Generalized Baire class functions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911373020102656 |
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| author | Ros, Luca Motto Pitton, Beatrice |
| author_facet | Ros, Luca Motto Pitton, Beatrice |
| contents | Let $λ$ be an uncountable cardinal such that $2^{< λ} = λ$. Working in the setup of generalized descriptive set theory, we study the structure of $λ^+$-Borel measurable functions with respect to various kinds of limits, and isolate a suitable notion of $λ$-Baire class $ξ$ function. Among other results, we provide higher analogues of two classical theorems of Lebesgue, Hausdorff, and Banach, namely:
(1) A function is $λ^+$-Borel measurable if and only if it can be obtained from continuous functions by iteratively applying pointwise $D$-limits, where $D$ varies among directed sets of size at most $λ$.
(2) A function is of $λ$-Baire class $ξ$ if and only if it is $\boldsymbolΣ^{0}_{ξ+1}$-measurable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17650 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized Baire class functions Ros, Luca Motto Pitton, Beatrice Logic Primary: 03E15, Secondary: 54E99 Let $λ$ be an uncountable cardinal such that $2^{< λ} = λ$. Working in the setup of generalized descriptive set theory, we study the structure of $λ^+$-Borel measurable functions with respect to various kinds of limits, and isolate a suitable notion of $λ$-Baire class $ξ$ function. Among other results, we provide higher analogues of two classical theorems of Lebesgue, Hausdorff, and Banach, namely: (1) A function is $λ^+$-Borel measurable if and only if it can be obtained from continuous functions by iteratively applying pointwise $D$-limits, where $D$ varies among directed sets of size at most $λ$. (2) A function is of $λ$-Baire class $ξ$ if and only if it is $\boldsymbolΣ^{0}_{ξ+1}$-measurable. |
| title | Generalized Baire class functions |
| topic | Logic Primary: 03E15, Secondary: 54E99 |
| url | https://arxiv.org/abs/2411.17650 |