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 I₁ ⊆ Λ₁ and I₂ ⊆ Λ₂ 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: • ω as a monotone non-decreasing and right-continuous function; • ω as an injective, absolutely continuous function; • ω as a bijection; • and ω as the identity on 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.