  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.
