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

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