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

In this work, we extend the concept of the Stieltjes derivative to encompass left-continuous derivators with locally 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.