Upper and Lower N-integrals

This paper provides alternative definitions of the N-integral using the set of discontinuity D-f 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 epsilon > 0, there exists an elementary set E with [a, b] \ E of measure smaller than epsilon and S-infinity subset of [a, b] \ (E) over bar such that f is Riemann integrable on (E) over bar and vertical bar(R) integral(E) f - A vertical bar < epsilon. Here S-infinity is the set of all points in [a, b] such that for every x is an element of S-infinity, there exists a sequence {x(n)} in [a,b] with vertical bar f(x(n))vertical bar -> as n -> infinity.