A Different Demonstration for Integral Identity Across Distinct 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 $\mathbb{R}$.