A categorical derivation of Lebesgue integration

TL;DR

This paper provides a universal property-based characterization of Lebesgue $L^p$ spaces, deriving integration theory from categorical principles and establishing unique, universal properties for various function spaces.

Contribution

It introduces universal properties that uniquely characterize Lebesgue $L^p$ spaces and related sequence spaces, offering a categorical foundation for integration theory.

Findings

Universal properties characterize $L^p$ spaces uniquely.

Categorical approach derives fundamental elements of integration.

Universal properties also characterize sequence spaces like $\,ell^p$ and $c_0$.

Abstract

We identify simple universal properties that uniquely characterize the Lebesgue $L^p$ spaces. There are two main theorems. The first states that the Banach space $L^p[0, 1]$, equipped with a small amount of extra structure, is initial as such. The second states that the $L^p$ functor on finite measure spaces, again with some extra structure, is also initial as such. In both cases, the universal characterization of the integrable functions produces a unique characterization of integration. Using the universal properties, we develop some of the basic elements of integration theory. We also state universal properties characterizing the sequence spaces $\ell^p$ and $c_0$, as well as the functor $L^2$ taking values in Hilbert spaces.