An elementary way to introduce a Perron-like integral
Hana Bendov\'a, Jan Mal\'y
TL;DR
This paper introduces a simple, alternative definition of the Perron integral, extending it to multidimensional and Stieltjes cases, and proves its equivalence to the Henstock-Kurzweil integral, simplifying the foundation of integral theory.
Contribution
It presents a new, elementary definition of the Perron integral that simplifies the foundational exposition and extends to multidimensional and Stieltjes integrals, showing equivalence to existing integrals.
Findings
New elementary definition of Perron integral
Extension to multidimensional and Stieltjes integrals
Proven equivalence to Henstock-Kurzweil integral
Abstract
We give an alternative definition of integral at the generality of the Perron integral and propose an exposition of the foundations of integral theory starting from this new definition. Both definition and proofs needed for the development are unexpectedly simple. We show how to adapt the definition to cover the multidimensional and Stieltjes case and prove that our integral is equivalent to the Henstock-Kurzweil(-Stieltjes) integral.
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