The N-Integral

Abraham Perral Racca, Emmanuel A. Cabral

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:

  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 [(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
  3. The function f is continuous almost everywhere.

 

A characterization of continuous almost everywhere functions was also given.

 


Keywords


N-integral, Continuity almost everywhere, Henstock-Kurzweil integral

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References


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DOI: https://doi.org/10.22342/jims.26.2.865.242-257

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