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.
General
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