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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2403.19681 |
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| _version_ | 1866914732826427392 |
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| author | Peterseim, Benedikt |
| author_facet | Peterseim, Benedikt |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_19681 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Monadic Vector-Valued Integration Peterseim, Benedikt Category Theory Functional Analysis 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. |
| title | On Monadic Vector-Valued Integration |
| topic | Category Theory Functional Analysis |
| url | https://arxiv.org/abs/2403.19681 |