Geodesic
The AP-Denjoy and AP-Henstock integrals revisited
Czechoslovak Mathematical Journal, Tome 62 (2012) no. 3, pp. 581-591
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The note is related to a recently published paper J. M. Park, J. J. Oh, C.-G. Park, D. H. Lee: The AP-Denjoy and AP-Henstock integrals. Czech. Math. J. 57 (2007), 689–696, which concerns a descriptive characterization of the approximate Kurzweil-Henstock integral. We bring to attention known results which are stronger than those contained in the aforementioned work. We show that some of them can be formulated in terms of a derivation basis defined by a local system of which the approximate basis is known to be a particular case. We also consider the relation between the $\sigma $-finiteness of variational measure generated by a function and the classical notion of the generalized bounded variation.
The note is related to a recently published paper J. M. Park, J. J. Oh, C.-G. Park, D. H. Lee: The AP-Denjoy and AP-Henstock integrals. Czech. Math. J. 57 (2007), 689–696, which concerns a descriptive characterization of the approximate Kurzweil-Henstock integral. We bring to attention known results which are stronger than those contained in the aforementioned work. We show that some of them can be formulated in terms of a derivation basis defined by a local system of which the approximate basis is known to be a particular case. We also consider the relation between the $\sigma $-finiteness of variational measure generated by a function and the classical notion of the generalized bounded variation.
DOI : 10.1007/s10587-012-0050-5
Classification : 26A39, 26A42, 26A46
Keywords: approximate Kurzweil-Henstock integral; approximate continuity; local system; variational measure 
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Skvortsov, Valentin A.; Sworowski, Piotr. The AP-Denjoy and AP-Henstock integrals revisited. Czechoslovak Mathematical Journal, Tome 62 (2012) no. 3, pp. 581-591. doi: 10.1007/s10587-012-0050-5

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