Henstock-Stieltjes integrals not induced by measure (Q1852422)

The authors define a \(\;\delta^k\)-fine division of \([a,b]\) as \(\{[x_i,x_{i+k}], \xi_i\}_{0\leq i\leq n-k}\), where \(a=x_0<\cdots<x_n=b\) and \( \xi\in[x_i,x_{i+k}]\subset]\xi_i - \delta(\xi_i),\xi_i+ \delta(\xi_i)[, 0\leq i\leq n-k\). Then if \(g:[a,b]^{k+1}\mapsto \mathbb R, f:[a,b]\mapsto \mathbb R\) the \(GR_k\) integral of \(f\) with respect to \(g\) is defined in the usual way using as Riemann sums \(\sum_0^{n-1}f (\xi_i)g(x_i,\ldots,x_{i+k})\). This integral includes, when \(k=1\), the Henstock-Stieltjes integral, and for all \(k\) the \(RS_k^*\) integral of \textit{A. M. Russell} [J. Aust. Math. Soc., Ser. A 20, 431-448 (1975; Zbl 0313.26012)], as special cases. The standard properties are proved, including an integration by parts formula in the case \(k=2\).