Measure and Integration: A First Course

This concise text is intended as an introductory course in measure and integration. It covers essentials of the subject, providing ample motivation for new concepts and theorems in the form of discussion and remarks, and with many worked-out examples.

The novelty of Measure and Integration: A First Course is in its style of exposition of the standard material in a student-friendly manner. New concepts are introduced progressively from less abstract to more abstract so that the subject is felt on solid footing. The book starts with a review of Riemann integration as a motivation for the necessity of introducing the concepts of measure and integration in a general setting. Then the text slowly evolves from the concept of an outer measure of subsets of the set of real line to the concept of Lebesgue measurable sets and Lebesgue measure, and then to the concept of a measure, measurable function, and integration in a more general setting. Again, integration is first introduced with non-negative functions, and then progressively with real and complex-valued functions. A chapter on Fourier transform is introduced only to make the reader realize the importance of the subject to another area of analysis that is essential for the study of advanced courses on partial differential equations.

Key Features

• Numerous examples are worked out in detail.
• Lebesgue measurability is introduced only after convincing the reader of its necessity.
• Integrals of a non-negative measurable function is defined after motivating its existence as limits of integrals of simple measurable functions.
• Several inquisitive questions and important conclusions are displayed prominently.
• A good number of problems with liberal hints is provided at the end of each chapter.

The book is so designed that it can be used as a text for a one-semester course during the first year of a master's program in mathematics or at the senior undergraduate level.

About the Author

M. Thamban Nair is a professor of mathematics at the Indian Institute of Technology Madras, Chennai, India. He was a post-doctoral fellow at the University of Grenoble, France through a French government scholarship, and also held visiting positions at Australian National University, Canberra, University of Kaiserslautern, Germany, University of St-Etienne, France, and Sun Yat-sen University, Guangzhou, China.

The broad area of Prof. Nair’s research is in functional analysis and operator equations, more specifically, in the operator theoretic aspects of inverse and ill-posed problems. Prof. Nair has published more than 70 research papers in nationally and internationally reputed journals in the areas of spectral approximations, operator equations, and inverse and ill-posed problems. He is also the author of three books: Functional Analysis: A First Course (PHI-Learning, New Delhi), Linear Operator Equations: Approximation and Regularization (World Scientific, Singapore), and Calculus of One Variable (Ane Books Pvt. Ltd, New Delhi), and he is also co-author of Linear Algebra (Springer, New York).

M Thamban Nair is a Professor of Mathematics at the Indian Institute of Technology Madras, Chennai, India. After completing his Ph.D. thesis in 1984 from the Indian Institute of Technology Bombay, Mumbai (India), he did his post-doctoral research at the University of Grenoble (France), for a year under a French Government Scholarship, and after returning from France, he worked as a Research Scientist at Indian Institute of Technology Bombay for a year. He taught at the Goa University almost for a decade, and from December 1995 onwards, he is a regular faculty member at the Indian Institute of Technology Madras. He 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). Besides, 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 (World Scientific, Singapore), Calculus of One Variable (Ane Books, New Delhi), and co-authored a book, Linear Algebra (Springer). He published over 75 research papers in nationally and internationally reputed journals, including the Journal of Indian Mathematical Society, Proceedings of Indian Academy of Sciences, Proceedings of the American Mathematical Society, Journal of Integral Equations and Operator Theory, Mathematics of Computation, Numerical Functional Analysis and Optimization, Journal of Inverse and Ill-Posed Problems, and Inverse Problems. He received many awards for his academic achievements, including the C.L. Chandna award of the Indo-Canadian Math Foundation for outstanding contributions in mathematics research and teaching for the year 2003, and Ganesh Prasad Memorial Award of the Indian Mathematical Society for the year 2015. He is a life member of academic bodies such as the Indian Mathematical Society and Ramanujan Mathematical Society.