably to overcome these d i f f i c u l t i e s , C. Jordan [30] (lo94) suggested a more r e f i n e d approach to the problem of measures. He f i r s t of a l l considered approximating a set not only from the outside but a l s o from the i n s i d e , u sing i n each .case a f i n i t e number of elementary f i g u r e s . He then c a l c u l a t e d l i m i t s as the s i z e of the f i g u r e s ; approached zero. He i l l u s t r a t -ed h i s procedure by c o n s i d e r i n g a set E i n the plane. ...Decompose t h i s plane by p a r a l l e l s to the coordinate axes, i n t o squares of sides r . The set of those squares which are. i n t e r i o r to E form a domain S i n t e r i o r to E; The set of those which are i n t e r i o r to E or which meet i t s boundary form a new domain S + S' to which E i s i n t e r i o r . We can represent the' areas of these domains by S and S + S' . 31. Let us now vary our decomposition i n t o squares i n such-a way that r tends to zero: the areas S and S+S' w i l l tend to some f i x e d l i m i t s . • ( [ 3 0 ] , p. 2 0 ) These l i m i t s A and a are c a l l e d r e s p e c t i v e l y the i n t e r i o r area and e x t e r i o r area of E . I f these two numbers are .equal the set E i s c a l l e d " quarrable" and has. area'or . measure a=A . Jordan then r e s t r i c t e d h i mself to these quarrable sets and showed that the measure has the a d d i t i v e property,. This was probably the f i r s t time t h a t , i n order to achieve t h i s a d d i t i v e property, the measure was r e s t r i c t e d to a f a m i l y of subsets r a t h e r than being c a l c u l a t e d f o r a l l subsets. Jordan a l s o mentioned t h a t t h i s procedure can be adopt-ed f o r se t s , of any number of dimensions. The i n t e r i o r and exter-i o r "content" (etendue) can be determined and i f these two numbers, are equal the s e t ' i s . c a l l e d measurable.: The measure i s a d d i t i v e on these measurable set s . ..•.'...-."•-.'•.':;•/. Apparently G-. Peano ( . [ 4 ] , . p . 2 4 9 ) ,• a t approximately the same time, developed, s i m i l a r ideas of measure and measura-b i l i t y .'.••'.';..' These ideas of Jordan and Peano, however, had l i m i t e d a p p l i c a b i l i t y because too many commonly used sets were not mea-surable. For example, the set of i r r a t i o n a l s i n [0,1] has Inner content equal to zero and outer content equal to one and 32. i s t h e r e f o r e not measurable. A l l of these p r e v i o u s l y mentioned measures, using .',. f i n i t e approximations from the outside,were very coarse. They would not, i n f a c t , d i s t i n g u i s h between a set and i t s c l o s u r e . . E. B o r e l ( 1696) , apparently ( [ 1 2 ] , p. 3^2) while studying s e r i e s of f u n c t i o n s , found the need f o r a measure w i t h the property t h a t the measure of countable sets was zero. To f u l f i l l t h i s need, he introduced ( [ 3 ] , PP • 46~r-50) a new pro-perty f o r a measure and a new method f o r c a l c u l a t i n g the measure of c e r t a i n s e t s . The new property was countable a d d i t i v e l y , i e . the measure of the union o f a countable number of d i s j o i n t sets i s equal to the sum of t h e i r measures. The new method Involved ' c o n s i d e r i n g how c e r t a i n sets were constructed and deducing what the measure should be. R e s t r i c t i n g h imself to subsets of the i n t e r v a l [ 0 , 1 ] B o r e l began by c o n s i d e r i n g an i n t e r v a l w i t h or without end p o i n t s . The measure should be i t s 1ength. Since an open set G can be expressed as the