Upper and Lower N-integrals

• Main Authors: Cabral, Emmanuel A, Racca, Abraham P
• Format: text
• Published: Archīum Ateneo 2019
• Subjects: N−integral; set of discontinuity; unbounded functions; nonabsolute integral; upper Darboux sums; lower Darboux sums; Mathematics
• Online Access: https://archium.ateneo.edu/mathematics-faculty-pubs/56; https://aip.scitation.org/doi/abs/10.1063/1.5139143
• Institution: Ateneo De Manila University

. 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] \ E such that f is Riemann integrable on E and (R) E f − 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 → ∞