Title: Normed modules and The Stieltjes integrations of functions defined on finite-dimensional algebras Show Chinese title

Title: 赋范模和定义在有限维代数上的Stieltjes积分

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. Show Chinese abstract

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. 我们探讨四个具体情形： $\bullet$ $\omega$ 作为单调不减且右连续的函数； $\bullet$ $\omega$ 作为单射、绝对连续的函数； $\bullet$ $\omega$ 作为双射； $\bullet$和$\omega$作为$\mathbb{R}$上的恒等映射。 这些情景分别对应于勒贝格-斯特尔jes积分框架、黎曼-斯特尔jes积分框架、勒贝格积分的替换规则以及传统的勒贝格或黎曼积分。