union of a countable number of d i s j o i n t i n t e r v a l s Ev, i = 1 ,2 , the measure of G,m(G), should be equal to the sum of the measures m(E^) i = l , 2 , . . . A closed set P i s the complement of an open set G. I t s mea-sure m(P) should t h e r e f o r e be l-m(G) . B o r e l continued to c a l c u l a t e the measure of a set by t h i s step by step procedure, using the f o l l o w i n g two p r o p e r t i e s : 33. (1) countable a d d i t i v i t y . (2) i f E 3 E' and m(E) = S, m(E«) - S', then m(E - E>) = S - S' Those sets which had a measure defined by t h i s proce-dure were c a l l e d measurable sets.. By a t r a n s f i n i t e procedure, B o r e l constructed a l l the sets which belong to what we now c a l l the B o r e l a - r i n g . He then showed th a t the measure defined on these sets was a non-negative countably a d d i t i v e set f u n c t i o n . I n 1902 Henri Lebesgue introdu c e d , i n h i s t h e s i s [ 3 ^ ] , some powerful, new ideas on measure and i n t e g r a t i o n . H i s work '.marks the beginning of a new era i n these f i e l d s , .Lebesgue was i n t e r e s t e d i n the problems of f i n d i n g a . f u n c t i o n knowing i t s . d e r i v a t i v e . This l e d him qu i t e n a t u r a l l y to a c o n s i d e r a t i o n of area under the curve, hence to.area i n the plane and to the more general problem of measures. He i n t r o -duced a new approach to measures, and a more general c l a s s of measurable s e t s . These ideas l e d d i r e c t l y to a d e f i n i t i o n of . t h e . i n t e g r a l f o r . a wider c l a s s of f u n c t i o n s . His i n t e g r a l pos-sessed some important, new p r o p e r t i e s and was used i n the s o l u t i o n of the o r i g i n a l problem of f i n d i n g a f u n c t i o n knowing i t s d e r i -v a t i v e . . 1 . Lebesgue began h i s work on measures by s t i p u l a t i n g the co n d i t i o n s B o r e l had apparently suggested a measure must s a t i s f y ( [ 3^]j P- 232) , the measure being r e s t r i c t e d to bounded s e t s : 34. We propose to a t t a c h to each hounded set a number, p o s i t i v e or zero, which we w i l l c a l l i t s measure and which w i l l s a t i s f y the f o l l o w i n g c o n d i t i o n s : ( 1 ) There e x i s t s some sets f o r which the measure i s not zero. (2) Two equal sets have the same measure ( s e t s are equal i f they can be made to coi n c i d e by displacement.) ; , (3) The measure of the sum of a f i n i t e or countable number of d i s j o i n t sets i s the , sum of the measures of the s e t s . ( [ 3 4 ] , p . 2 3 6 ) I n order to achieve t h i s goal Lebesgue used a much simpler and, as i t turned out, more general approach than had B o r e l . He amended the procedure of Jordan by approximating sets w i t h countable covers rather than j u s t f i n i t e ones. This idea i s Lebesgue's key c o n t r i b u t i o n to measure theory. He considered bounded sets E f i r s t on the r e a l l i n e and covered E w i t h a countable number of i n t e r v a l s . These i n t e r v a l s formed a set E^ . He defined the measure of an i n -t e r v a l as i t s l e n g t h arid defined m(E 1) as the sum of the lengths of the component i n t e r v a l s . He then defined the outer measure of .E, m e(E):, as the Inf. of, the numbers m(E^) taken over a l l p o s s i b l e countable covers by i n t e r v a l s . To get the i n n e r mea-sure of E he l e t I represent an i n t e r v a l c o n t a i n i n g E arid defined the i n n e r measure m.(E) by m.