Upper and lower N-integrals

Abstract

This paper provides alternative definitions of the N−integral using the set of discontinuity Df of the function f. Upper and lower Darboux sums are introduced so that a Darboux characterization of the N−integral similar to the Darboux definition of the Riemann integral is obtained. It is also shown that a function is N− integrable with integral A if and only if for every ∈ >0, there exists an elementary set E with [a, b] \E of measure smaller than ∈ and S∞ ⊂ [a, b] Ē such that f is Riemann integrable on Ē and | (R)∫Ēf−A |<∈ Here S∞ is the set of all points in [a, b] such that for every x ∈ S∞, there exists a sequence {xn} in [a,b] with | f (xn)|→∞ as n→∞.