Integral and Measure From Rather Simple to Rather Complex

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.

Vigirdas MACKEVIèIUS is Professor of the Department of Mathematical Analysis in the Faculty of Mathematics of Vilnius University in Lithuania. His research interests include stochastic analysis, limit theorems for stochastic processes, and stochastic numerics.

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

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