The Primitives of \(N-\)Integrable Functions Emmanuel Cabral (a*), Khaing Khaing Aye (b), Abraham Racca (c)
a) Ateneo de Manila University
Katipunan Avenue, Loyola Heights, Quezon City, Philippines
ecabral[at]ateneo.edu*
b) Yangon Technological University
Yangon, Myanmar
khaingkhaingaye[at]ytu.edu.mm
c) Adventist University of the Philippines
Puting Kahoy, Silang, Cavite, Philippines
APRacca[at]aup.edu.ph
Abstract
A function \(f:[a,b]\to\mathbb{R}\) is \(N-\)integrable with integral \(A\) if and only if for every \(\varepsilon>0\), there exists an elementary set \(E\) with \([a,b]\setminus E\) of measure smaller than \(\varepsilon\) and \(S_\infty\subset[a,b]\setminus \overline{E}\) such that \(f\) is Riemann integrable on \(\overline{E}\) and \[\left|(R)\int_{\overline{E}}f-A\right|<\varepsilon.\] Here, \(S_\infty\) is the set of all points in \([a,b]\) such that for every \(x\in S_\infty\), there is some sequence \(\left\{x_n\right\}\) in \([a,b]\) with \(\left|f(x_n)\right|\to\infty\) as \(n\to\infty\) and \((R)\int_\overline{E} f\) is a Riemann integral. It will be shown that if \(f\) is \(N-\)integrable, then there exists a sequence of Riemann integrable functions \(f_n\) pointwise convergent to \(f\) and with corresponding primitives \(F_n\) that are uniformly \(ACG^*\). The primitive of \(N-\)integrable functions are shown to be those almost everywhere slope continuous functions that satisfy the strong Lusin condition.
