The \(N\)-integral. (English) Zbl 1451.26012

Summary: In this paper we introduced a Henstock-type integral, named \(N\)-integral, of a real valued function \(f\) on a closed and bounded interval \([a,b]\). The set of \(N\)-integrable functions lies entirely between Riemann integrable functions and Henstock-Kurzweil integrable functions. Furthermore, this new integral integrates all improper Riemann integrable functions even if they are not Lebesgue integrable. It was shown that for a Henstock-Kurzweil integrable function \(f\) on \([a,b]\), the following are equivalent:
(1) The function \(f\) is \(N\)-integrable;
(2) There exists a null set \(S\) for which given \(\epsilon >0\) there exists a gauge \(\delta\) such that for any \(\delta\)-fine partial division \(D=\{(\xi,[u,v])\}\) of \([a,b]\) we have \((\varphi _S(D) \cap \Gamma_\epsilon ) \sum|f(v)-f(u)||v-u|< \epsilon\) where \(\varphi _S(D) = \{(\xi, [u,v])\in D: \xi\not\in S\}\) and \(\Gamma_\epsilon= \{(\xi,[u,v]) :|f(v)-f(u)|\ge \epsilon\}\);
(3) The function \(f\) is continuous almost everywhere.
A characterization of continuous almost everywhere functions was also given.


26A39 Denjoy and Perron integrals, other special integrals
26A42 Integrals of Riemann, Stieltjes and Lebesgue type
Full Text: DOI


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