Properties of a generalized Stieltjes integral defined on dense subsets of an interval (Q1308840)

Denote by \(\Delta\) the class of all sets \(M \subseteq [a,b]\) which are dense in \([a,b]\) and contain the endpoints \(a,b\). The author and \textit{J. F. Vance} [Riv. Mat. Univ. Parma, III. Ser. 1, 73-78 (1972; Zbl 0302.26007)] have introduced the concept of integral \(\int_ M f dg\) of Stieltjes type on sets \(M \in \Delta\). This paper is a contribution to the theory of this integral. It is shown here for instance that if \(\int_ M f dg\) exists and \(M\) is uncountable, then there is an uncountable class \(\Delta_ 1 \subset \Delta\) such that \(\int_ H f dg\) exists for each \(H \in \Delta_ 1\). Further, if \(\int_ M f dg\) exists and \(M\) is countable, then it is possible to choose two functions \(f,g\) such that the integral \(\int_ H f dg\) does not exist for any \(H \in \Delta\), \(H \neq M\).