A study of a Stieltjes integral defined on arbitrary number sets (Q2520039)

The authors study a generalized Stieltjes integral defined on a class of subsets of a closed real interval. This integral is compared with the partition-refinement Stieltjes integral. The main results are: I) if \(M \subseteq[a,b]\) 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 \(\int^b_af^*dg^*=\int_Mf\,dg\). II) Suppose that \(F\) and \(G\) are functions with domain including \([a,b]\) such that a) \(F\) is \(G\)-integrable on \([a,b]\) b) \(\overline M\subseteq [a,b]\), and \(a,b\in M\) c) if \(z\) belongs to \([a,b]-M\) and \(\varepsilon\) is a positive number, then there is an open interval \(s\) containing \(z\) such that \(|F(x)-Fz)|\,|G(v)-G(u)|<\varepsilon\) where each of \(u,v\) and \(x\) is in \(s\cap [a,b]\), \(u<z<v\), and \(u\leq x\leq v\). Then \(F\) is \(G\)-integrable on \(M\), and \(\int^b_aF\,dG=\int_MF\,dG\).