Preface
This text is intended to provide a student’s first encounter with the
concepts of measure theory and functional analysis. Its structure
and content were greatly influenced by my belief that good pedagogy
dictates introducing difficult concepts in their simplest and most con-
crete forms. For example, the study of abstract metric spaces should
come after the study of the metric and topological properties of
Rn.
Multidimensional calculus should not be introduced in Banach spaces
even if the proofs are identical to the proofs for
Rn.
And a course in
linear algebra should precede the study of abstract algebra.
Hence, 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 substantive graduate level analy-
sis course. I have emphasized the motivation of these concepts and
attempted to treat them in their simplest and most concrete form.
In particular, little mention is made of general measures other than
Lebesgue until the final chapter. Indeed, we restrict our attention
to Lebesgue measure on R and no treatment of measures on
Rn
for
n 1 is given. The emphasis is on real-valued functions but com-
plex functions are considered in the chapter on Fourier series and in
the final chapter on ergodic transformations. I consider the narrow
selection of topics to be an approach at one end of a spectrum whose
xi
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