A Characterization of Borel Measures which Induce Lipschitz-Free Space Elements
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866918215196606464 |
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| author | Raad, Lucas Maciel |
| author_facet | Raad, Lucas Maciel |
| contents | We will solve a problem by Aliaga and Pernecká about Lipschitz free spaces (denoted by $\mathcal F(M)$): $$\text{Does every Borel measure $μ$ on a complete metric space $M$ such that $\int d(m,0) d |μ|(m)< \infty$ induce a weak$^*$ continuous functional $\mathcal Lμ\in \mathcal F(M)$ by the mapping $\mathcal Lμ(f)=\int f d μ$ ? }$$ In particular, we will show a characterization of the measures such that $\mathcal Lμ\in \mathcal F(M)$, which indeed implies inner-regularity for complete metric spaces, and we will prove that every Borel measure on $M$ induces an element of $\mathcal F(M)$ if and only if the weight of $M$ is strictly less than the least real-valued measurable cardinal, and thus the existence of a metric space on which there is a measure $μ$ such that $\mathcal Lμ\in \mathcal F(M)^{**} \setminus \mathcal F(M)$ cannot be proven in ZFC. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_13319 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Characterization of Borel Measures which Induce Lipschitz-Free Space Elements Raad, Lucas Maciel Functional Analysis 46B26 (Primary), 46B10 (Primary), 46E27(Primary), 03E55 (Secondary) We will solve a problem by Aliaga and Pernecká about Lipschitz free spaces (denoted by $\mathcal F(M)$): $$\text{Does every Borel measure $μ$ on a complete metric space $M$ such that $\int d(m,0) d |μ|(m)< \infty$ induce a weak$^*$ continuous functional $\mathcal Lμ\in \mathcal F(M)$ by the mapping $\mathcal Lμ(f)=\int f d μ$ ? }$$ In particular, we will show a characterization of the measures such that $\mathcal Lμ\in \mathcal F(M)$, which indeed implies inner-regularity for complete metric spaces, and we will prove that every Borel measure on $M$ induces an element of $\mathcal F(M)$ if and only if the weight of $M$ is strictly less than the least real-valued measurable cardinal, and thus the existence of a metric space on which there is a measure $μ$ such that $\mathcal Lμ\in \mathcal F(M)^{**} \setminus \mathcal F(M)$ cannot be proven in ZFC. |
| title | A Characterization of Borel Measures which Induce Lipschitz-Free Space Elements |
| topic | Functional Analysis 46B26 (Primary), 46B10 (Primary), 46E27(Primary), 03E55 (Secondary) |
| url | https://arxiv.org/abs/2412.13319 |