12 Month Open Extension of TNS for ADHD.

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1975 ◽  
Vol 18 (1) ◽  
pp. 7-17 ◽  
Author(s):  
O. S. Bellamy ◽  
H. W. Ellis

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.


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pp. 1057-1064 ◽  
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Caroline Brumsen ◽  
Socrates E. Papapoulos ◽  
Paul Lips ◽  
Petronella H. L. M. Geelhoed-Duijvestijn ◽  
Neveen A. T. Hamdy ◽  
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Vol 44 (12) ◽  
pp. 1555-1558 ◽  
Author(s):  
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M. C. Muñoz-Villanueva ◽  
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J. Gratacós ◽  
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pp. 531-544 ◽  
Author(s):  
Robert L. Findling ◽  
Idil Çavuş ◽  
Elizabeth Pappadopulos ◽  
Douglas G. Vanderburg ◽  
Jeffrey H. Schwartz ◽  
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