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arXiv · 2509.05247

The Fundamental Theorem of Calculus for Lebesgue-Stieltjes integrals involving non-monotonic derivators

Lamiae Maia·F. Adrián F. Tojo
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Abstract

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.

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BibTeXRIS

Lamiae Maia, F. Adrián F. Tojo. 2025-09-05. The Fundamental Theorem of Calculus for Lebesgue-Stieltjes integrals involving non-monotonic derivators. https://arxiv.org/abs/2509.05247

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