The AP-Denjoy and AP-Henstock integrals revisited

Valentin A. Skvortsov; Piotr Sworowski

Czechoslovak Mathematical Journal (2012)

  • Volume: 62, Issue: 3, page 581-591
  • ISSN: 0011-4642

Abstract

top
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 σ -finiteness of variational measure generated by a function and the classical notion of the generalized bounded variation.

How to cite

top

Skvortsov, Valentin A., and Sworowski, Piotr. "The AP-Denjoy and AP-Henstock integrals revisited." Czechoslovak Mathematical Journal 62.3 (2012): 581-591. <http://eudml.org/doc/246290>.

@article{Skvortsov2012,
abstract = {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.},
author = {Skvortsov, Valentin A., Sworowski, Piotr},
journal = {Czechoslovak Mathematical Journal},
keywords = {approximate Kurzweil-Henstock integral; approximate continuity; local system; variational measure; approximate Kurzweil-Henstock integral; local system; variational measure; VBG-function},
language = {eng},
number = {3},
pages = {581-591},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The AP-Denjoy and AP-Henstock integrals revisited},
url = {http://eudml.org/doc/246290},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Skvortsov, Valentin A.
AU - Sworowski, Piotr
TI - The AP-Denjoy and AP-Henstock integrals revisited
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 3
SP - 581
EP - 591
AB - 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.
LA - eng
KW - approximate Kurzweil-Henstock integral; approximate continuity; local system; variational measure; approximate Kurzweil-Henstock integral; local system; variational measure; VBG-function
UR - http://eudml.org/doc/246290
ER -

References

top
  1. Bongiorno, B., Piazza, L. Di, Skvortsov, V. A., 10.1016/S0022-247X(02)00146-4, J. Math. Anal. Appl. 271 (2002), 506-524. (2002) Zbl1010.26006MR1923649DOI10.1016/S0022-247X(02)00146-4
  2. Bongiorno, D., Piazza, L. Di, Skvortsov, V. A., 10.1007/s10587-006-0037-1, Czech. Math. J. 56(131) (2006), 559-578. (2006) MR2291756DOI10.1007/s10587-006-0037-1
  3. Ene, V., Real Functions---Current Topics. Lecture Notes in Mathematics, Vol. 1603, Springer Berlin (1995). (1995) MR1369575
  4. Ene, V., 10.2307/44153985, Real Anal. Exch. 23 (1997), 611-630. (1997) MR1639992DOI10.2307/44153985
  5. Ene, V., 10.2307/44152978, Real Anal. Exch. 24 (1998), 523-565. (1998) Zbl0968.26007MR1704732DOI10.2307/44152978
  6. Gordon, R. A., The inversion of approximate and dyadic derivatives using an extension of the Henstock integral, Real Anal. Exch. 16 (1991), 154-168. (1991) Zbl0723.26005MR1087481
  7. Park, J. M., Oh, J. J., Kim, J., Lee, H. K., The equivalence of the {AP}-Henstock and {AP}-Denjoy integrals, J. Chungcheong Math. Soc. 17 (2004), 103-110. (2004) 
  8. Park, J. M., Kim, Y. K., Yoon, J. H., 10.4134/BKMS.2005.42.3.535, Bull. Korean Math. Soc. 42 (2005), 535-541. (2005) Zbl1090.26002MR2162209DOI10.4134/BKMS.2005.42.3.535
  9. Park, J. M., Oh, J. J., Park, C.-G., Lee, D. H., 10.1007/s10587-007-0106-0, Czech. Math. J. 57(132) (2007), 689-696. (2007) Zbl1174.26308MR2337623DOI10.1007/s10587-007-0106-0
  10. Pfeffer, W. F., The Riemann Approach to Integration: Local Geometric Theory. Cambridge Tracts in Mathematics, 109 Cambridge (1993). (1993) MR1268404
  11. Saks, S., Theory of the Integral, G. E. Stechert & Co New York (1937). (1937) Zbl0017.30004
  12. Sworowski, P., Skvortsov, V. A., Variational measure determined by an approximative differential basis, Mosc. Univ. Math. Bull. 57 (2002), 37-40. (2002) MR1933126
  13. Thomson, B. S., 10.2307/44151585, Real Anal. Exch. 8 (1983), 67-207, 278-442. (1983) Zbl0525.26003DOI10.2307/44151585
  14. Thomson, B. S., Real Functions. Lecture Notes in Mathematics, 1170 Springer Berlin (1985). (1985) MR0818744
  15. Wang, C., Ding, C. S., An integral involving Thomson's local systems, Real Anal. Exch. 19 (1994), 248-253. (1994) Zbl0802.26004

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.