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Algebraic theory of vector-valued integration
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.Comment: shortened, e.g. by citing references regarding basic lemmas; made
changes to ordering of some lemmas and section