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arXiv · 2403.19681

On Monadic Vector-Valued Integration

Benedikt Peterseim
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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.

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Benedikt Peterseim. 2024-02-26. On Monadic Vector-Valued Integration. https://arxiv.org/abs/2403.19681

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