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|

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N积分

本文引入了实值函数f在闭有界区间[a,b]上的henstock型积分n积分。集n可积函数完全介于黎曼可积函数和Henstock-Kurzweil可积函数之间。进一步地，这个新积分积分了所有反常黎曼可积函数，即使它们不是勒贝格可积函数。证明了对于Henstock-Kurzweil可积函数f on [a,b]，下列是等价的:函数f是n可积的;存在一个空集S，对于给定的epsilon >，存在一个规范函数，使得对于任意的[a,b]的精细偏分D={(xi，[u,v])}，我们有[(phi_S(D) Gamma_epsilon)和|f(v)-f(u)||v-u|

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