Normed modules and the categorization of Lebesgue integration

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