On measurability of Kurzweil--Stieltjes integrable functions on compact lines

TL;DR

This paper investigates the conditions under which functions are measurable with respect to the Kurzweil--Stieltjes integral on compact lines, extending classical measure theory results.

Contribution

It establishes the relationship between G-integrability and measurability for functions on compact lines, generalizing Lebesgue integration via the G-integral.

Findings

Every G-integrable function is μ_G-measurable when G is nondecreasing.

Bounded G-integrable functions are μ_G-measurable for any G of bounded variation.

The G-integral extends the Lebesgue integral and characterizes Lebesgue integrability via Radon measures.

Abstract

We continue the study on Kurzweil--Stieltjes integration on compact lines initiated in [doi:10.1007/s11117-025-01161-9]. Given a real valued function $G$ on a compact line, the presented integral is called the Kurzweil--Stieltjes integral with respect to $G$, or simply the $G$-integral. %Given a compact line $K$ and a right-continuous function $G:K\to\mathbb{R}$ of bounded variation, we consider the Radon measure $\mu_G$ naturally induced by $G$. Our main results concern the relationship between $G$-integrability and measurability. We prove that, whenever $G$ is nondecreasing, every $G$-integrable function is $\mu_G$-measurable, where $\mu_G$ is the natural Radon measure induced by $G$. We also show that, for an arbitrary $G$ of bounded variation, every bounded $G$-integrable function is $\mu_G$-measurable. %, where $|\mu_G|$ denotes the total variation measure of $\mu_G$. As an application, we provide a full characterization of Lebesgue integrablility with respect to Radon measures in terms of the $G$-integral, and demonstrate that the $G$-integral represents an extension of the Lebesgue integral with respect to $\mu_G$ for suitable $G$. In addition, we establish a version of Hake's theorem for the $G$-integral in this setting.