Functions of Bounded Variation and the Riemann–Stieltjes Integral
In the chapters ahead, we will study the Lebesgue integral. In this chapter, we introduce the Riemann-Stieltjes integral and, as a natural preliminary step, study functions of bounded variation. The justification for doing so is that Lebesgue integration is intimately connected with Riemann-Stieltjes integration, although this is not apparent from the definitions. We shall see in Theorem5.43 that Lebesgue integrals can be represented as Riemann-Stieltjes integrals.