Integral and measure : from rather simple to rather complex / Vigirdas Mackevičius.

Includes bibliographical references and index.

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Cover page; Half-title page; Title page; Copyright page; Contents; Preface; Note for the Teacher or Who is better, Riemann or Lebesgue?; Notation; Part 1: Integration of One-Variable Functions; 1: Functions without Second-kind Discontinuities; P.1. Problems; 2: Indefinite Integral; P.2. Problems; 3: Definite Integral; 3.1. Introduction; P.3. Problems; 4: Applications of the Integral; 4.1. Area of a curvilinear trapezium; 4.2. A general scheme for applying the integrals; 4.3. Area of a surface of revolution; 4.4. Area of curvilinear sector; 4.5. Applications in mechanics; P.4. Problems.

5: Other Definitions: Riemann and Stieltjes Integrals5.1. Introduction; P.5. Problems; 6: Improper Integrals; P.6. Problems; Part 2: Integration of Several-variable Functions; 7: Additional Properties of Step Functions; 7.1. The notion "almost everywhere"; P.7. Problems; 8: Lebesgue Integral; 8.1. Proof of the correctness of the definition of integral; 8.2. Proof of the Beppo Levi theorem; 8.3. Proof of the Fatou-Lebesgue theorem; P.8. Problems; 9: Fubini and Change-of-Variables Theorems; P.9. Problems; 10: Applications of Multiple Integrals; 10.1. Calculation of the area of a plane figure.

10.2. Calculation of the volume of a solid10.3. Calculation of the area of a surface; 10.4. Calculation of the mass of a body; 10.5. The static moment and mass center of a body; 11: Parameter-dependent Integrals; 11.1. Introduction; 11.2. Improper PDIs; P.11. Problems; Part 3: Measure and Integration in a Measure Space; 12: Families of Sets; 12.1. Introduction; P.12. Problems; 13: Measure Spaces; P.13. Problems; 14: Extension of Measure; P.14. Problems; 15: Lebesgue-Stieltjes Measures on the Real Line and Distribution Functions; P.15 Problems.

16: Measurable Mappings and Real Measurable FunctionsP. 16. Problems; 17: Convergence Almost Everywhere and Convergence in Measure; P.17. Problems; 18: Integral; P.18. Problems; 19: Product of Two Measure Spaces; P.19. Problems; Bibliography; Index.

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.