Fractal Measure Integral: An Axiomatic Framework
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| Formato: | Recurso digital |
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2025
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| _version_ | 1866901562643709952 |
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| author | ZHOU, changzheng 周, 子清 |
| author_facet | ZHOU, changzheng 周, 子清 |
| contents | <p> This paper systematically constructs an axiomatic theory for fractal measure integration<br> by introducing recursive self-similarity axioms, modified normalization conditions, and<br> compatible measurable space structures. It resolves traditional ambiguities in symbolic<br> representation and convergence issues through rigorous constraints on homogeneous self<br>similar fractals. We prove the uniqueness of measures under open set conditions (OSC)<br> and establish integral convergence theorems. Notably, σ-compactness of support sets and<br> Lipschitz decay conditions for integrands ensure mathematical coherence. Comparative<br> analysis reveals fundamental differences between fractal and Lebesgue integrals, while<br> highlighting limitations in heterogeneous multifractal systems and infinitely recursive<br> fractals.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_14954795 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Fractal Measure Integral: An Axiomatic Framework ZHOU, changzheng 周, 子清 Fractal measure, Self-similarity, Hausdorff dimension, σ-finiteness, Re cursive integration, Nonhomogeneous fractals <p> This paper systematically constructs an axiomatic theory for fractal measure integration<br> by introducing recursive self-similarity axioms, modified normalization conditions, and<br> compatible measurable space structures. It resolves traditional ambiguities in symbolic<br> representation and convergence issues through rigorous constraints on homogeneous self<br>similar fractals. We prove the uniqueness of measures under open set conditions (OSC)<br> and establish integral convergence theorems. Notably, σ-compactness of support sets and<br> Lipschitz decay conditions for integrands ensure mathematical coherence. Comparative<br> analysis reveals fundamental differences between fractal and Lebesgue integrals, while<br> highlighting limitations in heterogeneous multifractal systems and infinitely recursive<br> fractals.</p> |
| title | Fractal Measure Integral: An Axiomatic Framework |
| topic | Fractal measure, Self-similarity, Hausdorff dimension, σ-finiteness, Re cursive integration, Nonhomogeneous fractals |
| url | https://doi.org/10.5281/zenodo.14954795 |