On the validity of the Radon-Nikodym Theorem
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866909939286409216 |
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| author | Roselli, Paolo Willem, Michel |
| author_facet | Roselli, Paolo Willem, Michel |
| contents | This paper presents a new general formulation of the Radon-Nikodym theorem in the setting of abstract measure theory. We introduce the notion of weak localizability for a measure and show that this property is both necessary and sufficient for the validity of a Radon-Nikodym-type representation under a natural compatibility relation between measures. The proof relies solely on elementary tools, such as Markov's inequality and the monotone convergence theorem. In addition to establishing the main result, we provide a constructive approach to envelope functions for families of non-negative measurable functions supported on sets of finite measure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_12023 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the validity of the Radon-Nikodym Theorem Roselli, Paolo Willem, Michel General Mathematics This paper presents a new general formulation of the Radon-Nikodym theorem in the setting of abstract measure theory. We introduce the notion of weak localizability for a measure and show that this property is both necessary and sufficient for the validity of a Radon-Nikodym-type representation under a natural compatibility relation between measures. The proof relies solely on elementary tools, such as Markov's inequality and the monotone convergence theorem. In addition to establishing the main result, we provide a constructive approach to envelope functions for families of non-negative measurable functions supported on sets of finite measure. |
| title | On the validity of the Radon-Nikodym Theorem |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2506.12023 |