This book is devoted to integration, one of the two main operations in 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 approach allows us, on the one hand, to quickly develop the practical skills of integration as well as, on the other hand, in Part 2, to pass naturally to the more general Lebesgue integral. Based on the latter, in Part 2, the author develops a theory of integration for functions of several variables. In Part 3, within the same methodological scheme, the author presents 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 chapters are complemented with problems, mostly of the theoretical type.
The book is mainly devoted to students of mathematics and related specialities. However, Part 1 can be successfully used by any student as a simple introduction to integration calculus.
Author(s): Vigirdas Mackevičius
Series: Mathematics and Statistics Series
Publisher: ISTE-Wiley
Year: 2014
Language: English
Pages: 284
Tags: Analysis, Integral, Mathematics, Measure Theory
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. Area of a curvilinear trapezium 43
4.2. A general scheme for applying the integrals 51
4.3. Area of a surface of revolution 52
4.4. Area of curvilinear sector 53
4.5. Applications in mechanics 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. The notion “almost everywhere” 97
P.7. Problems 104
CHAPTER 8. LEBESGUE INTEGRAL 105
8.1. Proof of the correctness of the definition of integral 106
8.2. Proof of the Beppo Levi theorem 114
8.3. Proof of the Fatou–Lebesgue theorem 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. Calculation of the area of a plane figure 161
10.2. Calculation of the volume of a solid 162
10.3. Calculation of the area of a surface 162
10.4. Calculation of the mass of a body 165
10.5. The static moment and mass center of a body 166
CHAPTER 11. PARAMETER-DEPENDENT INTEGRALS 169
11.1. Introduction 169
11.2. Improper PDIs 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
CHAPTER 19. PRODUCT OF TWO MEASURE SPACES 267
P.19. Problems 275
BIBLIOGRAPHY 277
INDEX 279
