On a generalization of Henstock-Kurzweil integrals

Jan Malý; Kristýna Kuncová

Mathematica Bohemica (2019)

  • Volume: 144, Issue: 4, page 393-422
  • ISSN: 0862-7959

Abstract

top
We study a scale of integrals on the real line motivated by the M C α integral by Ball and Preiss and some recent multidimensional constructions of integral. These integrals are non-absolutely convergent and contain the Henstock-Kurzweil integral. Most of the results are of comparison nature. Further, we show that our indefinite integrals are a.e. approximately differentiable. An example of approximate discontinuity of an indefinite integral is also presented.

How to cite

top

Malý, Jan, and Kuncová, Kristýna. "On a generalization of Henstock-Kurzweil integrals." Mathematica Bohemica 144.4 (2019): 393-422. <http://eudml.org/doc/294370>.

@article{Malý2019,
abstract = {We study a scale of integrals on the real line motivated by the $MC_\{\alpha \}$ integral by Ball and Preiss and some recent multidimensional constructions of integral. These integrals are non-absolutely convergent and contain the Henstock-Kurzweil integral. Most of the results are of comparison nature. Further, we show that our indefinite integrals are a.e. approximately differentiable. An example of approximate discontinuity of an indefinite integral is also presented.},
author = {Malý, Jan, Kuncová, Kristýna},
journal = {Mathematica Bohemica},
keywords = {Henstock-Kurzweil integral},
language = {eng},
number = {4},
pages = {393-422},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On a generalization of Henstock-Kurzweil integrals},
url = {http://eudml.org/doc/294370},
volume = {144},
year = {2019},
}

TY - JOUR
AU - Malý, Jan
AU - Kuncová, Kristýna
TI - On a generalization of Henstock-Kurzweil integrals
JO - Mathematica Bohemica
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 144
IS - 4
SP - 393
EP - 422
AB - We study a scale of integrals on the real line motivated by the $MC_{\alpha }$ integral by Ball and Preiss and some recent multidimensional constructions of integral. These integrals are non-absolutely convergent and contain the Henstock-Kurzweil integral. Most of the results are of comparison nature. Further, we show that our indefinite integrals are a.e. approximately differentiable. An example of approximate discontinuity of an indefinite integral is also presented.
LA - eng
KW - Henstock-Kurzweil integral
UR - http://eudml.org/doc/294370
ER -

