  We explore the assignment of norms to $\Lambda$-modules over a
finite-dimensional algebra $\Lambda$, resulting in the establishment of normed
$\Lambda$-modules. Our primary contribution lies in constructing a new category
$\mathscr{N}\!\!or^p$ related to normed modules along with its full subcategory
$\mathscr{A}^p$. By examining the objects and morphisms in these categories, we
establish a framework for understanding the categorization of Lebesgue
integration.
