Algebraic theory of vector-valued integration

Details Algebraic theory of vector-valued integration Algebraic theory of vector-valued integration

2011-08-15

arxiv.org/abs/1108.2913v3

Mathematics

Category Theory

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

Scientific paper

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.

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