The N-Integral

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}] andThe function f is continuous almost everywhere. A characterization of continuous almost everywhere functions was also given.