(E) = m(l) - m (I-E) . The Important sets considered were those f o r which the two measures were equal: 35. We c a l l sets measurable i f the outer measure and the i n n e r measure are equal. ([ 3'0 ,p. 238) Lebesgue showed that t h i s c l a s s of measurable sets •was closed under countable unions and i n t e r s e c t i o n s and included the c l a s s e s of Jordan and B o r e l measurable se t s . He a l s o showed ..that the measure r e s t r i c t e d to these sets had the d e s i r e d pro-. p e r t i e s f o r a measure. Lebesgue then stated that these c o n s i d e r a t i o n s could e a s i l y be extended to bounded sets E of any dimension. He contented h i m s e l f , however, w i t h c o n s i d e r i n g only dimension two and suggested a procedure which was completely analogous to the procedure f o r dimension one. I t i s i n t e r e s t i n g to.note t h a t he used t r i a n g l e s to cover the plane s e t s . They would be more u s e f u l i n the extension to surface area. '.Having s e t t l e d the problem of measures, Lebesgue was l e d q uite n a t u r a l l y to the f o l l o w i n g d e f i n i t i o n of the. i n t e g r a l f o r -bounded- function's ([ 34 ], p. 250) .':•• . (This i s a p a r a - - v.-. phrase of the a c t u a l d e f i n i t i o n ) : L e t f be a bounded f u n c t i o n defined on [a,b], •let m be the plane measure, and l e t : •: :• \ {(x, y) i a < x < b 0 < y < f ( x ) } , E 2 - {(x,y) i a < x < b f ( x ) < y < o}. : I f E^ and E ? are measurable sets then the i n t e g r a l of f i s defined as the quantity m(E^) -. m(E;2) , and the f u n c t i o n f i s c a l l e d summable. • • 36. The next step •was.-to t r y . to define the i n t e g r a l f o r unbounded f u n c t i o n s . One procedure, which Lebesgue acknowledged but d i d not f o l l o w , was to extend;the measure to unbounded s e t s . Instead he used a new'procedure which i n v o l v e d s u b d i v i d i n g the y - a x i s .••;,• ' H i s i n s p i r a t i o n . f o r t h i s procedure came from consider-i n g a continuous monotone•increasing f u n c t i o n f defined on [oc,S] w i t h range [a,b] (a<b) . Corresponding to a subdivision a. = x Q < x^ < x 2 ... <x n = 3 of . [ a , 8 ] was a s u b d i v i s i o n a' = a Q'<a^ < ... <an = b of [a,b ] . Lebesgue noted that the c l a s s i c a l i n t e g r a l of the f u n c t i o n which was u s u a l l y d e f i n -ed as the common l i m i t of the two. sums n n E(x^ - x x - l ^ a i _ l ' > j^x± ~ x i - l ^ a I as max (x^ - x j _ - l J approached zero could a l s o be defined as the common l i m i t as max ( a ^ - a^ "'^). approached zero. Gener-a l i z i n g t h i s idea he associated- the f o l l o w i n g , sums w i t h an a r b i t r a r y bounded f u n c t i o n f and anv s u b d i v i s i o n a = a < • • • ' ' • . ' • • • • • . " ° al ^ a 2 • • • < a n '"' D of an i n t e r v a l [a,b ] 'Containing .the range: ' ' • • n . ' ' n-l a -• S a.m(.e) + £ a m(e. ».) n n-l S = ' E a.m(e. ) + E a. ,-,m(e, '.) where e^ - [x: f (x) •= a., ] ; V = t X ! a i < f< x) < a i + l } ; m i s the measure on the l i n e . . 37. These sums w i l l be defined o n l y • i f m(e i) and m(e' i) are de-f i n e d . Consequently, Lebesgue considered the sums only f o r the functions, f o r which, given any a and b, the set [x: a < f ( x ) < b} i s measurable. This c o n d i t i o n , i t turned out, i s equivalent to the c o n d i t i o n that the f u n c t i o n i s summ-abl e . Therefore, for: these f u n c t i o n s the sums , a and E are defined and, as Lebesgue showed, they have the same l i m i t as max ( a ^ - a^^) approaches zero. This l i m i t i s equal to the i n t e g r a l of the f u n c t i o n . T h i s procedure thus suggested another d e f i n i t i o n of summable f u n c t i o n and i n t e g r a l which i s a p p l i c a b l e to unbounded f u n c t i o n s as w e l l : A (bounded or unbounded) f u n c t i o n f i s c a l l e d , summable i f f o r any a and b the set , .: (x: a <.