This volume is not part of this online collection.
This is a graduate text introducing the fundamentals of measure theory
and integration theory, which is the foundation of modern real analysis.
The text focuses first on the concrete setting of Lebesgue measure and
the Lebesgue integral (which in turn is motivated by the more classical
concepts of Jordan measure and the Riemann integral), before moving on
to abstract measure and integration theory, including the standard
convergence theorems, Fubini's theorem, and the Carathéodory extension
theorem. Classical differentiation theorems, such as the Lebesgue and
Rademacher differentiation theorems, are also covered, as are
connections with probability theory. The material is intended to cover
a quarter or semester's worth of material for a first graduate course in
real analysis.
There is an emphasis in the text on tying together the abstract and the
concrete sides of the subject, using the latter to illustrate and
motivate the former. The central role of key principles (such as
Littlewood's three principles) as providing guiding intuition to the
subject is also emphasized. There are a large number of exercises
throughout that develop key aspects of the theory, and are thus an
integral component of the text.
As a supplementary section, a discussion of general problem-solving
strategies in analysis is also given. The last three sections discuss
optional topics related to the main matter of the book.
Readership
Graduate students interested in analysis, in particular, measure
theory.