Integration – A Functional Approach (Birkhäuser Advanced Texts Basler Lehrbücher) 1998th Edition by Klaus Bichteler (PDF)

Description

This book covers Lebesgue integration and its generalizations from Daniell’s point of view, modified by the use of seminorms. Integrating functions rather than measuring sets is posited as the main purpose of measure theory. From this point of view Lebesgue’s integral can be had as a rather straightforward, even simplistic, extension of Riemann’s integral; and its aims, definitions, and procedures can be motivated at an elementary level. The notion of measurability, for example, is suggested by Littlewood’s observations rather than being conveyed authoritatively through definitions of (sigma)-algebras and good-cut-conditions, the latter of which are hard to justify and thus appear mysterious, even nettlesome, to the beginner. The approach taken provides the additional benefit of cutting the labor in half. The use of seminorms, ubiquitous in modern analysis, speeds things up even further. The book is intended for the reader who has some experience with proofs, a beginning graduate student for example. It might even be useful to the advanced mathematician who is confronted with situations – such as stochastic integration – where the set-measuring approach to integration does not work.

User’s Reviews

Editorial Reviews: Review “One particularly valuable aspect of the book is that it gives a much richer discussion of Daniell’s approach to integration than is usual. A generous supply of exercises is provided for each chapter. Answers to selected problems are given in an appendix.” –ZAA From the Back Cover This book covers Lebesgue integration and its generalizations from Daniell’s point of view, modified by the use of seminorms. Integrating functions rather than measuring sets is posited as the main purpose of measure theory. From this point of view Lebesgue’s integral can be had as a rather straightforward, even simplistic, extension of Riemann’s integral; and its aims, definitions, and procedures can be motivated at an elementary level. The notion of measurability, for example, is suggested by Littlewood’s observations rather than being conveyed authoritatively through definitions of (sigma)-algebras and good-cut-conditions, the latter of which are hard to justify and thus appear mysterious, even nettlesome, to the beginner. The approach taken provides the additional benefit of cutting the labor in half. The use of seminorms, ubiquitous in modern analysis, speeds things up even further. The book is intended for the reader who has some experience with proofs, a beginning graduate student for example. It might even be useful to the advanced mathematician who is confronted with situations – such as stochastic integration – where the set-measuring approach to integration does not work.—— Reviews This book provides a complete and rapid introduction to Lebesgue integration and its generalizations from Daniell’s point of view, (…) The development is clear and it contains interesting historical notes and motivations, abundant exercises and many supplements. The connection with the historical development of integration theory is also pointed out.- Zentralblatt MATHThe material is well motivated and the writing is pleasantly informal. (…) There are numerous exercises, many destined to be used later in the text, and 15 pages of solutions/hints.- Mathematical Reviews About the Author Klaus Bichteler is a Professor of mathematics at the University of Texas. Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book is a great reference for Daniell’s method of integration. The exercises in each section help the reader understand the material more deeply. The approach to integration developed in the book makes the construction of vector valued integrals and the stochastic theory integration more direct. Connections and equivalence to the Lebesgue Caratheodory approach are also presented.

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