Measure and Integration

Designed for senior undergraduate and graduate students in mathematics, this textbook offers a comprehensive exploration of measure theory and integration. It acts as a pivotal link bridging the Riemann integral and the Lebesgue integral, with a primary focus on tracing the evolution of measure and integration from their historical roots. A distinctive feature of the book is meticulous guidance, providing a step-by-step journey through the subject matter, thus rendering complex concepts more accessible to beginners. A fundamental grasp of differential and integral calculus, as well as Riemann integration, is recommended to ensure a smoother comprehension of the material.

This textbook comprises 10 well-structured chapters, each thoughtfully organized to lead students from fundamental principles to advanced complexities. Beginning with the establishment of Lebesgue’s measure on the real line and an introduction to measurable functions, the book then delves into exploring the cardinalities of various set classes. As readers progress, the subtleties of the Lebesgue integral emerge, showcasing its generalization of the Riemann integral and its unique characteristics in higher dimensions.

One of the book’s distinctive aspects is its indepth comparison of the Lebesgue integral, improper Riemann integral, and Newton integral, shedding light on their distinct qualities and relative independence. Subsequent chapters delve into the realm of general measures, Lebesgue-Stieltje's measure, Hausdorff ’s measure, and the concept of measure and integration in product spaces. Furthermore, the book delves into function spaces, such as 𝘓𝘱 spaces, and navigates the intricacies of signed and complex measures, providing students with a comprehensive foundation in this vital area of mathematics.

Satya N. Mukhopadhyay is Emeritus Professor at the Department of Mathematics, the University of Burdwan, West Bengal, India. A renowned mathematician, after earning his master’s degree in Mathematics in 1962, he began teaching at the University of Burdwan in 1963. With over a remarkable 38 years, his passion for research led him to explore different types of derivatives and integrals, and how they connect to trigonometric series. In 1967, he earned his Ph.D. from the University of Calcutta. He was invited twice as Visiting Professor to the University of British Columbia in Canada, first in the early 1970s and then in the mid-1980s, during which he teamed up with Prof. P.S. Bullen for research. With more than 100 research papers to his credit, most of them appearing in international journals, he has reviewed more than 150 research papers for the Mathematical Reviews. Author of two books, Real Analysis and Higher-Order Derivatives, he has guided many students in achieving their Ph.D. degrees.

Subhasis Ray is Professor at the Department of Mathematics, Visva-Bharati University, West Bengal, India, since 2006. Earlier, he worked as Assistant Professor at Kalna College, affiliated with the University of Burdwan, West Bengal, from 2000–2006. A student of Prof. Satya N. Mukhopadhyay, Prof. Ray embarked on his mathematical journey and joined the University of Burdwan as a CSIR scholar in 1997, after completing his M.Sc. degree. He earned his Ph.D. in Mathematics under the supervision of Prof. Mukhopadhyay, in 2004.

His areas of interest are in real function theory, generalized derivatives on real lines, the theory of non-absolute integration, and its application to trigonometric series. Additionally, he has displayed a keen interest in soft set theory and fuzzy set theory. His prowess as a guide is evident from the Ph.D. students he has supervised, with five of them having been awarded doctorates and three others currently pursuing research under his guidance. Some of his notable works include “On Laplace derivative”, “Soft set and soft group from the classical viewpoint”, and “Soft measure theory”. He also reviewed many research papers for the Mathematical Reviews.