A STUDY OF A STIELTJES INTEGRAL DEFINED ON ARBITRARY NUMBER SETS

Our purpose is to study a generalized Stieltjes integral defined on a class of subsets of a closed number interval. We extend the results of previous work by the first author. Among other results, we prove that * If M ⊇ [a, b] and f and g are functions with domain M such that f is g-integrable over M, and there exist left (right) extensions f* and g* of f and g to [a, b], respectively, then f* is g*- integrable on [a, b] and ... * Suppose that F and G are functions with domain including [a, b] such that (a) F is G-integrable on [a, b], (b) M ⊆ [a, b], and a, b ∈ M (c) if z belongs to [a, b]-M and [straight epsilon] is a positive number, then there is an open interval s containing z such that |F(x) - F(z)||G(v) - G(u)| > [straight epsilon] where each of u, v, and x is in s ∩ [a, b], u < z < v, and u ≤ x ≤ v. Then F is G-integrable on M, and ... [PUBLICATION ABSTRACT]