1. Normed modules and The Stieltjes integrations of functions defined on finite-dimensional algebras
- Author
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Gao, Hanpeng, Liu, Shengda, Liu, Yu-Zhe, and Wang, Yucheng
- Subjects
Mathematics - Classical Analysis and ODEs ,Mathematics - Category Theory ,16G10, 46B99, 46M40 - Abstract
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 $\mathbb{I}_1 \subseteq {\mathit{\Lambda}}_1$ and $\mathbb{I}_2 \subseteq {\mathit{\Lambda}}_2$ of two finite-dimensional algebras, under the influence of a mapping $\omega$, which can be an injection or a bijection. We explore four specific cases: $\bullet$ $\omega$ as a monotone non-decreasing and right-continuous function; $\bullet$ $\omega$ as an injective, absolutely continuous function; $\bullet$ $\omega$ as a bijection; $\bullet$ and $\omega$ 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., Comment: 23 pages
- Published
- 2024