-'• f (x) < b} i s measurable. ([34], p . 256) Thus Lebesgue•s concept of a summable f u n c t i o n i s equivalent to our present concept of a measurable f u n c t i o n . To define,the i n t e g r a l he considered a s u b d i v i s i o n ..,.31 2 < m_.j < m Q < m1 < mp . . . of the y - a x i s , v a r y i n g be-tween -» and +00 and such that m. - m. , i s bounded, and 1 1 - 1 he. l e t : a = I m im(e i) + 2 m im(e i') .... S = £ rrum(e^) + £ m^+^m(e^') 3b He then showed that i f one of these sums i ^ f i n i t e then both w i l l converge to the same f i n i t e l i m i t e as..max (m^ ~ ^ . x ) a P ~ proaches zero. This l i m i t , i f i t e x i s t s , i s defined, as t h e : i n -t e g r a l of the f u n c t i o n . '. Lebesgue noted, however, th a t the i r i t e g r a l does not n e c e s s a r i l y e x i s t f o r unbounded summable f u n c t i o n •hence the term measurable i n s t e a d . This i n t e g r a l of Lebesgue has some i n t e r e s t i n g proper . t i e s . • Although i t i s a g e n e r a l i z a t i o n of the proper Riemann i n t e g r a l , i t i s not a g e n e r a l i z a t i o n of the improper one. For ,-..,,'-V, v , . '/ .' •• ' -: :. example, the f u n c t i o n f ( x ) = v — f o r r - l <_ x < r r ^ 1,2.. has.an improper Riemann i n t e g r a l but i s not Lebesgue i n t e g r a b l e .However, u n l i k e any.other i n t e g r a l considered before, i t possesses the f o l l o w i n g .important property, which, i s of para-mount i n t e r e s t i n a n a l y s i s . ([>'-!-], p. 259) I f a', sequence of summable f u n c t i o n s f ^ , f p , f having i n t e g r a l s , hasalimit f, and i f |f - f n i < M, Vn, where M i s some f i x e d number,, then ,f .' has an i n t e g r a l which i s ."the' l i m i t of the. i n t e g r a l s of f u n c t i o n s f . Moreover, the i n t e g r a l can be used to f i n d p r i m i t i v e s f o r a wider c l a s s of f u n c t i o n s than those.considered h e r e t o f o r e 39. CHAPTER POUR A Modern Glimpse I n .this part of .the t h e s i s , we w i l l g ive some i n d i c a -t i o n of the developments in. i n t e g r a t i o n i n the period a f t e r Lebesgue... The .amount of m a t e r i a l on t h i s period i s tremendous;; we w i l l , confine ourselves to a very b r i e f coverage. The notions of measure and i n t e g r a l are i n t i m a t e l y connected. Measure, .assigns numbers to sets while the i n t e g r a l assigns numbers to f u n c t i o n s , t h a t i s , i t i s a f u n c t i o n a l . Given a measure, one can define an i n t e g r a l by a procedure l i k e Lebesgue's or one devised by ¥. H. Young (1905) which uses Darboux sums. S i m i l a r l y , given.an i n t e g r a l , one can a s s i g n a measure to a set by co n s i d e r i n g the i n t e g r a l . of i t s . c h a r a c t e r i s t i c , f u n c t i o n i f . i t i s i n t e g r a b l e . These p o i n t s of view are r e f l e c t e d . ..in developments, along two broad l i n e s , , a set t h e o r e t i c approach and a f u n c t i o n a l approach. The work of Radon (1913), Prechet (1915), and Carath£odory (1914, 19l6). stressed the measure theory approach. T h e i r work represents, a n a t u r a l g e n e r a l i z a t i o n of the works of •Lebesgue and S t i e l t j e s . Radon suggested r e p l a c i n g the n-dimensional Lebesgue measure by any completely a d d i t i v e set f u n c t i o n . d e f i n e d oh the Lebesgue measurable s e t s . 