On the connection between the Hilger and Radon--Nikodym derivatives

TL;DR

This paper demonstrates that the Hilger derivative on time scales is a specific instance of the Radon--Nikodym derivative, establishing a connection between time scale calculus and measure theory, and clarifying the concept of delta absolute continuity.

Contribution

It reveals that the Hilger derivative is a special case of the Radon--Nikodym derivative, bridging time scale calculus with measure theory.

Findings

Hilger derivative is a Radon--Nikodym derivative

Delta absolute continuity aligns with measure theory

Establishes a measure-theoretic foundation for time scale calculus

Abstract

We show that the Hilger derivative on time scales is a special case of the Radon--Nikodym derivative with respect to the natural measure associated with every time scale. Moreover, we show that the concept of delta absolute continuity agrees with the one from measure theory in this context.