A First Course in Integration

506 Pages·1966·6.288 MB·English

by  Edgar Asplund, Lutz Bungart

A COURSE IN
INTEGRATION
/
xM ?Iunddrl
'
4'

A
FIRST COURSE
IN
INTEGRA TION
EDGAR ASPLUND
UNIVERSITY OF STOCKHOLM
LUTZ BUNGART
UNIVERSITY OF CALIFORNIA, BERKELEY
HOLT, RINEHART AND WINSTON
NEW YORK• CHICAGO• SAN FRANCISCO• TORONTO• LONDON
COPYRIGHT © 1966 BY HOLT, RINEHART AND WINSTON, INC
ALL RIGHTS RESERVED
LIBRARY OF CONGRESS CATALOG CARD NUMBER 66—10122
20550—0116
PRINTED IN THE UNITED STATES OF AMERICA
Preface
This undergraduate textbook on the theory ofLebesgue integration is designed
for use in a one-semester or two-quarter undergraduate course in the junior or
senioryear.
Quite soon after the appearance of Lebesgue's famous theory of integration,
it was felt that his thesis should be taught in the regular university mathematics
courses, preferably to replace the Riemann integration theory, which it super
seded at the professional level. The initial difficulties in presenting Lebesgue's
material were rapidly smoothed out, but inertia in the educational system worked
against carrying out the program. Indeed, one could notexpecta majority ofthe
colleges and universities to offer an undergraduate course in integration theory
before a well-established need for this theory was felt in important domains of
application. This need is now growing, and we hope that this book will help to
satisfy it.
By "integration theory" we mean that the technique ofintegrating functions is
developed first and is then used for the measuring of sets, rather than the other
way around. The choice ofthe "integration first" approach was mostly a matter
oftaste, butalso madeitpossibleto arriveatmeaningfuland interestingtheorems
onintegrationasearlyas possiblein thecourse. This is theapproach to Lebesgue
integration pioneered by F. Riesz (see Bibliogr.aphical Comments on Chapters 1
and 2 at the end of the book), an approach that~~as been partially forgotten in
recent years with the appearance.of sO many graduate-level books favoring a
measure theoreticapproach. Becausethe studentatthe stagefor which this book
isintendedhashadmuchinstructiononfunctieJ"ns butiittle ornoinstructiononset
theory (subsequent developments i~ high-·school tnat~ematics may change this,
ofcourse), hecangettothecoreof~hemattermor~.fapidlybyfuncti~nal methods
than by set theoretic ones. The fl:l,nttiQ~s play"toe role of"prefabricated parts,"
withwhich thestudentisalreadyfamiliar: .F01-'\be1same kind of unmathematical
reasons, which are nevertheless of tangiliie"i~portance, we have emphas,ized
series more than sequences. Series are the practical tools ofthe applied scientist
and hence have an air of being more concrete than sequences, although the step
from the oneto the otheris trivial. Itis also truethatthe use ofseries gives some
statementsaparticularlyniceappearance(compareLemma2.1.10,byJ.Mikusinski).
Theamountofmaterialincludedislargeenoughtomakethebookameaningful
first textbook for those who plan to go on to graduate studies in mathematics.
The only prerequisite is the material usually covered in a two-year calculus se
quence, thoughitmaybeusefulto precedeacourseinLebesgueintegrationwitha
v
vi PREFACE
course that gives a rigorous foundation for calculus. Within this scope we have
tried to organize the material so that parts of the book can be used for other
purposes. For instance, the following sections can be omitted without causing
gaps later on: 2.0, 2.6, 3.4, 3.5, 4.1, 4.4, 5.3, 5.4, 5.5, 5.6, 6.7, 9.3, and 9.4. Of
these, several sections are not intended to be covered in class, but to be used as
additional reading material for the interested student. These sections are: 2.0,
2.6, 3.4, 5.4, and 6.7.
Special attention has been given to a kind of "minimal course," consisting of
Chapters 1and 2. These two chapters-with the exception, perhaps, of Section
2.6-couldevenreplacethecustomary Riemann integrationtheoryinthestandard
analysis courses. Chapters 1 to 4 constitute a course that could be meaningful
for certain engineering students and which could be supplemented with parts of
Chapter 5 and/or Chapter 8-the last alternative may be especially attractive to
students ofstatistics. The rest ofthebook, Chapters6, 7, and 9, comprising what
one may call the "differentiation theory," is slightly more difficult than the other
chapters, so that its omission would lighten the course more than would appear
