The Fundamental Theorem of Calculus for Lebesgue-Stieltjes integrals involving non-monotonic derivators
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866911298571206656 |
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| author | Maia, Lamiae Tojo, F. Adrián F. |
| author_facet | Maia, Lamiae Tojo, F. Adrián F. |
| contents | In this work, we extend the concept of the Stieltjes derivative to encompass left-continuous derivators with bounded variation, thereby relaxing the monotonicity constraint. This generalization necessitates a refined definition of the Stieltjes derivative applicable across the entire domain, accommodating derivators that may change sign. We establish a generalized Fundamental Theorem of Calculus for the Lebesgue-Stieltjes integral in this broader context, presenting both "almost-everywhere" and "everywhere" versions. The latter requires a specific condition relating the derivator to its variation function, which we prove to be optimal through a density theorem. Our framework bridges the gap between Stieltjes differential equations and measure differential equations, offering a tool for modeling complex systems with non-monotonic dynamics. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_05247 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Fundamental Theorem of Calculus for Lebesgue-Stieltjes integrals involving non-monotonic derivators Maia, Lamiae Tojo, F. Adrián F. Classical Analysis and ODEs 26A24 (Primary) 28A25, 26A36 (Secondary) In this work, we extend the concept of the Stieltjes derivative to encompass left-continuous derivators with bounded variation, thereby relaxing the monotonicity constraint. This generalization necessitates a refined definition of the Stieltjes derivative applicable across the entire domain, accommodating derivators that may change sign. We establish a generalized Fundamental Theorem of Calculus for the Lebesgue-Stieltjes integral in this broader context, presenting both "almost-everywhere" and "everywhere" versions. The latter requires a specific condition relating the derivator to its variation function, which we prove to be optimal through a density theorem. Our framework bridges the gap between Stieltjes differential equations and measure differential equations, offering a tool for modeling complex systems with non-monotonic dynamics. |
| title | The Fundamental Theorem of Calculus for Lebesgue-Stieltjes integrals involving non-monotonic derivators |
| topic | Classical Analysis and ODEs 26A24 (Primary) 28A25, 26A36 (Secondary) |
| url | https://arxiv.org/abs/2509.05247 |