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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.
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.
About the Author
The Darboux Integral
Bounded, Continuous, and Monotonic Functions
Step Functions
The Darboux Integral
Properties of the Darboux Integral
Limits and the Integral
The Fundamental Theorem of Calculus
Exercises
Further Properties of the Integral
The Lebesgue Criterion
The Riemann Integral
Integrable Functions as a Normed Vector Space
Exercises
Functions of Bounded Variation
Monotonic Functions
Functions of Bounded Variation
Properties of Functions of Bounded Variation
Limits and Bounded Variation
Discontinuities and the Saltus Decomposition
BV[a, b] as a Normed Vector Space
Exercises
The Stieltjes Integral
The Stieltjes Integral of Step Functions
The Stieltjes Integral with Increasing Integrator
The Stieltjes Integral with BV Integrator
Existence of the Stieltjes Integral
Limits and the Stieltjes Integral
Exercises
Further Properties of the Stieltjes Integral
The Stieltjes Integral and Integration by Parts
The Lebesgue-Stieltjes Criterion
The Riemann-Stieltjes Integral
The Riesz Representation Theorem
Exercises
Index of Symbols