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.