Real Analysis

The philosophy of the book, which makes it quite distinct from many existing texts on the subject, is based on treating the concepts of measure and integration starting with the most general abstract setting and then introducing and studying the Lebesgue measure and integration on the real line as an important particular case. The book consists of nine chapters and appendix, with the material flowing from the basic set classes, through measures, outer measures and the general procedure of measure extension, through measurable functions and various types of convergence of sequences of such based on the idea of measure, to the fundamentals of the abstract Lebesgue integration, the basic limit theorems, and the comparison of the Lebesgue and Riemann integrals. Also, studied are Lp spaces, the basics of normed vector spaces, and signed measures. The novel approach based on the Lebesgue measure and integration theory is applied to develop a better understanding of differentiation and extend the classical total change formula linking differentiation with integration to a substantially wider class of functions. Being designed as a text to be used in a classroom, the book constantly calls for the student's actively mastering the knowledge of the subject matter. There are problems at the end of each chapter, starting with Chapter 2 and totaling at 125. Many important statements are given as problems and frequently referred to in the main body. There are also 358 Exercises throughout the text, including Chapter 1 and the Appendix, which require of the student to prove or verify a statement or an example, fill in certain details in a proof, or provide an intermediate step or a counterexample. They are also an inherent part of the material. More difficult problems are marked with an asterisk, many problems and exercises are supplied with ``existential'' hints. The book is generous on Examples and contains numerous Remarks accompanying definitions, examples, and statements to