\(D^{\#}\) derivation basis and the Lebesgue-Stieltjes integral (Q1113303)

The following question was raised by Thomson: The basis \(D^{\#}\) can be used to characterize the Lebesgue integral. The corresponding Stieltjes integral \(D^{\#}-\int f(x)dg(x)\) seems not to have been investigated, apart from several remarks in McShane, which asserts that if the \(D^{\#}\)-Stieltjes integral exists for continuous functions with respect to a function g, then g must be of bounded variation. In this paper, it is shown that the \(D^{\#}-\int f(x)dg(x)\) is the Lebesgue- Stieltjes integral for functions g of bounded variation. Recall that the sharp derivation basis, essentially introduced by McShane, is defined by \(D^{\#}=\{\beta_{\delta}^{\#}:\quad \delta \quad is\quad a\quad positive\quad function\quad on\quad R\}\) and \(\beta_{\delta}^{\#}=\{(I,x):\) I is an interval in R, \(x\in R\), and I belongs to \((x-\delta (x),x+\delta (x))\},\) and that the link between those derivation bases and the construction of a Kurzweil-Henstock type integral has been developed by Thomsom.