Normed modules and The Stieltjes integrations of functions defined on finite-dimensional algebras

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 ${\mathbb{I}}_1\subseteq {\Lambda }_1$ and ${\mathbb{I}}_2\subseteq {\Lambda }_2$ 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.