Kurzweil--Stieltjes integration on compact lines

TL;DR

This paper develops a Kurzweil--Stieltjes integral on compact lines, generalizing Lebesgue integration and establishing a Fundamental Theorem of Calculus in this setting.

Contribution

It introduces a new integral on compact lines, proves its key properties, and connects it with derivation to extend classical calculus results.

Findings

The integral generalizes Lebesgue measure-based integration.

Convergence theorems are established for regular integrators.

A version of the Fundamental Theorem of Calculus is formulated on compact lines.

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.