2024/02/26 by Benedikt Peterseim, Peterseim, Benedikt
Computer Science · Engineering · #Advanced Control Systems Optimization #Advanced Numerical Analysis Techniques #Category Theory (math.CT) #FOS: Mathematics #Functional Analysis (math.FA) #Rough Sets and Fuzzy Logic
paper · pdf · doi:10.48550/arxiv.2403.19681
openalex publication_date 2024/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.