Geodesic
On extensions of the Riemann and Lebesgue Integrals by Nets
Canadian mathematical bulletin, Tome 18 (1975) no. 1, pp. 7-17

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In this note our principal interest is in using nets to give spaces of non-absolutely convergent integrals as extensions of the spaces of absolutely convergent Riemann and Lebesgue integrals. For this purpose we develop a general theory of extensions, by nets, of functions defined on the open intervals with closures in the complement of a fixed closed set, the nets being directed by inclusion for finite disjoint collections of such intervals. Two cases are considered leading to open extension (OE-) and conditional open extension (COE-) nets, the latter being subnets of the former. Necessary and sufficient conditions for the convergence of the OE- and COE-nets are given, those for the COE-nets being similar to conditions that arise in the definition of the restricted Denjoy integral. Properties of inner continuity, weak additivity and the existence of a continuous integral are defined and studied. These relate to the more specialized nets that are suitable for the extension of integrals. 
Bellamy, O. S.; Ellis, H. W. On extensions of the Riemann and Lebesgue Integrals by Nets. Canadian mathematical bulletin, Tome 18 (1975) no. 1, pp. 7-17. doi: 10.4153/CMB-1975-002-1
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     title = {On extensions of the {Riemann} and {Lebesgue} {Integrals} by {Nets}},
     journal = {Canadian mathematical bulletin},
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     year = {1975},
     volume = {18},
     number = {1},
     doi = {10.4153/CMB-1975-002-1},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1975-002-1/}
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