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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.22503 |
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| _version_ | 1866917359163277312 |
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| author | Aguilera, Juan P. Kouptchinsky, Thibaut Yokoyama, Keita |
| author_facet | Aguilera, Juan P. Kouptchinsky, Thibaut Yokoyama, Keita |
| contents | A classical theorem of Lusin states that all analytic sets are Lebesgue-measurable. In this article we established the reverse mathematical strength of Lusin's theorem, which depends on how precisely it is formalized. By doing so, we answer to a question of Simpson. Our main proof is motivated towards proving a specific version of that result, namely that analytic sets are Lesbesgue-regular, which requires the equality of the outer and inner measures of the set in question. We prove this statement to be equivalent to $Σ^{1}_{1}$-$\mathrm{IND}$ over $\mathrm{ATR}_{0}$. The full statement of the theorem, that is the one implying the existence of the measure as a real number, is equivalent to $Π^{1}_{1}$-$\mathrm{CA}_{0}$, again provably over $\mathrm{ATR}_{0}$. In our main proof, we draw inspiration from Solovay's construction of a model of Zermelo-Fraenkel set theory where every set is Lebesgue measurable. In our case the argument requires the use of class forcing over a family of standard and non-standard models of a very weak set theory obtained through the method of pseudohierarchies. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_22503 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Reverse Mathematics of Analytic Measurability Aguilera, Juan P. Kouptchinsky, Thibaut Yokoyama, Keita Logic A classical theorem of Lusin states that all analytic sets are Lebesgue-measurable. In this article we established the reverse mathematical strength of Lusin's theorem, which depends on how precisely it is formalized. By doing so, we answer to a question of Simpson. Our main proof is motivated towards proving a specific version of that result, namely that analytic sets are Lesbesgue-regular, which requires the equality of the outer and inner measures of the set in question. We prove this statement to be equivalent to $Σ^{1}_{1}$-$\mathrm{IND}$ over $\mathrm{ATR}_{0}$. The full statement of the theorem, that is the one implying the existence of the measure as a real number, is equivalent to $Π^{1}_{1}$-$\mathrm{CA}_{0}$, again provably over $\mathrm{ATR}_{0}$. In our main proof, we draw inspiration from Solovay's construction of a model of Zermelo-Fraenkel set theory where every set is Lebesgue measurable. In our case the argument requires the use of class forcing over a family of standard and non-standard models of a very weak set theory obtained through the method of pseudohierarchies. |
| title | The Reverse Mathematics of Analytic Measurability |
| topic | Logic |
| url | https://arxiv.org/abs/2603.22503 |