Nair M. Thamban - Measure and Integration

Measure and Integration

A First Course

Measure and Integration

A First Course

M. Thamban Nair

CRC Press

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Contents

The concepts from the theory of measure and integration are vital to any advanced courses in analysis and its applications, specifically in the applications of functional analysis to other areas such as harmonic analysis, partial differential equations and integral equations, and in the theoretical investigations in applied mathematics. Therefore, an early introduction to such concepts becomes essential in the master's program in mathematics. This book is an attempt toward that goal, requiring minimal background in mathematical analysis.

It is essentially an updated version of the notes the author has been using for teaching courses on measure and integration for the last thirty years. The topics covered in this book are standard ones. However, the reader will definitely find that the presentation of the concepts and results differ from the standard texts, in the sense that it is more student-friendly.

It starts with a short introduction on Riemann integration to motivate the necessity of the concept of integration of functions more general than those allowed in the theory of Riemann integration, and then, in , on Fourier transform, is included only to show how the basic concepts of measure and integration are useful in proving results in another branch of analysis, which has lots of applications in partial differential equations and many engineering subjects. Since the probability space is a particular case of a measure space, at a few places, the implications of certain concepts to probability theory are also included, such as the concepts of random variable, distribution measure, distribution function, and conditional expectation.

Although the theory of integration is vast, the attempt in this book is to introduce the students to this modern subject in a simple and natural manner so that they can pursue the subject further with confidence, and also apply the concepts to other branches of mathematics such as those mentioned earlier. Thus, as the subtitle shows, the book is meant only as a first course on measure and integration. Advanced topics involving measures in the context of topological spaces and topological groups and so on are beyond the scope of this text.

This book can be used for a one-semester course of about 45 lectures for the first- or second-semester of a master's programme in mathematics. The book can also be used for the final year of a bachelor's program, perhaps, omitting the last two chapters.

To use this book for a course on measure and integration, no pre-requisite is assumed, except the mathematical maturity to appreciate and grasp concepts in analysis, though it is recommended that it be taught after a course on real analysis.

Acknowledgments:

While teaching this course, as well as during the preparation of the notes, I have greatly benefited from the contributions of my students in terms of their questions in classes and also during the clarification of their doubts. One of those students, Rama Seshan, a research scholar from the electrical engineering department at IIT Madras, read the notes carefully and made suggestions from the students' points of view. Dr. S. Sivanandan (IIT Delhi) and my former research scholar Dr. Ajoy Jana read some of the chapters and brought to my attention some typos and corrections. Dr. P. Sam Johnson (NIT Karnataka) and my research scholar Subhankar Mondal read all the chapters thoroughly and critically and suggested many corrections in the text. I am thankful to all of them.

Finally, I thank my wife Sunita for her forbearance and encouragement.

M. Thamban Nair

M. Thamban Nair is professor of mathematics at the Indian Institute of Technology (IIT) Madras, Chennai, India. After completing his PhD thesis in 1984 at IIT Bombay, Mumbai (India), he did his post-doctoral research at the University of Grenoble (France), for a year under a French government scholarship. Following his return to India, he worked as a research scientist at IIT Bombay for a year. He taught at the Goa University for nearly a decade, and from December 1995 onward, he has been a faculty member at IIT Madras. He has also held visiting positions at the Australian National University, Canberra (Australia), University of Kaiserslautern (Germany), Sun Yat-sen University, Guangzhou (China), University of Saint-Etienne (France), Weierstrass Institute for Applied Analysis and Stochastics, Berlin (Germany), and University of Chemnitz (Germany). In addition, he has given many invited talks at various institutes in India and abroad.

The broad area of Professor Nair's research is in functional analysis and operator theory, more specifically, spectral approximation, the approximate solution of integral and operator equations, regularization of inverse and ill-posed problems. He has authored three books, Functional Analysis: A First Course (PHI-Learning, New Delhi), Linear Operator Equations: Approximation and Regularization