On some variational measures related to the wide Denjoy integral and its counterparts

2019 ◽  
Vol 142 ◽  
pp. 9-36
Author(s):  
Piotr Sworowski
Keyword(s):  
Denjoy Integral

scholarly journals Definability aspects of the Denjoy integral

2017 ◽  
Vol 237 (1) ◽  
pp. 1-29
Author(s):  
Sean Walsh
Keyword(s):  
Denjoy Integral

On the Generality of the AP-Integral

1971 ◽  
Vol 23 (3) ◽  
pp. 557-561 ◽  
Author(s):  
G. E. Cross
Keyword(s):  
Trigonometric Series ◽  
Denjoy Integral ◽  
Integrable Functions ◽  
Definition Of ◽  
General Conditions

In 1955 Taylor [6] constructed an AP-integral sufficiently strong to integrate Abel summable series with coefficients o(n). He showed that the AP-integral includes the special Denjoy integral and further that, when applied to trigonometric series, the AP-integral is more powerful than the SCP-integral of Burkill [1] and the P2-integral of James [3]. The present paper shows that the AP-integral includes the SCP-integral, and, under natural assumptions, the P2-integral.After completing this manuscript I was advised by Skvorcov that he had shown [5] under more general conditions that the P2-integral is included in the AP-integral. The proof in the present paper seems to have some value in its own right and is considerably shorter.Since the definition of the AP-integral is essentially for a function defined in (0, 2π] and elsewhere by 2π-periodicity, we shall consider SCP-integrable and P2-integrable functions defined similarly.

On extensions of the Riemann and Lebesgue Integrals by Nets

1975 ◽  
Vol 18 (1) ◽  
pp. 7-17 ◽  
Author(s):  
O. S. Bellamy ◽  
H. W. Ellis
Keyword(s):  
General Theory ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Closed Set ◽  
Denjoy Integral ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Open Extension

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.

scholarly journals The preponderantly continuous Denjoy integral

1978 ◽  
Vol 30 (4) ◽  
pp. 537-541
Author(s):  
Yôto Kubota
Keyword(s):  
Denjoy Integral
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