4 0 . .. Frechet g e n e r a l i z e d t h i s i d e a by c o n s i d e r i n g any com-p l e t e l y a d d i t i v e set f u n c t i o n defined on the subsets of any ab-s t r a c t space. He postulated the measurable subsets to be a 0 - f i e l d . The corresponding i n t e g r a l s . i n both these cases are defined i n any of the usual ways using.sums. Caratheodory next, devised a. procedure f o r generating a measure r a t h e r than assuming i t s existence on a a - f i e l d . . S t a r t i n g w i t h any nonnegatlve f u n c t i o n defined on a given c l a s s of s e t s , he determined.an outer measure defined on a l l sets of the .space.considered. T h i s . o u t e r measure, i n g e n e r a l , i s only sub a d d i t i v e . He then i s o l a t e d s e t s . c a l l e d measurable which form a, a - f i e l d and on which the outer measure i s completely a d d i t i v e , t h a t i s , i t i s a measure. I n the. d e f i n i t i o n of.the i n t e g r a l , Car.atheodory continued h i s s t r e s s on measures by pursuing the i d e a of area under the curve. To t h i s end, he defined product measure ( t o take the.place of area i n the plane) and defined the i n t e g r a l , i n terms of. t h i s product measure. The i d e a of the i n t e g r a l as a f u n c t i o n a l , s p e c i f i c a l l y a l i n e a r f u n c t i o n a l , was stressed by F. Riesz (1909) and D a n i e l l ( l 9 l o ) . T h e i r work e s t a b l i s h e d fundamental connections between i n t e g r a t i o n and f u n c t i o n a l a n a l y s i s . . Riesz solved a problem posed p r e v i o u s l y by J . Hadamard when he showed that the S t i e l t j e s i n t e g r a l f eg was the.most 41 general l i n e a r continuous f u n c t i o n a l on the space C(I) of .continuous f u n c t i o n s on [a,b] . That i s , given a l i n e a r con-t i n u o u s ; f u n c t i o n a l • S' on C ( l ) , he showed there e x i s t s a f u n c t i o n g of bounded v a r i a t i o n such that S (f) = : f°f dg , V f € C ( l ) , thereby e s t a b l i s h i n g a fundamental connection between l i n e a r continuous .functionals. and measures . D a n i e l l d i s a s s o c i a t e d the i n t e g r a l from I t s dependence on a measure by a b s t r a c t i n g the e s s e n t i a l ' p r o p e r t i e s of the Lebesgue i n t e g r a l . . He began by p o s t u l a t i n g a f u n c t i o n a l defined on a c e r t a i n ..class of f u n c t i o n s P , f o r example the .continuous •functions or step f u n c t i o n s . This f u n c t i o n a l i s as-rt f» rt sumed to be l i n e a r ( af + bg = a: f + b g ) } nonnegative ( f _> o := > j f >. o) , and to have the monotone convergence property (f„t f =••> f — f ) .. D a n i e l l then devised a procedure f o r . n J n J ' •extending-this f u n c t i o n a l . to a l a r g e r c l a s s of f u n c t i o n s i n such a way.that i t s t i l l - s a t i s f i e s , the given c o n d i t i o n s . . I f the c l a s s P I s the- continuous f u n c t i o n s and . i s the Riemann i n t e g r a l , then the.extension procedure w i l l y i e l d the Lebesgue i n t e g r a l f o r the Lebesgue-integrable f u n c t i o n s . . The ide a of the i n t e g r a l as a l i n e a r f u n c t i o n a l was f u r t h e r extended beginning i n t h e . 1 9 3 0 ' s w i t h the study of i n -t e g r a l s of f u n c t i o n s w i t h values i n a Banach space. The i n t e g r a l . , • . ,{ now maps f u n c t i o n s i n t o a more general space than the r e a l l i n e . 