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

Lamiae Maia, F. Adrián F. Tojo

TL;DR

This work extends Stieltjes calculus to left-continuous, non-monotonic derivators of bounded variation, formulating a comprehensive framework that unifies Stieltjes derivatives with measure-theoretic tools. It develops a g-topology and g-absolute continuity, enabling a robust Lebesgue–Stieltjes integration theory for non-monotone derivators. The authors prove generalized Fundamental Theorems of Calculus in both almost-everywhere and everywhere forms, the latter requiring a precise positivity condition that is shown to be optimal. The results bridge Stieltjes differential equations and measure differential equations, offering a rigorous tool for modeling systems with signed, non-monotone dynamics.

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.

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

TL;DR

This work extends Stieltjes calculus to left-continuous, non-monotonic derivators of bounded variation, formulating a comprehensive framework that unifies Stieltjes derivatives with measure-theoretic tools. It develops a g-topology and g-absolute continuity, enabling a robust Lebesgue–Stieltjes integration theory for non-monotone derivators. The authors prove generalized Fundamental Theorems of Calculus in both almost-everywhere and everywhere forms, the latter requiring a precise positivity condition that is shown to be optimal. The results bridge Stieltjes differential equations and measure differential equations, offering a rigorous tool for modeling systems with signed, non-monotone dynamics.

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.

Paper Structure

This paper contains 8 sections, 18 theorems, 206 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Monotonic derivators
  4. General non-monotonic derivators of bounded variation
  5. The Stieltjes derivative for non-monotonic derivators
  6. $g$-topology and $g$-continuity
  7. $g$-absolute continuity
  8. Generalized Fundamental Theorem of Calculus

Key Result

Proposition 2.2

Let $[a,b]\subset \mathbb{R}$ be a closed interval and $g:[a,b]\to\mathbb{R}$ a left-continuous nondecreasing derivator such that $b\notin N_g^+$ and let $t\in [a,b]$. Then the following statements are equivalent such that, $f(s)=f(t^*)+[d+h(s)][g(s)-g(t^*)]$ for $s\in[a,b]$, $g(s)\ne g(t^*)$, with $t^*$ as in eq:notation of t^*. When these properties hold, $d=f'_g(t)$.

Figures (5)

  • Figure 5.1: The graph of the derivator $g$ associated to the sequence \ref{['eq:x_n']}.
  • Figure 5.2: Graph of f (left) and zoomed-in view (right).
  • Figure 5.3: Illustration of the fact that $F$ cannot have a $g$-derivative at $0$.
  • Figure 5.4: The continuous function $f$ with $g^{\dagger}(g(a))<a$, $g^{\dagger}(g(b))=b$, $g(a)-g(t)<\frac{\varepsilon}{2}$.
  • Figure 5.5: The continuous function $f$ in the case $a\geqslant g^{\dagger}(g(b))$ with $a\in D_g$, $t=g^\dagger(g(a))$, and $g(g^{\dagger}(g(a)))<g(a)$.

Theorems & Definitions (48)

  • Definition 2.1: FMarTo-OnFirstandSec
  • Proposition 2.2
  • proof
  • Theorem 2.3: PR
  • Theorem 2.4: Fundamental Theorem of Calculus for the Lebesgue--Stieltjes Integral PR
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Remark 2.8
  • Theorem 2.9: Jordan decomposition theorem
  • ...and 38 more