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A history of the definite integral Kallio, Bruce Victor 1966

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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 .. 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