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: Cabral, Emmanuel A, Racca, Abraham P
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出版: Archīum Ateneo 2019
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在線閱讀:https://archium.ateneo.edu/mathematics-faculty-pubs/56
https://aip.scitation.org/doi/abs/10.1063/1.5139143
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spelling ph-ateneo-arc.mathematics-faculty-pubs-10552020-03-09T02:01:42Z Upper and Lower N-integrals Cabral, Emmanuel A Racca, Abraham P . 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 → ∞ 2019-01-01T08:00:00Z text https://archium.ateneo.edu/mathematics-faculty-pubs/56 https://aip.scitation.org/doi/abs/10.1063/1.5139143 Mathematics Faculty Publications Archīum Ateneo N−integral set of discontinuity unbounded functions nonabsolute integral upper Darboux sums lower Darboux sums Mathematics
institution Ateneo De Manila University
building Ateneo De Manila University Library
country Philippines
collection archium.Ateneo Institutional Repository
topic N−integral
set of discontinuity
unbounded functions
nonabsolute integral
upper Darboux sums
lower Darboux sums
Mathematics
spellingShingle N−integral
set of discontinuity
unbounded functions
nonabsolute integral
upper Darboux sums
lower Darboux sums
Mathematics
Cabral, Emmanuel A
Racca, Abraham P
Upper and Lower N-integrals
description . 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 → ∞
format text
author Cabral, Emmanuel A
Racca, Abraham P
author_facet Cabral, Emmanuel A
Racca, Abraham P
author_sort Cabral, Emmanuel A
title Upper and Lower N-integrals
title_short Upper and Lower N-integrals
title_full Upper and Lower N-integrals
title_fullStr Upper and Lower N-integrals
title_full_unstemmed Upper and Lower N-integrals
title_sort upper and lower n-integrals
publisher Archīum Ateneo
publishDate 2019
url https://archium.ateneo.edu/mathematics-faculty-pubs/56
https://aip.scitation.org/doi/abs/10.1063/1.5139143
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