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). |Measure and Integration A First Course | Mathematics
