A new proof of the Lebesgue-Stieltjes representation theorem for cumulative distribution functions

Abstract

and lim f(x)= 0.) Conversely, if we begin with a cumulative distribution X--¢. -- CO function f, then the Lebesgue-Stieltjes representation theorem states that there is a unique probability measure v on R such that f(x) = v(- oo, x] for each x in R. (See [6], Section 12.3.) The main purpose of this article is to present a new proof of this result. Our proof depends on the Krein-Milman theorem and on a result (Theorem 2.5) of independent interest. Theorem 2.5 improves an integral representation theorem for monotone functions originally due to Choquet by giving a necessary and sufficient condition for the uniqueness of the representing measure. A sketch of the contents of this article follows. The first and second sections contain background material. We begin in Section 1 by recalling the statement of the integral foml of the Krein-Milman theorem. Section 2 deals with an application of that result. Specifically, if A denotes a partially ordered set with a largest element, we consider the family M(A) of all normalized, nondecreasing, and nonnegative real-valued functions on A. The Krein-Milman theorem is then applied to show that functions in M(A) can be represented by probability measures on F(A), where F(A) denotes the space of all nonempty final sets in A, equipped with a suitable compact Hausdorff topology. This representation is shown to be unique if and only ifA is a totally ordered set. In the third and final section we apply this representation theorem to obtain our new proof of the Lebesgue-Stieltjes theorem. We accomplish this by choosing A to be (~, where ~) denotes the totally ordered set of rational numbers together with a largest element + ~. If f is a cumulative distribution function and g denotes the restriction of f to Q, then the (unique) probability measure which represents g on F((~) is restricted to a copy of R to obtain the desired (unique) probability measure v.