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).
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