  In recent times, there has been a growing interest in a structuralist
understanding of probability, measure and integration theory. The present
thesis contributes to this programme in three ways. First, we construct a
commutative probability monad on the cartesian closed category of hk-spaces
(also known as CGWH spaces, or weak Hausdorff k-spaces in the literature).
Secondly, in order to achieve this in a seamless way, we develop the theory of
paired linear hk-spaces, a functional-analytic category tailored to the duality
between measures and functionals. Finally, vector-valued integration emerges
naturally from the free-forgetful adjunction between paired linear hk-spaces
and hk-spaces, inducing a commutative monad of compactly supported measures and
leading to a theory of monadic vector-valued integration.
