Integrable and absolutely continuous vector-valued functions

Abstract

The theory of integration for functions with values in a topological vector space is a bit of a messy subject. There are many different choices for the integral, most distinct from one another in general, and with no compelling reason to always adopt one definition or another as the “right” one. No attempt is made here to resolve this question of which integral is the “right” integral, but we overview a few of the common notions and prove a couple of useful properties for the notion of integrability by seminorm. For this notion of integrability, the completeness of the space of integrable functions with values in a complete locally convex space is established. This space is then easily seen to be isomorphic to the completion of the projective tensor product of the usual ${\text{L}}^1$ space of scalar functions with the vector space. Absolute continuity for this notion of integrability is also considered. Here, it is shown that the familiar properties of differentiation for absolutely continuous scalar functions holds in the vector-valued case.