Integration - A Functional Approach / Modern Birkhäuser Classics (PDF)

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 MATH

The 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