The Stieltjes Integral

269 Pages·2023·6.487 MB·English

by  Gregory Convertito, David Cruz-Uribe

The Stieltjes Integral
The Stieltjes Integral provides a detailed, rigorous treatment of the Stieltjes in-
tegral. This integral is a generalization of the Riemann and Darboux integrals
of calculus and undergraduate analysis and can serve as a bridge between clas-
sical and modern analysis. It has applications in many areas, including number
theory, statistics, physics, and finance. This book begins with the Darboux inte-
gral, builds the theory of functions of bounded variation, and then develops the
Stieltjes integral. It culminates with a proof of the Riesz representation theorem
as an application of the Stieltjes integral.
For much of the 20th century, the Stjeltjes integral was a standard part of the
undergraduate or beginning graduate student sequence in analysis. However,
the typical mathematics curriculum has changed at many institutions, and the
Stieltjes integral has become less common in undergraduate textbooks and analy-
sis courses. This book seeks to address this by offering an accessible treatment
of the subject to students who have had a one-semester course in analysis. This
book is suitable for a second-semester course in analysis, and also for indepen-
dent study or as the foundation for a senior thesis or Master’s project.
Features:
•
Written to be rigorous without sacrificing readability.
•
Accessible to undergraduate students who have taken a one-semester course
on real analysis.
•
Contains a large number of exercises from routine to challenging.

The Stieltjes Integral
Gregory Convertito
DePaul University, United States of America
David Cruz-Uribe, OFS
University of Alabama, United States of America
First edition published 2023
by CRC Press
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Library of Congress Cataloging‑in‑Publication Data
Names: Convertito, Gregory, author. | Cruz-Uribe, David V., author.
Title: The Stieltjes integral / Gregory Convertito, DePaul University,
United States of America, David Cruz-Uribe, OFS, University of Alabama,
United States of America.
Description: First edition. | Boca Raton : C&H/CRC Press, 2023. | Includes
bibliographical references and index.
Identifiers: LCCN 2022041648 (print) | LCCN 2022041649 (ebook) | ISBN
9780815374008 (hardback) | ISBN 9781032439136 (paperback) | ISBN
9781351242813 (ebook)
Subjects: LCSH: Integrals, Stieltjes.
Classification: LCC QA311 .C66 2023 (print) | LCC QA311 (ebook) | DDC
515/.43--dc23/eng20221121
LC record available at https://lccn.loc.gov/2022041648
LC ebook record available at https://lccn.loc.gov/2022041649
ISBN: 978-0-815-37400-8 (hbk)
ISBN: 978-1-032-43913-6 (pbk)
ISBN: 978-1-351-24281-3 (ebk)

DOI: 10.1201/9781351242813
Typeset in CMR10
by KnowledgeWorks Global Ltd.
Publisher’s note: This book has been prepared from camera-ready copy provided by the authors.
Dedication
I want to dedicate this book to the members of the secretarial staff in the
Department of Mathematics at the University of Alabama: Michele Farley,
Marcia Black, and Natalie Lau. Without their unwavering support, I would
have never survived my time as Chair of the department, and their hard work
freed up the time that made the writing of this book possible.
—DCU
For my grandad (1934–2022), one of many who cultivated my interest in
mathematics, and who I had hoped would hold a copy.
—GC

Contents
Preface ix
About the Author xvii
1 The Darboux Integral 1
1.1 Bounded, Continuous, and Monotonic Functions . . . . . . . 2
1.2 Step Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 The Darboux Integral . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Properties of the Darboux Integral . . . . . . . . . . . . . . . 17
1.5 Limits and the Integral . . . . . . . . . . . . . . . . . . . . . 26
1.6 The Fundamental Theorem of Calculus . . . . . . . . . . . . 30
1.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2 Further Properties of the Integral 47
2.1 The Lebesgue Criterion . . . . . . . . . . . . . . . . . . . . . 47
2.2 The Riemann Integral . . . . . . . . . . . . . . . . . . . . . . 56
2.3 Integrable Functions as a Normed Vector Space . . . . . . . 67
2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3 Functions of Bounded Variation 89
3.1 Monotonic Functions . . . . . . . . . . . . . . . . . . . . . . 91
3.2 Functions of Bounded Variation . . . . . . . . . . . . . . . . 93
3.3 Properties of Functions of Bounded Variation . . . . . . . . . 99
3.4 Limits and Bounded Variation . . . . . . . . . . . . . . . . . 103
3.5 Discontinuities and the Saltus Decomposition . . . . . . . . . 110
3.6 BV[a,b] as a Normed Vector Space . . . . . . . . . . . . . . 119
3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4 The Stieltjes Integral 137
4.1 The Stieltjes Integral of Step Functions . . . . . . . . . . . . 138
4.2 The Stieltjes Integral with Increasing Integrator . . . . . . . 142
vii
viii Contents
4.3 The Stieltjes Integral with BV Integrator . . . . . . . . . . . 148
4.4 Existence of the Stieltjes Integral . . . . . . . . . . . . . . . 154
4.5 Limits and the Stieltjes Integral . . . . . . . . . . . . . . . . 162
4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
5 Further Properties of the Stieltjes Integral 177
5.1 The Stieltjes Integral and Integration by Parts . . . . . . . . 177
5.2 The Lebesgue-Stieltjes Criterion . . . . . . . . . . . . . . . . 189
5.3 The Riemann-Stieltjes Integral . . . . . . . . . . . . . . . . . 196
5.4 The Riesz Representation Theorem . . . . . . . . . . . . . . 209
5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
Bibliography 237
Index of Symbols 241
Index 245
Preface
The purpose of this book is to provide a detailed, rigorous treatment of the
Stieltjesintegral.ThisintegralisageneralizationoftheRiemannandDarboux
integrals of calculus and undergraduate analysis and can serve as a bridge
between classical and modern analysis. It has applications in many areas,
including number theory, statistics, physics, and finance. For much of the
twentieth century, it was a standard part of the undergraduate or beginning
graduate student sequence in analysis.
The typical math major has changed at many institutions, and the Stielt-
jesintegralhasbecomelesscommoninundergraduatetextbooksandanalysis
courses. To address this, we have written a book accessible to students who
havehadaonesemestercourseinanalysisthatwouldintroducetheDarboux
and Riemann integrals and then fully develop the Stieltjes integral. It is suit-
able for a second semester course in analysis, and also for independent study
or as the foundation for a senior thesis or Masters project.
How this Project Began
Before describing the aims, structure, and organization of our book, we
firstwanttodescribethegenesisofthisproject,asthisverymuchshapedthe
final product. In 2013 the second author (hereafter, DCU), then a professor
atTrinityCollege,supervisedareadingcourseontheLebesgueintegralusing
the book by Franks, A (Terse) Introduction to Lebesgue Integration [16]. To
motivate the definition of the Lebesgue integral, this book gives a brief intro-
duction to the Darboux integral in terms of approximation by step functions.
DCU found this approach fascinating since it used ideas that are standard in
graduateanalysiscourses:definingtheintegralonasimplerclassoffunctions,
and extending the definition using approximation, the integral as a linear op-
erator, and the interaction of pointwise and uniform convergence with the
integral.
In 2014 DCU taught the introductory course on real analysis at Trinity
and used the approach sketched in Franks to develop the integral. The first
author(hereafter,GC)wasastudentintheclass.Thefollowingyearhetooka
readingcourseoutofFranks.DCUsuggestedthatforhisseniorhonorsthesis,
ix