Algebraic theory of vector-valued integration

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.

shortened, e.g. by citing references regarding basic lemmas; made changes to ordering of some lemmas and sections

References in corpus (2)

• Monads and extensive quantities
• Calculus of extensive quantities

Cited by in corpus (4)

• Codensity and the ultrafilter monad
• Codensity and the Giry monad
• Riesz-Schwartz extensive quantities and vector-valued integration in closed categories
• A general Fubini theorem for the Riesz paradigm