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2. The AP-Denjoy and AP-Henstock integrals revisited
- Creator:
- Skvortsov, Valentin A. and Sworowski, Piotr
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- approximate Kurzweil-Henstock integral, approximate continuity, local system, and variational measure
- Language:
- English
- Description:
- 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.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
3. Variational measures related to local systems and the Ward property of $\scr P$-adic path bases
- Creator:
- Bongiorno, Donatella , Di Piazza, Luisa, and Skvortsov, Valentin A.
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- local system, ${\mathcal{P}}$-adic system, differentiation basis, variational measure, and Ward property
- Language:
- English
- Description:
- Some properties of absolutely continuous variational measures associated with local systems of sets are established. The classes of functions generating such measures are described. It is shown by constructing an example that there exists a $\mathcal{P}$-adic path system that defines a differentiation basis which does not possess Ward property.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public