Integration through Generalized Sequences

TL;DR

This paper introduces generalized Riemann and Lebesgue integrals using generalized sequences, aiming to unify and extend classical integration theories with a simple, transparent approach.

Contribution

It presents a novel framework for defining integrals through generalized sequences, enhancing understanding and applicability of integration methods.

Findings

Unified approach to Riemann and Lebesgue integrals

Extended class of integrable functions

Simplified, transparent integral definitions

Abstract

The process of integration was a subject of significant development during the last century. Despite that the Lebesgue integral is complete and has many good properties, its inability to integrate all derivatives prompted the introduction of new approaches - Denjoy, Perron and others introduced new ways of integration aimed at preserving the good properties of the Lebesgue integral but extending the set of functions to which it could be applied. The goal was achieved but neither of the new approaches was elegant or simple or transparent. In the 50s a new integral was introduced, independently by Kurzweil and Henstock,in a very simple, Riemann like way, but it turned out that it was more powerful than the Lebesgue integral. There are many names attached to this integral, I will use here the name Henstock integral. The goal of this article is to introduce the generalized Riemann integral and the Lebesgue integral using generalized sequences.