Abstract
We locate Musial and Sagher’s concept of \( \operatorname{HK}_{r} \)-integration within the approximate Henstock–Kurzweil integral theory. If we restrict the \( \operatorname{HK}_{r} \)-integral by the requirement that the indefinite \( \operatorname{HK}_{r} \)-integral is continuous, then it becomes included in the classical Denjoy–Khintchine integral. We provide a direct argument demonstrating that this inclusion is proper.
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Notes
Slightly abusing notation here and in what follows, we identify the intervals \( u_{n},v_{n},r_{n} \) and their lengths, so that there is no distinction made between different intervals of the same kind and the same rank.
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Skvortsov, V.A., Sworowski, P. On the Relation between Denjoy–Khintchine and \( \operatorname{HK}_{r} \)-Integrals. Sib Math J 65, 441–447 (2024). https://doi.org/10.1134/S0037446624020162
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DOI: https://doi.org/10.1134/S0037446624020162
Keywords
- \( L^{r} \)-derivative
- \( \operatorname{HK}_{r} \)-integral
- variational measure
- Denjoy–Khintchine integral