Kurzweil--Stieltjes integration on compact lines
Leandro Candido, Pedro L. Kaufmann
TL;DR
This work builds a Kurzweil–Stieltjes integration theory on compact lines, defining $G$-integrals via gauges and tagged partitions and proving key properties (linearity, additivity, convergence) and a Saks–Henstock-type lemma. By restricting to integrators of bounded variation (NBV), the theory aligns with Radon measures, yielding an isometric identification NBV$(K)\cong\mathcal{M}(K)$ and a direct link $\,\int f\,dG=\int f\,d\mu_G$ for continuous $f$. The framework supports $G$-differentiability and a Fundamental Theorem of Calculus, with higher regularity (amenable, NBV, positive nondecreasing) enabling $G$-null sets, $G$-differentiation, and standard convergence theorems (Monotone, Dominated, Fatou). These results generalize Lebesgue integration with respect to Radon measures on compact lines and offer a robust calculus for nonabsolute integration in ordered spaces, with broad implications for analysis on time scales and related nonclassical domains.
Abstract
We develop a version of the Kurzweil--Stieltjes integral on compact lines and establish its fundamental properties. For sufficiently regular integrators, we obtain convergence theorems and show that the presented integration process generalizes Lebesgue integration with respect to positive Radon measures. Additionally, we introduce a notion of derivation on compact lines which, when paired with the proposed integral, yields a formulation of the Fundamental Theorem of Calculus in this context.