**Student Mathematical Library**

Volume: 48;
2009;
202 pp;
Softcover

MSC: Primary 28; 42;

**Print ISBN: 978-0-8218-4862-3
Product Code: STML/48**

List Price: $42.00

Individual Price: $33.60

**Electronic ISBN: 978-1-4704-1219-7
Product Code: STML/48.E**

List Price: $39.00

Individual Price: $31.20

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#### Supplemental Materials

# A (Terse) Introduction to Lebesgue Integration

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*John Franks*

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.

#### Readership

Undergraduate and graduate students interested in analysis or its applications to other areas of mathematics.

#### Reviews & Endorsements

The book is suitable for an advanced undergraduate course or for the start of a graduate course. Each chapter contains a suitable number of exercises.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## A (Terse) Introduction to Lebesgue Integration

- Cover Cover11 free
- Title page iii5 free
- Contents vii9 free
- Preface xi13 free
- The regulated and Riemann integrals 117 free
- Lebesgue measure 2541
- The Lebesgue integral 4157
- The integral of unbounded functions 6379
- The Hilbert space 𝐿² 8399
- Classical Fourier series 111127
- Two ergodic transformations 129145
- Background and foundations 141157
- Lebesgue measure 173189
- A non-measurable set 193209
- Bibliography 197213
- Index 199215 free
- Back Cover Back Cover1219