Mackevicius V. Integral and Measure

New York: Wiley-ISTE, 2014. — 302 p.This textbook is devoted to integration, an important part of 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 practical skills of integration and, on the other hand, later, in Part 2, to pass naturally to the more general Lebesgue integral. Based on the latter, in Part 2, we develop a theory of integration for functions of several variables. In Part 3, within the same methodological scheme, we present 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 the chapters are complemented with problems, mostly of the theoretical type.
Although the three parts of the book are methodically related to each other, they are somewhat independent. For example, any reader accustomed to the Riemann integral and wishing to get into the theory of the Lebesgue integral is encouraged to begin with Part 2. Those who
feel they are lacking in the basics of general theory of measure and integration should open the book from Part 3.
The book is mainly devoted to students of mathematics and related specialities. However, Part 1 can be successfully used by any students as a simple introduction to integration calculus.