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

Racca, Abraham Perral; Cabral, Emmanuel A.

J. Indones. Math. Soc. 26, No. 2, 242-257 (2020).

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 \(\Gamma_\epsilon= \{(\xi,[u,v]) :|f(v)-f(u)|\ge \epsilon\}\);
and
(3) The function \(f\) is continuous almost everywhere.
A characterization of continuous almost everywhere functions was also given.

MSC:

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

Keywords:

\(N\)-integral; continuity almost everywhere; Henstock-Kurzweil integral

References:

[1]	bibitem{Bar} Bartle, R.G., A Modern Theory of Integration, textit{Graduate Studies in Math. 32}, Amer Math. Soc., 2001.
[2]	bibitem{Cabral}
[3]	Cabral, E. A. and Lee, P.Y.,A Fundamental Theorem of Calculus for the Kurzweil-Henstock Integral in \(mathbb{R}^m\), {em Real Analysis Exchange}, textbf{26} \((2001slash2002), 867-876\). · Zbl 1024.26005
[4]	bibitem{Fenecios}
[5]	Fenecios, J.P., Cabral, E.A., and Racca, A.P., Baire One Functions and their Sets of Discontinuity, {em Mathematica Behemica }textbf{141} (2016), 109-114. · Zbl 1389.26009
[6]	bibitem{Gor}
[7]	Gordon, R. A., The Integrals of Lebegue, Denjoy, Perron, and Henstock, textit{Graduate Studies in Math. 4}, Amer. Math. Soc., 1994.
[8]	bibitem{Car}
[9]	Lee, C.S.Y., On Baire One Functions and Baire One Integration, Undergraduate Thesis, Nanyang Technological University, Singapore, 2001.
[10]	bibitem{Lee}
[11]	Lee, P.Y., Lanzhou Lectures on Henstock Integration, World Scientific Publishing, 1989. · Zbl 0699.26004
[12]	bibitem{LPY}
[13]	Lee, P.Y., The Integral A La Henstock, {em Scientiae Mathematicae Japonicae Online}, e-2007, 763-771.
[14]	bibitem{Vy}
[15]	Lee, P.Y. and Vyborny, R., The Integral: An Easy Approach after Kurzweil and Henstock, Cambridge University Press, 2000. · Zbl 0941.26003
[16]	bibitem{Racca}
[17]	Racca, A.P. and Cabral, E.A., On The Double Lusin Condition and Convergence Theorem for Kurzweil-Henstock Type Integrals, textit{Mathematica Bohemica} textbf{141} (2016), 153-168. · Zbl 1389.26015
[18]	bibitem{TBB}