Normed modules and The Stieltjes integrations of functions defined on finite-dimensional algebras
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866916269183205376 |
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| author | Gao, Hanpeng Liu, Shengda Liu, Yu-Zhe Wang, Yucheng |
| author_facet | Gao, Hanpeng Liu, Shengda Liu, Yu-Zhe Wang, Yucheng |
| contents | We define integrals for functions on finite-dimensional algebras, adapting methods from Leinster's research. This paper discusses the relationships between the integrals of functions defined on subsets $\mathbb{I}_1 \subseteq {\mathitΛ}_1$ and $\mathbb{I}_2 \subseteq {\mathitΛ}_2$ of two finite-dimensional algebras, under the influence of a mapping $ω$, which can be an injection or a bijection. We explore four specific cases:
$\bullet$ $ω$ as a monotone non-decreasing and right-continuous function;
$\bullet$ $ω$ as an injective, absolutely continuous function;
$\bullet$ $ω$ as a bijection;
$\bullet$ and $ω$ as the identity on $\mathbb{R}$.
These scenarios correspond to the frameworks of Lebesgue-Stieltjes integration, Riemann-Stieltjes integration, substitution rules for Lebesgue integrals, and traditional Lebesgue or Riemann integration, respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_00161 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Normed modules and The Stieltjes integrations of functions defined on finite-dimensional algebras Gao, Hanpeng Liu, Shengda Liu, Yu-Zhe Wang, Yucheng Classical Analysis and ODEs Category Theory 16G10, 46B99, 46M40 We define integrals for functions on finite-dimensional algebras, adapting methods from Leinster's research. This paper discusses the relationships between the integrals of functions defined on subsets $\mathbb{I}_1 \subseteq {\mathitΛ}_1$ and $\mathbb{I}_2 \subseteq {\mathitΛ}_2$ of two finite-dimensional algebras, under the influence of a mapping $ω$, which can be an injection or a bijection. We explore four specific cases: $\bullet$ $ω$ as a monotone non-decreasing and right-continuous function; $\bullet$ $ω$ as an injective, absolutely continuous function; $\bullet$ $ω$ as a bijection; $\bullet$ and $ω$ as the identity on $\mathbb{R}$. These scenarios correspond to the frameworks of Lebesgue-Stieltjes integration, Riemann-Stieltjes integration, substitution rules for Lebesgue integrals, and traditional Lebesgue or Riemann integration, respectively. |
| title | Normed modules and The Stieltjes integrations of functions defined on finite-dimensional algebras |
| topic | Classical Analysis and ODEs Category Theory 16G10, 46B99, 46M40 |
| url | https://arxiv.org/abs/2406.00161 |