On Monadic Vector-Valued Integration
Abstract
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.
Cite
@article{arxiv.2403.19681,
title = {On Monadic Vector-Valued Integration},
author = {Benedikt Peterseim},
journal= {arXiv preprint arXiv:2403.19681},
year = {2024}
}