42. A t h i r d post Lebesgue approach to integration was toward the u n i f i c a t i o n of the ideas of ant i d e r i v a t i v e and l i m i t of a sum. The Lebesgue i n t e g r a l did not completely combine these two ideas. For example, the derivative of x sin — 2 has an an t i d e r i v a t i v e x but i s not integrable i n the Lebesgue sense. To overcome such d i f f i c u l t i e s Denjoy (1912) and Perron (1914) devised new i n t e g r a l s . 43-1. E. T. B e l l : 2. G.A. B l i s s : 3. E. Borel:: ... 4. N. Bourbaki: 5. C . B. Boyer: 6. F. C a j o r i : G. Cantor: 6. A. Cauchy: 9. P. J . Danie.ll: BIBLIOGRAPHY The Development of Mathematics. New York: McGraw H i l l , 1940. I n t e g r a l s of Lebesgue. Amer. Math. Soc. B u l l . , v o l . 24, 1917-16, pp 1-47. Le con's Sur La Theorie Des Fonctions, 2nd\ed. P a r i s : G a u t h i e r - V i l l a r s , . 1914. Elements D ' H i s t o i r e Des Mathematiques. . P a r i s : Hermann, i960. The H i s t o r y of the C a l c u l u s . New York: Dover, 1949. A H i s t o r y of Mathematics. •' . •New York: . MacMi.llan, 1931. Ueber unendliche., l i n e a r e Punktmannich-f a l t i g k e l t e n . Mathemat.• Anna1en, v o l . 2 3 , lb&4, pp.. 453-456. . ! Oeuvres Completes, 2nd s e r i e s , v o l . 4. P a r i s : G a u t h i e r - V i l l a r s , 1&99. A General Form of the I n t e g r a l . Ann. of Math., v o l . 19, 1917-1&, pp. 279-294. 44. 10. G . M. Darboux: Memoire sur l e s f o n c t i o n s d i s c o n t i n -ues. Ann. E C Norm. Sup. number 2, v o l . 4, 1675, pp. 57-112. 11. G. Lejeune D i r i c h l e t : Werke I . B e r l i n : Reimer, I&69. 12. 'Encyclopaedia B r i t a n n i c a . Volume 12. Chicago: Benton, 1965, 13. Encyclopedic Des Sciences Mathematlques. P a r i s : Gauthler-; . ; : . / : : y ; ; : / 1912. . 14. H. Eves: An I n t r o d u c t i o n to the H i s t o r y of Mathematics. New York: Rinehart, 1963. 15. J . F o u r i e r : The A n a l y t i c Theory of Heat. Trans-l a t e d by.A. Freeman. ,. New York: Dover; 1955. 16. A. Harnack: Die allgemeinen Satze liber den .. Zusammenhang, der Funktionen einen r e e l l e n ' V a r i a b e l n . mit i h r e n -Ableltungen. ., Mathemat Annalen, v o l 24,. I 0 7 4 , pp. 217-252. 17. A. Harnack: Ueber den I n h a l t von Punktmengen. Mathemat. Annalen, v o l . 25,'-l.S&5, pp.. 241-25&. l o . T. L. Heath: A H i s t o r y of Greek Mathematics. v o l . I I . Oxford: Clarendon P r e s s , 19.21. 19. T. L. Heath: A Manual of Greek Mathematics, vol.11. Oxford: Clarendon. Pr e s s , 1921. • 45. 20. T. L. Heath: The Method of Archimedes. Cambridge: U n i v e r s i t y P r e s s , 1912. 21. T. L. Heath: The T h i r t e e n Books of E u c l i d ' s E l e -ments. 2nd. ed. v o l . I I I . Cambridge U n i v e r s i t y P r e s s , 1926. 22. T. L . Heath: The Work of Archimedes. New York: Dover, 1912. 23. E. Heine: Die Elements der Functionenlehre. L. C r e l l e J o u r n a l , v o l . 74, 1S72, PP. 172-185.. , 24. R. Henstock: Theory of I n t e g r a t i o n . London: Butterworths, 1963. '. 25. T.E. Hildebrandt: On I n t e g r a l s Related to and Exten-sions of Lebesgue I n t e g r a l s . Am. Math. Soc. B u l l . , v o l . 24, 1917-1&, pp. 113-144, 177-202. 2b. T. H. Hildebrant: I n t e g r a t i o n i n A b s t r a c t Spaces. ; B u l l . ' Am. Math. Soc. v o l . 59, number 2, March, 1953, PP• 111-159. 27. T. H.;; Hi Id ebrandt: I n t r o d u c t i o n to the Theory of I n t e -g r a t i o n . New York: Academic. P r e s s , 1963. "•• 2o. E ... Wv Hobson: The Theory of Functions of a Real V a r i a b l e , v o l . 1, 3rd ed. Cambridge, 1927. 29.. A. Hooper: Makers of Mathematics.. New York: Random House, 1945. 46. 