Algebrability and Riemann integrability of the composite function
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866907811501309952 |
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| author | D'Aniello, E. Fernández-Sánchez, J. Maiuriello, M. Sepúlveda, J. B. Seoane |
| author_facet | D'Aniello, E. Fernández-Sánchez, J. Maiuriello, M. Sepúlveda, J. B. Seoane |
| contents | In this note we show that there exist a $2^\mathfrak{c}$-generated free algebra $\mathcal{S} \subset \mathbb{R}^\mathbb{R}$ of Riemann integrable functions and a free algebra $\mathcal{C} \subset \mathbb{R}^{[0,1]}$ of continuous functions, having $\mathfrak{c}$-generators, such that $r \circ c$ is not Riemann integrable for any $r \in \mathcal{S}$ and $c \in \mathcal{C}$. This result is the best possible one in terms of lineability within these families of functions and, at the same time, an improvement of a precious result (\cite[Theorem 2.7]{A}). In order to achieve our results we shall employ set theoretical tools such as the Fichtenholz-Kantorovich-Hausdorff theorem, Cantor-Smith-Volterra--type sets, and classical real analysis techniques. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_16545 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Algebrability and Riemann integrability of the composite function D'Aniello, E. Fernández-Sánchez, J. Maiuriello, M. Sepúlveda, J. B. Seoane Functional Analysis In this note we show that there exist a $2^\mathfrak{c}$-generated free algebra $\mathcal{S} \subset \mathbb{R}^\mathbb{R}$ of Riemann integrable functions and a free algebra $\mathcal{C} \subset \mathbb{R}^{[0,1]}$ of continuous functions, having $\mathfrak{c}$-generators, such that $r \circ c$ is not Riemann integrable for any $r \in \mathcal{S}$ and $c \in \mathcal{C}$. This result is the best possible one in terms of lineability within these families of functions and, at the same time, an improvement of a precious result (\cite[Theorem 2.7]{A}). In order to achieve our results we shall employ set theoretical tools such as the Fichtenholz-Kantorovich-Hausdorff theorem, Cantor-Smith-Volterra--type sets, and classical real analysis techniques. |
| title | Algebrability and Riemann integrability of the composite function |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2402.16545 |