Perron-Stieltjes integrability with respect to gap functions (Q1075449)

Let \(Q=\{q_ n\}\) be a countable subset of ]a,b[, and for all n, \(\alpha (q_ n)=a_ n\geq 0\) with \(\sum a_ n<\infty:\) then \(\alpha: [a,b]\to {\mathbb{R}}_+\) is called a gap function (determined by Q and \(\{a_ n\})\) if \(\alpha (x)=\sum_{q_ n<x}a_ n,\) \(x\in Q\), and if \(x\not\in Q\) then \(\alpha (x)=\alpha\), for same \(\alpha\), \(\sum_{q_ n<x}a_ n\leq \alpha \leq \sum_{q_ n\leq x}a_ n.\) The author proves that f: [a,b]\(\to {\mathbb{R}}\) is Perron-Stieltjes integrable with respect to \(\alpha\), Q having a finite derived set, iff every monotone sequence in Q is a subsequence of some monotone sequence in Q, \(\{q_ n'\}\) say for which \(\sum f(q_ n')\alpha (q_ n')<\infty.\) An example is given of a Q, with countably many limit points, and associated \(\alpha\), and an f that is not Perron-Stieltjes integrable with respect to this \(\alpha\), but for which the above property holds. The methods are those of the Riemann theory of Henstock-Kurzweil.