Chapter 1 Setting the Stage This chapter contains a mix of topics needed to study the remainder of this book. It has a little of metric space theory and some basics on normed vector spaces. Many readers will be familiar with some of these topics, but few will have seen them all. We start with a definition of the Riemann–Stieltjes integral, which I suspect is new to most. 1.1. Riemann–Stieltjes integrals For a fixed closed, bounded interval J = [a, b] in R, the set of all real numbers, we want to define an extension of the usual Riemann integral from calculus. This extended integral will also assign a number to each continuous function on the interval, though later we will see how to extend it even further so we can integrate more general functions than the continuous ones. This more general integral, however, will be set in a far broader context than intervals in R. As in calculus, a partition of J is a finite, ordered subset P = {a = x0 x1 · · · xn = b}. Say that P is a refinement of a partition Q if Q P so P adds additional points to Q. 1.1.1. Definition. A function α : J R is of bounded variation if there is a constant M such that for every partition P = {a = x0 · · · xn = b} of J, n j=1 |α(xj) α(xj−1)| M. 1
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