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By Ole A. Nielsen

This publication describes integration and degree idea for readers drawn to research, engineering, and economics. It provides a scientific account of Riemann-Stieltjes integration and deduces the Lebesgue-Stieltjes degree from the Lebesgue-Stieltjes quintessential.

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3 interval [a,b]. Suppose that m is a nondecreasing function on a closed (a) A bounded function / defined on [a, 6] is said to be Riemann-Stieltjes integrable over [a,i>] with respect to u if suppS(/,w;P) = infp5(/,w;P), where the infimum and the supremum are taken over all partitions P of [a, b], and in this case the common value of this supremum and infimum is called the Riemann-Stieltjes integral of f over [a, fc] with respect to u and is denoted by either / du or f{x)du{x). (b) The set of all those bounded functions on [a,b] which are RiemannStieltjes integrable over [a, b'] with respect to u will be denoted by J^{u; a, b).

Before going on to discuss the calculation of expectations, it will be helpful to consider a more general problem. This problem is, in fact, only slightly more complicated, but its solution will turn out to be very illuminating. Namely, suppose that /2 is a real-valued function defined on U and consider the composi­ tion h<^X; this is, of course, a real-valued function defined on Q— that is, a random variable on Q— and the more general problem is that of finding an expression for the expectation Elh^X'] of h^X in terms of h and F.

E) Show that if u is strictly increasing, if infpS(/;P) = 0, where the infimum is over all partitions P of [a, b], and if ¡ ^ fd u = 0, then every open subinterval of la, ft] contains a point x with /(x) = 0. iSee U. V- Satyanarayana, A note on Riemann-Stieltjes integrals. Am. Math. Monthly, 87 (1980), 477-478. Chapter Four Characterization o f Riemann—Stieltjes Integrability The discussion in the previous chapter left open the question of which bounded functions are Riemann-Stieltjes integrable with respect to a given nondecreas­ ing function over a given closed interval.

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