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.
