Measure and Integration
An Introduction
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Beschreibung
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 ������� spaces, and navigates the intricacies of signed and complex measures, providing students with a comprehensive foundation in this vital area of mathematics.
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.
Features the detailed comparison among the Lebesgue integral, improper Riemann integral, and Newton integral Acts as a bridge between Riemann integral and the more sophisticated Lebesgue integral Serves as a core textbook for senior undergraduate and beginning graduate students of mathematics
Autor*in
Satya N. Mukhopadhyay
Themen in »Measure and Integration«
Lebesgue measure Lebesgue integral Dini derivatives Lebesgue-Stieltje's measure Hausdroff measure Lp-spaces signed measure Radon-Nikodym derivative
Stimmen zu »Measure and Integration«
"The authors have written a careful and rigorous exposition of the foundational topics usually taught in a second undergraduate course in real analysis ... . The mode of presentation evokes the classic 'definition-theorem-proof' ... lecture format, interspersed with illustrative examples and supported by well-constructed exercises at the end of each chapter. The authors take care not to gloss over technical details in the proofs, which makes the text especially accessible to less experienced readers." (Tushar Das, zbMATH 1570.28001, 2026)
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