from the number ofpages involved. In Chapter 10, Lebesgue integration, as well
asthedifferentiationtheory,areappliedtothestudyofFourierseries. Thesections
ofthis chapter are not meant to exhaust the topic ofFourier series, but rather to
give some applications of what has been done in the preceding chapters. The
sections of Chapter 10 can be fitted in at various places in the book (see the
interdependence diagram at the end ofthis preface).
Thebook isdivided intosectionsthataregroupedin ten chapters. With a few
exceptionseachsectionisdesignedforaone-hourlecture. Eachchapterstartswith
a shortintroduction in which thematerialofthatchapteris outlined and in which
commentson itsdependenceonsectionsofotherchaptersaremade. Each section
endswithafewexercises,usuallyofa simple nature,andthereisa largercollection
ofexercises at the end ofeach chapter. The book contains enough exercises for
bothclassandhomework. Asummaryofthemainresultscompleteseachchapter.
At the speed of approximately one section per lecture, the book will serve as a
textbook for a one-term (15-week) course ofthree lectures a week. The authors
haveusedthematerialofthebookforsucha courseattheUniversityofCalifornia
at Berkeley. However, by omitting some ofthe sections or chapters, the book is
also suited to serve as a textbook for a one-quarter or two-quarter course. To
facilitate the selection ofmaterial for such shorter courses, we have included two
interdependence diagrams at the end ofthis preface.
>/
We have included Section 1.0 to specify our terminology; here a theorem on
double series is proved, sinceit may not be well knownto all students. Whenever
this unfamiliarity with a theorem ofcalculus appeared likely, we have included a
proofin the body ofthe text. We havealso endeavored to make this book some
what repetitive in certain places to achieve a maximum independence among
chapters. We have not hesitated to repeat similar proofs rather than use devices
likegenerallemmasthatwouldmakethepresentationmoreelegant, butalso more
obscure, to the beginner. We have chosen to develop the theory of integration
for the Lebesgue integral, but we have done it in such a way that the proofs are
PPRREEFFAACCEE Vviiii
aallssoo vvaalliidd ffoorr tthhee DDaanniieellll iinntteeggrraall.. SSuucchh ppoossssiibbllee ggeenneerraalliizzaattiioonnss aarree ppooiinntteedd oouutt
aatt aapppprroopprriiaattee ppllaacceess;; tthheeyy aarree aapppplliieedd iinn CChhaapptteerrss 44 aanndd 88 toto sesveveeraral lvvaarriaiabblleess
aanndd SSttiieellttjjeess iinntteeggrraattiioonn,, rreessppeeccttiivveellyy..
TThhee bbeesstt wwaayy ttoo lleeaarrnn aabboouutt aa ssuubbjjeecctt iiss,, ooff ccoouurrssee,, ttoo rreeaadd nnoott oonnllyy oonnee bbooookk,,
bbuutt sseevveerraal.l. IInn tthhee lliitteerraattuurree sseeccttiioonn aatt tthhee eenndd ooff tthhiiss tteexxtt wwee hhaavvee ininddiiccaatteedd
ssoommee bbooookkss tthhaatt mmaayy bbee uusseedd ffoorr ccoollllaatteerraall rreeaaddiinngg.. TThheeyy aarree aallll mmooddeerrnn aanndd
wweellll wwrriitttetenn,, ssoo tthhaatt tthhee sseerriioouuss ssttuuddeenntt wwiillll cceerrttaaiinnllyy nnoott wwaassttee tthhee ttiimmee hhee mmaayy
ddeevvoottee ttoo tthheemm.. VVaalluuaabbllee aallssoo wwiillll bbee tthhee ttiimmee ssppeenntt oonn ssoommee oof ftthhee ffiirrsstt ppaappeerrss
ooff tthhee ppiioonneeeerrss iinn tthhiiss ffiieelldd..
TThhaannkkss aanndd aapppprreecciiaattiioonn aarree eexxtteennddeedd ttoo DDaavviidd GGoohheeeenn,, wwhhoo rreeaadd tthhee mmaannuu-
ssccrriipptt ssoo ccaarreeffuullllyy aanndd mmaaddee mmaannyy ssuuggggeessttioiol1nSs ffoorr iimmpprroovviinngg tthhee pprreesseennttaattiioonn..
AAmmoonngg oouurr ccoolllleeaagguuees,s,p paartritcicuulalarrt hthaannkkss aarree dduuee ttoo PPrrooffeessssoorr JJoohhnn TToodddd,, wwhhoo
ssuuggggeesstteedd aaddddiinngg ssoommee mmaatteerriiaall ((CChhaapptteerr 1100)) aanndd ttoo PPrrooffeessssoorr MMiicchheell LLooeèvvee,, wwhhoo
ppaattiieennttllyy ddiissccuusssseeddm maannyyp prroobblelemms.s. FFiinnaallllyy,, oouurr tthhaannkkss ggoo ttoo MMrrss.. OOllliliee CCuulllleerrss
aanndd ttoo MMrrss.. JJeeaannnnee RRoobbiinnssoonn ffoorr tthheeiirr ppaattiieennccee aanndd sskkiillll iinn ttyyppiinngg tthhee mmaannuussccrriipptt..
SSttoocckkhhoollmm,, SSwweeddeenn EEDDGGAARR AASSPPLLUUNNDD
BBeerrkkeelleeyy,, CCaalliiffoorrnniiaa
LLUUTTZZ BBUUNNGGAARRTT
FFeebbrruuaarryy 11996666