On measurability of Kurzweil--Stieltjes integrable functions on compact lines
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866910159023898624 |
|---|---|
| author | Candido, Leandro Kaufmann, Pedro L. |
| author_facet | Candido, Leandro Kaufmann, Pedro L. |
| contents | 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 $μ_G$ 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 $μ_G$-measurable, where $μ_G$ 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 $μ_G$-measurable. %, where $|μ_G|$ denotes the total variation measure of $μ_G$. 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 $μ_G$ for suitable $G$.
In addition, we establish a version of Hake's theorem for the $G$-integral in this setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_21141 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On measurability of Kurzweil--Stieltjes integrable functions on compact lines Candido, Leandro Kaufmann, Pedro L. Functional Analysis 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 $μ_G$ 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 $μ_G$-measurable, where $μ_G$ 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 $μ_G$-measurable. %, where $|μ_G|$ denotes the total variation measure of $μ_G$. 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 $μ_G$ for suitable $G$. In addition, we establish a version of Hake's theorem for the $G$-integral in this setting. |
| title | On measurability of Kurzweil--Stieltjes integrable functions on compact lines |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2604.21141 |