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 ofnine chapters andappendix, with the material flowing from thebasic set classes, throughmeasures,outer measuresand the general procedure ofmeasure extension,throughmeasurable functionsandvarious types of convergenceof sequences of such based on the idea of measure, to the fundamentals of the abstract Lebesgue integration, the basiclimit theorems, and the comparison of the Lebesgue and Riemann integrals. Also, studied areLp spaces,the basics of normed vector spaces, andsignedmeasures. The novel approach based on the Lebesgue measure and integration theory is applied to develop a better understanding of differentiation and extend the classicaltotal change formulalinking 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 discuss certain subtleties, raise questions on whether the converse assertions are true, whenever appropriate, or whether the conditions are essential.
Withplenty of examples, problems, and exercises, this well-designed text is ideal for a one-semester Master's level graduate course on real analysis with emphasis on the measure and integration theory for students majoring in mathematics, physics, computer science, and engineering.
A concise but profound and detailed presentation of the basics of real analysis with emphasis on the measure and integration theory.Designedfor a one-semester graduate course, with plethora of examples, problems, and exercises.Is of interest to students and instructors in mathematics, physics, computer science, and engineering.Prepares the students for more advanced courses in functional analysis and operator theory.
Contents
Preliminaries
Basic Set Classes
Measures
Extension of Measures
Measurable Functions
Abstract Lebesgue Integral
Lp Spaces
Differentiation and Integration
Signed Measures
The Axiom of Choice and Equivalents
