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 discuss certain subtleties, raise questions on
whether the converse assertions are true, whenever appropriate, or
whether the conditions are essential. With plenty 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. Designed for 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
General
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