Functions of Bounded Variation and Riemann–Stieltjes Integrals

Abstract

In Section 11.1, we introduce a special class of functions, namely, functions of bounded variation. In Section 11.1, we shall also discuss several nice properties of functions in the class BV ([a, b]) of functions of bounded variation on [a, b]. Monotone functions on [a, b] have nice properties. For example, they are integrable on [a, b] and have only a countable number of jump discontinuities. In this section, we shall also show that every monotone function is a function of bounded variation, and hence the class BV ([a, b]) contains the class of monotone functions on [a, b]. We shall show that increasing functions are in some sense the only functions of bounded variation. More precisely (see Theorem 11.19), every function of bounded variation is the difference of two increasing functions. As an application of functions of bounded variation, in Section 11.2 we shall consider important generalizations of the Darboux and Riemann integrals called the Darboux–Stieltjes and Riemann–Stieltjes integrals. The theory of Stieltjes integrals is almost identical to that of Riemann integrals, except that the notion of length of an integral is replaced by a more general concept of α-length. Stieltjes integrals are particularly useful in probability theory.