Integral and Measure: From Rather Simple to Rather Complex

284 Pages·2014·1.087 MB·English

by  Vigirdas Mackevicius

Integral and Measure
To my beloved wife Eugenija
Series Editor
Nikolaos Limnios

Integral and Measure

From Rather Simple to Rather Complex

Vigirdas Mackevičius
First published 2014 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
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Library of Congress Control Number: 2014945514

British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-84821-769-0
Contents
PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
NOTE FOR THE TEACHER OR WHO IS BETTER, RIEMANN
OR LEBESGUE? . . . . . . . . . . . . . . . . . . . . . . . . . . xi
NOTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
PART 1. INTEGRATION OF ONE-VARIABLE
FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
CHAPTER 1. FUNCTIONS WITHOUT SECOND-KIND
DISCONTINUITIES . . . . . . . . . . . . . . . . . . . . . . . . 3
P.1. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 9
CHAPTER 2. INDEFINITE INTEGRAL . . . . . . . . . . . . . 11
P.2. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 16
CHAPTER 3. DEFINITE INTEGRAL . . . . . . . . . . . . . . 19
3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 19
P.3. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 38
CHAPTER 4. APPLICATIONS OF THE INTEGRAL . . . . . . 43
4.1. Areaofacurvilineartrapezium. . . . . . . . . . . . . . 43
4.2. Ageneralschemeforapplyingtheintegrals . . . . . . 51
vi IntegralandMeasure
4.3. Areaofasurfaceofrevolution . . . . . . . . . . . . . . 52
4.4. Areaofcurvilinearsector . . . . . . . . . . . . . . . . . 53
4.5. Applicationsinmechanics . . . . . . . . . . . . . . . . 54
P.4. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 56
CHAPTER 5. OTHER DEFINITIONS: RIEMANN AND
STIELTJES INTEGRALS . . . . . . . . . . . . . . . . . . . . . 59
5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 59
P.5. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 75
CHAPTER 6. IMPROPER INTEGRALS . . . . . . . . . . . . . 79
P.6. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 88
PART 2. INTEGRATION OF SEVERAL-VARIABLE
FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
CHAPTER 7. ADDITIONAL PROPERTIES OF STEP
FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.1. Thenotion“almosteverywhere” . . . . . . . . . . . . . 97
P.7. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 104
CHAPTER 8. LEBESGUE INTEGRAL . . . . . . . . . . . . . . 105
8.1. Proofofthecorrectnessofthedefinitionof
integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
8.2. ProofoftheBeppoLevitheorem . . . . . . . . . . . . . 114
8.3. ProofoftheFatou–Lebesguetheorem . . . . . . . . . . 119
P.8. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 133
CHAPTER 9. FUBINI AND CHANGE-OF-VARIABLES
THEOREMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
P.9. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 157
CHAPTER 10. APPLICATIONS OF MULTIPLE
INTEGRALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
10.1.Calculationoftheareaofaplanefigure . . . . . . . . 161
10.2.Calculationofthevolumeofasolid . . . . . . . . . . . 162
Contents vii
10.3.Calculationoftheareaofasurface . . . . . . . . . . . 162
10.4.Calculationofthemassofabody . . . . . . . . . . . . 165
10.5.Thestaticmomentandmasscenterofabody . . . . . 166
CHAPTER 11. PARAMETER-DEPENDENT INTEGRALS . . . 169
11.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . 169
11.2.ImproperPDIs . . . . . . . . . . . . . . . . . . . . . . . 177
P.11.Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 187
PART 3. MEASURE AND INTEGRATION IN A MEASURE
SPACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
CHAPTER 12. FAMILIES OF SETS . . . . . . . . . . . . . . . 193
12.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . 193
P.12.Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 197
CHAPTER 13. MEASURE SPACES . . . . . . . . . . . . . . . 199
P.13.Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 206
CHAPTER 14. EXTENSION OF MEASURE . . . . . . . . . . . 209
P.14.Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 220
CHAPTER 15. LEBESGUE–STIELTJES MEASURES ON THE
REAL LINE AND DISTRIBUTION FUNCTIONS . . . . . . . . 223
P.15.Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 229
CHAPTER 16. MEASURABLE MAPPINGS AND REAL
MEASURABLE FUNCTIONS . . . . . . . . . . . . . . . . . . . 233
P.16.Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 239
CHAPTER 17. CONVERGENCE ALMOST EVERYWHERE
AND CONVERGENCE IN MEASURE . . . . . . . . . . . . . . 241
P.17.Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 246
CHAPTER 18. INTEGRAL . . . . . . . . . . . . . . . . . . . . 249
P.18.Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 263
viii IntegralandMeasure
CHAPTER 19. PRODUCT OF TWO MEASURE SPACES . . . 267
P.19.Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 275
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . 277
INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
Preface
This textbook is devoted to integration, an important part of
calculus. In Part 1, the definition of the integral of a one-variable
function is different (not essentially, but rather methodically) from
traditional definitions of Riemann or Lebesgue integrals. Such an
approachallowsus,ontheonehand,toquicklydeveloppracticalskills
of integration and, on the other hand, later, in Part 2, to pass naturally
tothemoregeneralLebesgueintegral.Basedonthelatter,inPart2,we
develop a theory of integration for functions of several variables. In
Part 3, within the same methodological scheme, we present the
elements of theory of integration in an abstract space equipped with a
measure; we cannot do without this in functional analysis, probability
theory, etc. The majority of the chapters are complemented with
problems,mostlyofthetheoreticaltype.
Although the three parts of the book are methodically related to
each other, they are somewhat independent. For example, any reader
accustomed to the Riemann integral and wishing to get into the theory
oftheLebesgueintegralisencouragedtobeginwithPart2.Thosewho
feel they are lacking in the basics of general theory of measure and
integrationshouldopenthebookfromPart3.
The book is mainly devoted to students of mathematics and related
specialities. However, Part 1 can be successfully used by any students
asasimpleintroductiontointegrationcalculus.
x IntegralandMeasure
Weusethedoublenumberingofstatements(theorems,propositions,
etc.):thefirstnumberdenotesthenumberofthechapter,andthesecond
indicatesthenumberofthestatementwithinthechapter.Thoughalarge
number of books and papers have influenced the contents of this book,
theshortreferencelistincludesonlythosedirectlyusedbytheauthor.
The book is essentially a revised translation of the author’s book
Integral and Measure (TEV, Vilnius, 1998) from Lithuanian. The
author would like to thank Vilijandas Bagdonavicˇius and a large
number of students of the Faculty of Mathematics and Informatics of
Vilnius University. Thanks to them, the book contains significantly
fewermisprints.
VigirdasMackevicˇius
Vilnius,July2014