The Stieltjes Integral
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The Stieltjes Integral provides a detailed, rigorous treatment of the Stieltjes integral. This integral is a generalization of the Riemann and Darboux 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. It begins with the Darboux integral, 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 analysis 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 independent study or as the foundation for a senior thesis or Masters project.
- 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.
Table of Contents
1. The Darboux Integral. 1.1. Bounded, Continuous, and Monotonic Functions. 1.2. Step Functions. 1.3. The Darboux Integral. 1.4. Properties of the Darboux Integral. 1.5. Limits and the Integral. 1.6. The Fundamental Theorem of Calculus. 1.7. Exercises. 2. Further Properties of the Integral. 2.1. The Lebesgue Criterion. 2.2. The Riemann Integral. 2.3. Integrable Functions as a Normed Vector Space. 2.4. Exercises. 3. Functions of Bounded Variation. 3.1. Monotonic Functions. 3.2. Functions of Bounded Variation. 3.3. Properties of Functions of Bounded Variation. 3.4. Limits and Bounded Variation. 3.5. Discontinuities and the Saltus Decomposition. 3.6. BV [a; b] as a Normed Vector Space. 3.7. Exercises. 4. The Stieltjes Integral. 4.1. The Stieltjes Integral of Step Functions. 4.2. The Stieltjes Integral with Increasing Integrator. 4.3. The Stieltjes Integral with BV Integrator. 4.4. Existence of the Stieltjes Integral. 4.5. Limits and the Stieltjes Integral. 4.6. Exercises. 5. Further Properties of the Stieltjes Integral. 5.1. The Stieltjes Integral and Integration by Parts. 5.2. The Lebesgue-Stieltjes Criterion. 5.3. The Riemann-Stieltjes Integral. 5.4. The Riesz Representation Theorem. 5.5. Exercises.
Gregory Convertito is a Ph.D. candidate in philosophy at DePaul University in Chicago. He attended Trinity College in Hartford, CT as an undergraduate, where he studied both mathematics and philosophy. He then earned an M.A. in philosophy at Boston College. His primary interest is in social and political philosophy and he continues to have an abiding interest in mathematics.
David Cruz-Uribe, OFS is a Mexican-American mathematician, born and raised in Green Bay, Wisconsin. He attended the University of Chicago as an undergraduate, and got his PhD at the University of California, Berkeley, under Donald Sarason. After a postdoc at Purdue where he worked with Chris Neugebauer, he joined the faculty of Trinity College, Hartford, teaching there for 19 years. He then moved to the math department at the University of Alabama to become department chair; he wrote this book while serving in this position. His research interests are in harmonic analysis, particularly weighted norm inequalities and variable Lebesgue spaces, and partial differential equations. He is the author and translator of three books in harmonic analysis. He is married with three adult children, and is a professed member of the Secular Franciscan Order.