Integral and Measure
From Rather Simple to Rather Complex
Publication Date: August 2014 Hardback 300 pp.
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.
Part 1. Integration of One-Variable Functions
1. Functions Without Second-Kind Discontinuities.
2. Indefinite Integral.
3. Definite Integral.
4. Applications of the Integral.
5. Other Definitions: Riemann and Stieltjes Integrals.
6. Improper Integrals.
Part 2. Integration of Several-Variable Functions
7. Additional Properties of Step Functions.
8. Lebesgue Integral.
9. Fubini and Change-of-Variables Theorems.
10. Applications of Multiple Integrals.
11. Parameter-Dependent Integrals.
Part 3. Measure and Integration in a Measure Space
12. Families of Sets.
13. Measure Spaces.
14. Extension of Measure.
15. Lebesgue–Stieltjes Measures on the Real Line and Distribution Functions.
16. Measurable Mappings and Real Measurable Functions.
17. Convergence Almost Everywhere.
19. Product of Two Measure Spaces.
About the Authors
Vigirdas Mackevicius 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.