On Stieltjes integration in Euclidean space

Abstract

As is well known, Riemann integrability in Euclidean space is characterized by continuity almost everywhere, with respect to Lebesgue measure. The main purpose of this paper is to generalize this classical theorem to Stieltjes integrals in Euclidean space. This paper is the sequel to an earlier paper [2], which dealt with characterizations of Riemann-Stieltjes integrability on the real line. Let us first briefly review the results of the previous paper [Z]. The above-mentioned classical theorem shows that with the Riemann integral, the (bounded) function to be integrated may be discontinuous at any point in the domain of integration, as long as the Lebesgue measure of the set of all points of discontinuity does not exceed zero. With the usual definitions of the Riemann-Stieltjes integral on the real line, the situation is different: if the integrator function has discontinuities, i.e., if there are points of nonzero Lebesgue-Stieltjes measure, then the integrand has to satisfy certain continuity conditions at these particular points (see [2, Theorems B and Cl). These problems of simultaneous discontinuity do not appear when one uses either the definition of the Riemann-Stieltjes integral given by Ross [3], or the definition given in [2, Sect. 61. The latter definition is stated in terms of a “premeasure, ” i.e., a certain restriction of a Lebesgue-Stieltjes measure on the real line. For these definitions, Riemann-Stieltjes integrability is equivalent to “continuity p-almost everywhere on the set of points of p-measure zero,” where p is the associated Lebesgue-Stieltjes measure (see [2, Theorems D and E] ). The definition of the Riemann-Stieltjes integral proposed in the previous paper [2] can also be applied to situations other than the real line. In this paper, the extension to n-dimensional Euclidean space (Rn) will be treated. In Section 2, a premeasure will be defined for [w”. In terms of this premeasure, the definition of the Riemann-Stieltjes integral given in [2] can be applied to [w”. 57 0022-247X/86 $3.00