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Keywords:
wide Denjoy integral; Kurzweil-Henstock integral; Kubota integral; local system; porosity; intersection conditions
Summary:
Kurzweil-Henstock integrals related to local systems and the wide Denjoy integral are discussed in the frame of their comparability and compatibility.
References:
[1] B. Bongiorno, L. Di Piazza and V. A. Skvortsov: On dyadic integrals and some other integrals associated with local systems. Journal of Mathematical Analysis and Applications 271 (2002), 506–524. DOI 10.1016/S0022-247X(02)00146-4 | MR 1923649
[2] B. Bongiorno, L. Di Piazza and V. A. Skvortsov: The Ward property for a $\mathcal{P}$-adic basis and the $\mathcal{P}$-adic integral. Journal of Mathematical Analysis and Applications 285 (2003), 578–592. DOI 10.1016/S0022-247X(03)00426-8 | MR 2005142
[3] D. Bongiorno, L. Di Piazza and V. A. Skvortsov: Variational measures related to local systems and the Ward property of $\mathcal{P}$-adic path bases. Czech. Math. J. 56 (2006), 559–578. DOI 10.1007/s10587-006-0037-1 | MR 2291756
[4] A. M. Bruckner: Differentiation of Real Functions. Lecture Notes in Mathematics 659, Springer-Verlag, 1978. MR 0507448 | Zbl 0382.26002
[5] V. Ene: On Borel measurable functions that are $\text{VBG}$ and ${\mathcal{N}}$. Real Anal. Exch. 22 (1996/97), 688–695. MR 1460981
[6] V. Ene: Hake-Alexandroff-Looman type theorems. Real Anal. Exch. 23 (1997/98), 491–524. MR 1639960
[7] V. Ene: A study of some general integrals that contains the wide Denjoy integral. Real Anal. Exch. 26 (2000/01), 51–100. MR 1825497
[8] T. Filipczak: Intersection conditions for some density and ${\mathcal{I}}$-density local systems. Real Anal. Exch. 15 (1989/90), 170–192. MR 1042535
[9] R. A. Gordon: The inversion of approximate and dyadic derivatives using an extension of the Henstock integral. Real Anal. Exch. 16 (1990/91), 154–168. MR 1087481
[10] R. A. Gordon: Some comments on an approximately continuous Khintchine integral. Real Anal. Exch. 20 (1994/95), 831–841. MR 1348106
[11] Y. Kubota: On the approximately continuous Denjoy integral. Tôhoku Mathematical Journal 15 (1963), 253–264. DOI 10.2748/tmj/1178243808 | MR 0153811 | Zbl 0129.27003
[12] Y. Kubota: An integral of the Denjoy type. Proceedings of the Japan Academy 40 (1964), 713–717. DOI 10.3792/pja/1195522601 | MR 0178113 | Zbl 0141.24601
[13] C. M. Lee: An analogue of the theorem of Hake-Alexandroff-Looman. Fundamenta Mathematicae 100 (1978), 69–74. MR 0486362 | Zbl 0384.26004
[14] C. M. Lee: On Baire one Darboux functions with Lusin’s condition ${\mathcal{N}}$. Real Anal. Exch. 7 (1981/82), 61–64. MR 0646637
[15] S. P. Lu: Notes on the approximately continuous Henstock integral. Real Anal. Exch. 22 (1996/97), 377–381. MR 1433622
[16] W. Poreda, E. Wagner-Bojakowska and W. Wilczyński: A category analogue of the density topology. Fundamenta Mathematicae 125 (1985), 167–173. MR 0813753
[17] J. Ridder: Über approximativ stetige Denjoy-Integrale. Fundamenta Mathematicae 21 (1933), 136–162. Zbl 0008.10901
[18] S. Saks: Theory of Integral. New York, 1937.
[19] D. N. Sarkhel: A wide constructive integral. Mathematica Japonica 32 (1987), 295–309. MR 0895552 | Zbl 0623.26007
[20] D. N. Sarkhel and A. K. De: The proximally continuous integrals. Journal of the Australian Mathematical Society (Series A) 31 (1981), 26–45. DOI 10.1017/S1446788700018462 | MR 0622811
[21] D. N. Sarkhel and A. B. Kar: $[\text{PVB}]$ functions and integration. Journal of the Australian Mathematical Society (Series A) 36 (1984), 335–353. DOI 10.1017/S1446788700025398 | MR 0733906
[22] V. A. Skvortsov: Some properties of dyadic primitives. In: New Integrals, Lecture Notes in Mathematics 1419, Springer-Verlag (1988), 167–179. MR 1051928
[23] V. Skvortsov and P. Sworowski: Characterization of approximate and ${\mathcal{I}}$-approximate Kurzweil-Henstock integrals. Real Analysis Conference Łeba (2001), 121–129.
[24] P. Sworowski: An answer to some questions of Ene. Real Anal. Exch. 30 (2004/05), 183–192. MR 2127524
[25] B. S. Thomson: Real Functions. Lecture Notes in Mathematics 1170, Springer-Verlag, 1985. MR 0818744 | Zbl 0581.26001
[26] B. S. Thomson: Derivation bases on the real line. Real Anal. Exch. 8 (1982/83), 67–207, 278–442.
[27] G. P. Tolstov: Sur l’intégrale de Perron. Matematicheskij Sbornik 5 (1939), 647–659. Zbl 0022.21203
[28] C. Wang and C. S. Ding: An integral involving Thomson’s local systems. Real Anal. Exch. 19 (1993/94), 248–253. MR 1268851
[29] L. Zajíček: Porosity and $\sigma $-porosity. Real Anal. Exch. 13 (1987/88), 314–350. MR 0943561
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