Main Article Content
Abstract
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 N-integrable functions lie 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:
- The function f is N-integrable;
- 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 [(phi_S(D) Gamma_epsilon) sum |f(v)-f(u)||v-u|<epsilon] where phi_S(D)={(xi,[u,v])in D:xi not in S} and [Gamma_epsilon={(xi,[u,v]): |f(v)-f(u)|<= epsilon}] and
- The function f is continuous almost everywhere.
A characterization of continuous almost everywhere functions was also given.
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Article Details
References
- bibitem{Bar} Bartle, R.G., A Modern Theory of Integration, textit{Graduate Studies in Math. 32}, Amer Math. Soc., 2001.
- bibitem{Cabral}
- 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} (2001$slash$2002), 867--876.
- bibitem{Fenecios}
- 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.
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- Gordon, R. A., The Integrals of Lebegue, Denjoy, Perron, and Henstock, textit{Graduate Studies in Math. 4}, Amer. Math. Soc., 1994.
- bibitem{Car}
- Lee, C.S.Y., On Baire One Functions and Baire One Integration, Undergraduate Thesis, Nanyang Technological University, Singapore, 2001.
- bibitem{Lee}
- Lee, P.Y., Lanzhou Lectures on Henstock Integration, World Scientific Publishing, 1989.
- bibitem{LPY}
- Lee, P.Y., The Integral A La Henstock, {em Scientiae Mathematicae Japonicae Online}, e-2007, 763-771.
- bibitem{Vy}
- Lee, P.Y. and Vyborny, R., The Integral: An Easy Approach after Kurzweil and Henstock, Cambridge University Press, 2000.
- bibitem{Racca}
- 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.
- bibitem{TBB}
- Thomson, B.S., Bruckner, J.B., and A.M. Bruckner, Real Analysis, textit{Prentice-Hall, Inc}. United States of America, 1997.
References
bibitem{Bar} Bartle, R.G., A Modern Theory of Integration, textit{Graduate Studies in Math. 32}, Amer Math. Soc., 2001.
bibitem{Cabral}
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} (2001$slash$2002), 867--876.
bibitem{Fenecios}
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.
bibitem{Gor}
Gordon, R. A., The Integrals of Lebegue, Denjoy, Perron, and Henstock, textit{Graduate Studies in Math. 4}, Amer. Math. Soc., 1994.
bibitem{Car}
Lee, C.S.Y., On Baire One Functions and Baire One Integration, Undergraduate Thesis, Nanyang Technological University, Singapore, 2001.
bibitem{Lee}
Lee, P.Y., Lanzhou Lectures on Henstock Integration, World Scientific Publishing, 1989.
bibitem{LPY}
Lee, P.Y., The Integral A La Henstock, {em Scientiae Mathematicae Japonicae Online}, e-2007, 763-771.
bibitem{Vy}
Lee, P.Y. and Vyborny, R., The Integral: An Easy Approach after Kurzweil and Henstock, Cambridge University Press, 2000.
bibitem{Racca}
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.
bibitem{TBB}
Thomson, B.S., Bruckner, J.B., and A.M. Bruckner, Real Analysis, textit{Prentice-Hall, Inc}. United States of America, 1997.