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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2604.10670 |
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| _version_ | 1866908957578100736 |
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| author | Schuricht, Friedemann |
| author_facet | Schuricht, Friedemann |
| contents | The paper, that continuous some previous work of Schönherr & Schuricht, treats density measures on ${\mathbb R}^n$ that concentrate in any neighborhood of a Lebesgue null set. Such measures are typical for purely finitely additive measures. We study their basic properties and investigate related integrals. Measures taking only the values 0 and 1 are considered as special case. The results are first applied to weak convergence in $\mathcal{L}^\infty(Ω)$. Then we derive integral representations by means of such measures for several notions of differentiability for integrable functions and we show a kind of mean value theorem for some class of Sobolev functions. Finally we provide a new approach to the generalized Jacobians in the sense of Clarke. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_10670 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Density measures and applications Schuricht, Friedemann Analysis of PDEs Optimization and Control 28A12, 46E35, 26B30, 49J52 The paper, that continuous some previous work of Schönherr & Schuricht, treats density measures on ${\mathbb R}^n$ that concentrate in any neighborhood of a Lebesgue null set. Such measures are typical for purely finitely additive measures. We study their basic properties and investigate related integrals. Measures taking only the values 0 and 1 are considered as special case. The results are first applied to weak convergence in $\mathcal{L}^\infty(Ω)$. Then we derive integral representations by means of such measures for several notions of differentiability for integrable functions and we show a kind of mean value theorem for some class of Sobolev functions. Finally we provide a new approach to the generalized Jacobians in the sense of Clarke. |
| title | Density measures and applications |
| topic | Analysis of PDEs Optimization and Control 28A12, 46E35, 26B30, 49J52 |
| url | https://arxiv.org/abs/2604.10670 |