Measure theory via Locales
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915569054253056 |
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| author | Lehner, Georg |
| author_facet | Lehner, Georg |
| contents | We present an approach to measure theory using the theory of locales. This includes concrete constructions of measure algebras associated to Radon measures, such as the Lebesgue measure on $\mathbb{R}^n$, via Grothendieck topologies constructed from valuations, that circumvent the classical approach via $σ$-algebras. As an application we obtain a functorial construction of the induced measure $μ_*$ on the locale of sublocales $\mathfrak{Sl}(X)$ of a Hausdorff space $X$ equipped with a Radon measure $μ$, which in particular shows that $μ_*$ is invariant under measure-preserving homeomorphisms. We furthermore give a construction of the measurable locale associated to a smooth manifold, functorial in submersions, as well as comparison results to classical measure theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_08826 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Measure theory via Locales Lehner, Georg General Topology Category Theory Logic Primary: 28A60, Secondary: 28C15, 18F10, 18F70 We present an approach to measure theory using the theory of locales. This includes concrete constructions of measure algebras associated to Radon measures, such as the Lebesgue measure on $\mathbb{R}^n$, via Grothendieck topologies constructed from valuations, that circumvent the classical approach via $σ$-algebras. As an application we obtain a functorial construction of the induced measure $μ_*$ on the locale of sublocales $\mathfrak{Sl}(X)$ of a Hausdorff space $X$ equipped with a Radon measure $μ$, which in particular shows that $μ_*$ is invariant under measure-preserving homeomorphisms. We furthermore give a construction of the measurable locale associated to a smooth manifold, functorial in submersions, as well as comparison results to classical measure theory. |
| title | Measure theory via Locales |
| topic | General Topology Category Theory Logic Primary: 28A60, Secondary: 28C15, 18F10, 18F70 |
| url | https://arxiv.org/abs/2510.08826 |