Kurzweil–Stieltjes Integral

The following sections are included:

• Preface
• Contents
• Conventions and notation

Stieltjes integrals were originally introduced in order to describe the moments for a given distribution of mass. However, there exist many other problems in mathematics and physics that lead to Stieltjes integrals. Some of them are treated in the present chapter. The exposition here is rather informal; precise statements of definitions and theorems will be given in Chapters 5 and 6.

The concept of a function of bounded variation was originally discovered in the context of Fourier series, but it turned out to be useful in other areas of mathematics, including integration theory and differential equations. A major part of this chapter deals with the variation of a real function defined on a compact interval. We establish some basic properties of bounded variation functions, such as the decomposition into a difference of two monotone functions, approximation by finite step functions, or decomposition into continuous and jump parts. The final section is concerned with the notion of variation over the so-called elementary sets, i.e., finite unions of bounded (not necessarily closed) intervals.

Absolutely continuous functions represent an important special case of continuous functions with bounded variation. Here we summarize their basic properties and recall their importance in integration theory.

Functions of bounded variation play a crucial role in the Stieltjes integration theory. Of similar importance is the class of regulated functions, which are functions whose one-sided limits at all points exist and are finite. Thus, regulated functions represent a very natural generalization of both continuous functions and functions of bounded variation. The present chapter provides an overview of the most important facts about regulated functions…

This chapter focuses on the simplest Stieltjes-type integral – the Riemann-Stieltjes integral, which makes it possible to deal with the problems mentioned in Chapter 1. It is well known that there are three equivalent definitions of the classical Riemann integral: the first is based on partitions whose norm approaches zero (Riemann’s original definition), the second involves refinements of partitions, and the third rests on upper and lower sums (Darboux's definition). The situation becomes more complicated when we consider Stieltjes integrals. The analogues of the first two definitions are no longer equivalent, and the third makes sense only if the integrator is nondecreasing. The goal of this chapter is to clarify the relations between the three definitions, to obtain necessary and sufficient conditions for the existence of the integral ${\displaystyle {\int }_a^{b}fdg}$, and to establish some basic results including integration by parts, substitution, and various convergence theorems.

The following sections are included:

• Introduction
• Definition and basic properties
• Existence of the integral
• Integration by parts
• The indefinite integral
• Substitution
• Absolute integrability
• Convergence theorems
• Integration over elementary sets
• Integrals of vector, matrix and complex functions
• Relation to the Perron-Stieltjes integral
• Relation to the Lebesgue-Stieltjes integral
• Relation to other Stieltjes-type integrals

The following sections are included:

• Introduction
• Differential equations with impulses
• Linear operators
• Existence and uniqueness of solutions
• A priori estimates of solutions
• Continuous dependence of solutions on parameters
• Fundamental matrices
• Variation of constants formula

This chapter presents several applications of the Riemann-Stieltjes and Kurzweil-Stieltjes integrals. We begin with topics related to functional analysis, such as the representation of continuous linear functionals, or the theory of distributions. Then we return to generalized linear differential equations and show how they lead to a generalization of the classical elementary functions. The final part of the chapter is devoted to the relation between Kurzweil-Stieltjes integrals, time scale integrals, and dynamic equations on time scales.

The following sections are included:

• Bibliography
• Subject index
• Symbol index