A Different Demonstration for Integral Identity Across Distinct Time Scales

TL;DR

This paper explores a novel approach to integral identities on time scales by demonstrating that for integrable functions, delta integrals can be converted to classical Riemann integrals without relying on Lebesgue theory, simplifying calculations.

Contribution

It shows that integral identities on time scales can be derived using Riemann sums alone, avoiding the need for Lebesgue integral development on the scale.

Findings

Delta-integrals on time scales can be converted to Riemann integrals of real functions.

The approach simplifies calculations by avoiding Lebesgue integral construction.

The method applies to integrable functions on time scales.

Abstract

In the theory of time scales, given $\mathbb{T}$ a time scale with at least two distinct elements, an integration theory is developed using ideas already well known as Riemann sums. Another, more daring, approach is to treat an integration theory on this scale from the point of view of the Lebesgue integral, which generalizes the previous perspective. A great tool obtained when studying the integral of a scale $\mathbb{T}$ as a Lebesgue integral is the possibility of converting the ``$\Delta$-integral of $\mathbb{T}$'' to a classical integral of $\mathbb{R}$. In this way, we are able to migrate from a calculation that is sometimes not so intuitive to a more friendly calculation. A question that arises, then, is whether the same result can be obtained just using the ideas of integration via Riemann sums, without the need to develop the Lebesgue integral for $\mathbb{T}$. And, in this article, we answer this question affirmatively: In fact, for integrable functions an analogous result is valid by converting a $\Delta$-integral over $\mathbb{T}$ to a riemannian integral of $\mathbb{R}$.