Algebraic theory of vector-valued integration

TL;DR

The paper develops a monadic framework for vector-valued integration in measurable bornological sets, connecting it to Pettis integrals and characterizing Banach spaces as algebras of this monad.

Contribution

It introduces a monad-based theory of vector-valued integration related to Pettis integrals, providing a new algebraic perspective on Banach spaces.

Findings

Banach spaces with Pettis integrals are M-algebras

Separable and reflexive Banach spaces are M-algebras

The monad framework unifies vector-valued integration concepts

Abstract

We define a monad M on a category of measurable bornological sets, and we show how this monad gives rise to a theory of vector-valued integration that is related to the notion of Pettis integral. We show that an algebra X of this monad is a bornological locally convex vector space endowed with operations which associate vectors \int f dm in X to incoming maps f:T --> X and measures m on T. We prove that a Banach space is an M-algebra as soon as it has a Pettis integral for each incoming bounded weakly-measurable function. It follows that all separable Banach spaces, and all reflexive Banach spaces, are M-algebras.