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A (Terse) Introduction to Lebesgue Integration (Student Mathematical Library) John Franks,
Publisher: American Mathematical Society
ISBN: 0821848623
Edition: Paperback; 2009-07-08
Summary:
This book provides a student's first encounter with the concepts of
measure theory and functional analysis. Its structure and content
reflect the belief that difficult concepts should be introduced in
their simplest and most concrete forms. Despite the use of the word
``terse'' in the title, this text might also have been called A
(Gentle) Introduction to Lebesgue Integration. It is terse in the
sense that it treats only a subset of those concepts typically found
in a substantial graduate-level analysis course. The book emphasizes
the motivation of these concepts and attempts to treat them simply
and concretely. In particular, little mention is made of general
measures other than Lebesgue until the final chapter and attention is
limited to $R$ as opposed to $R^n$. After establishing the primary
ideas and results, the text moves on to some applications. Chapter 6
discusses classical real and complex Fourier series for $L^2$
functions on the interval and shows that the Fourier series of an
$L^2$ function converges in $L^2$ to that function. Chapter 7
introduces some concepts from measurable dynamics. The Birkhoff
ergodic theorem is stated without proof and results on Fourier series
from Chapter 6 are used to prove that an irrational rotation of the
circle is ergodic and that the squaring map on the complex numbers of
modulus 1 is ergodic. This book is suitable for an advanced
undergraduate course or for the start of a graduate course. The text
presupposes that the student has had a standard undergraduate course
in real analysis.
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