Integral and Measure 1st Edition by Vigirdas Mackevicius ISBN 1848217692 978-1848217690
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ISBN 10: 1848217692
ISBN 13: 978-1848217690
Author: Vigirdas Mackevicius
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.
Integral and Measure 1st Table of contents:
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 Integrals
5.1. 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 solid
10.3. 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 Functions
P.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
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