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Integral and Measure
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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

Vigirdas MACKEVIeIUS 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.

#### Reviews

"We have to mention here that since some of the problems are of a theoretical nature, while others are direct computations or applications of the theory, the book addresses both students who want a quick introduction to integrals and their use in real analysis and students who want to understand the mechanism behind an integration theory built on some measure space." (Mathematical Reviews, 1 October 2015)  