30. C. Jordan: 31. P. E. B. J o u r d a i n : 32. P. E. B. J o u r d a i n : 33. H. Kestelman: 34. H. Lebesgue: 35. H.: Lebesgue:, 36. S. Saks: 37. V. Sanford: 35. J . P. S c o t t : Cours d'Analyse,de l ' E c o l e P o l y -technique, v o l . 1. P a r i s : G a u t h i e r - V i l l a r s , 1693. Note on F o u r i e r ' s I n f l u e n c e on the Conceptions of Mathematics. Piro-ceedings I n t e r n a t i o n a l Congress of Mathematics, v o l . 2. Cambridge,; 1912, pp. 526-527. The O r i g i n s of Cauchy's Concept of the D e f i n i t e I n t e g r a l and of the C o n t i n u i t y of a Function.< I s i s 1, 1913, PP. 661-703. Modern T h e o r i e s of I n t e g r a t i o n . New York: Dover, i 9 6 0 . Integrale,; Longeur, A i r e . A n n a l i Di Mathematica. s e r i e s 3/ v o l . 7, 1902, pp. 231-359. Lecbns Sur L ' I n t e g r a t i o n . P a r i s : G a u t h i e r - V i l l a r s , 1926. Theory of the I n t e g r a l . E n g l i s h ' T r a n s l a t i o n . New York: Hafner, 1937. A Short H i s t o r y of Mathematics. London: Harrap. A H i s t o r y of: Mathematics. London: T a y l o r and F r a n c i s , 195&. 47. 39. D. E. Smith: 40. T. J . S t i e l t j e s : 41. 0. S t o l z : 42. D. J . S t r u i k : 43. 0. T o e p l i t z : 44. E. J . Townsend: 45. E. B. Van V l e c k : 46. H. Weber: H i s t o r y of Math. v o l . I I . Boston: Ginn, 1925. Recherches Sur Les F r a c t i o n s Con-t i n u e s . Ann. Fac. S c i . de Toulous, v o l . V I I I , 1894, pp. J 1 - J 122. Ueber einen zu e i n e r u n e n d l i c h e n Punktmenge gehorigen Grenzwerth. Mathemat. Annalen, v o l . 23, ; 1&S4, pp. 152-156. A Concise H i s t o r y of Math. London: B e l l , arid Sons, 195^. The C a l c u l u s . Chicago: U n i v e r s i t y of Chicago P r e s s , 1963. F u n c t i o n s of a Real V a r i a b l e . New York: H o l t , 1925. The I n f l u e n c e of F o u r i e r S e r i e s Upon the Development of Math. Science, v o l . 39, 1914, pp. 113-124. The C o l l e c t e d Works of Bernhard Riemann. 2nd. ed. New York: 1 1 Dover, 1953-
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A history of the definite integral Kallio, Bruce Victor 1966
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Title | A history of the definite integral |
Creator |
Kallio, Bruce Victor |
Publisher | University of British Columbia |
Date Issued | 1966 |
Description | The definite integral has an interesting history. In this thesis we trace its development from the time of ancient Greece (500-200 B. C.) until the modern period. We place special emphasis on the work done in the nineteenth century and on the work of Lebesgue (1902). The thesis is divided into four parts arranged roughly chronologically. The first part traces the developments in the period from the fifth century B. C. until the eighteenth century A. D. Secondary sources were used in writing this history. The second part recounts the contributions of the nineteenth century. The original works of Cauchy, Dirichlet, Riemann, Darboux, and Stieltjes are examined, the third part is concerned with the development of measures in the latter part of the nineteenth century. This work leads to the Lebesgue integral. The final part is a brief survey of modern ideas. |
Subject |
Lebesgue, Henri Léon, 1875-1941 Definite integrals -- History |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-08-26 |
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. |
IsShownAt | 10.14288/1.0080597 |
URI | http://hdl.handle.net/2429/36941 |
Degree |
Master of Arts - MA |
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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