Upper and Lower N-integrals
. 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 sho...
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| Main Authors: | , |
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| Format: | text |
| Published: |
Archīum Ateneo
2019
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| Subjects: | |
| Online Access: | https://archium.ateneo.edu/mathematics-faculty-pubs/56 https://aip.scitation.org/doi/abs/10.1063/1.5139143 |
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| Institution: | Ateneo De Manila University |
| Summary: | . 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 → ∞ |
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