References

top
  1. Ball, T., Preiss, D., 10.1142/9789813237315_0005, Mathematics Almost Everywhere. In Memory of Solomon Marcus World Scientific Publishing, Hackensack A. Bellow et al. (2018), 69-92. (2018) MR3838200DOI10.1142/9789813237315_0005
  2. Bendová, H., Malý, J., 10.5186/aasfm.2011.3609, Ann. Acad. Sci. Fenn., Math. 36 (2011), 153-164. (2011) Zbl1225.26016MR2797688DOI10.5186/aasfm.2011.3609
  3. Bongiorno, B., 10.1016/B978-044450263-6/50014-2, Handbook of Measure Theory. Vol. I and II North-Holland, Amsterdam (2002), 587-615 E. Pap. (2002) Zbl1024.26004MR1954623DOI10.1016/B978-044450263-6/50014-2
  4. Burkill, J. C., 10.1007/BF01180588, Math. Z. 34 (1932), 270-278. (1932) Zbl0002.38604MR1545252DOI10.1007/BF01180588
  5. Pauw, T. De, Pfeffer, W. F., 10.1002/cpa.20204, Commun. Pure Appl. Math. 61 (2008), 230-260. (2008) Zbl1137.35014MR2368375DOI10.1002/cpa.20204
  6. Denjoy, A., Une extension de l'intégrale de M. Lebesgue, C. R. Acad. Sci., Paris Sér. I Math. 154 (1912), 859-862 French. (1912) Zbl43.0360.01
  7. Denjoy, A., Sur la dérivation et son calcul inverse, C. R. Acad. Sci., Paris Sér. I Math. 162 (1916), 377-380 French. (1916) Zbl46.0381.02MR0062821
  8. Evans, L. C., Gariepy, R. F., Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics. CRC Press, Boca Raton (1992). (1992) Zbl0804.28001MR1158660
  9. Gordon, R. A., 10.1090/gsm/004, Graduate Studies in Mathematics 4. AMS, Providence (1994). (1994) Zbl0807.26004MR1288751DOI10.1090/gsm/004
  10. Gordon, R. A., 10.2307/44152566, Real Anal. Exch. 20 (1994-95), 831-841. (1994) Zbl0847.26011MR1348106DOI10.2307/44152566
  11. Hencl, S., 10.4064/fm173-2-5, Fundam. Math. 173 (2002), 175-189. (2002) Zbl1002.26007MR1924813DOI10.4064/fm173-2-5
  12. Henstock, R., 10.1112/plms/s3-11.1.402, Proc. London Math. Soc., III. Ser. 11 (1961), 402-418. (1961) Zbl0099.27402MR0132147DOI10.1112/plms/s3-11.1.402
  13. Henstock, R., Theory of Integration, Butterworths Mathematical Texts. Butterworths & Co., London (1963). (1963) Zbl0154.05001MR0158047
  14. Henstock, R., The General Theory of Integration, Oxford Mathematical Monographs. Clarendon Press, Oxford (1991). (1991) Zbl0745.26006MR1134656
  15. Honzík, P., Malý, J., 10.1007/s13348-013-0103-6, Collect. Math. 65 (2014), 367-377. (2014) Zbl1305.26022MR3241000DOI10.1007/s13348-013-0103-6
  16. Jarník, J., Kurzweil, J., A non-absolutely convergent integral which admits C 1 -transformations, Čas. Pěst. Mat. 109 (1984), 157-167. (1984) Zbl0555.26005MR0744873
  17. Jarník, J., Kurzweil, J., A non absolutely convergent integral which admits transformation and can be used for integration on manifolds, Czech. Math. J. 35 (1985), 116-139. (1985) Zbl0614.26007MR0779340
  18. Jarník, J., Kurzweil, J., A new and more powerful concept of the PU-integral, Czech. Math. J. 38 (1988), 8-48. (1988) Zbl0669.26006MR0925939
  19. Jarník, J., Kurzweil, J., Schwabik, Š., On Mawhin's approach to multiple nonabsolutely convergent integral, Čas. Pěst. Mat. 108 (1983), 356-380. (1983) Zbl0555.26004MR0727536
  20. Jurkat, W. B., The divergence theorem and Perron integration with exceptional sets, Czech. Math. J. 43 (1993), 27-45. (1993) Zbl0789.26005MR1205229
  21. Khintchine, A., Sur une extension de l'intégrale de M. Denjoy, C. R. Acad. Sci. Paris, Sér. I Math. 162 (1916), 287-291 French. (1916) Zbl46.0381.01
  22. Kubota, Y., 10.3792/pja/1195521564, Proc. Japan Acad. 40 (1964), 713-717. (1964) Zbl0141.24601MR0178113DOI10.3792/pja/1195521564
  23. Kuncová, K., B V -packing integral in ℝ n , Available at https://arxiv.org/abs/1903.04908 (2019). (2019) 
  24. Kuncová, K., Malý, J., 10.1016/j.jmaa.2012.12.044, J. Math. Anal. Appl. 401 (2013), 578-600. (2013) Zbl1264.26015MR3018009DOI10.1016/j.jmaa.2012.12.044
  25. Kurzweil, J., Generalized ordinary differential equations and continuous dependence on a parameter, Czech. Math. J. 7 (1957), 418-449. (1957) Zbl0090.30002MR0111875
  26. Kurzweil, J., 10.1112/blms/16.4.432, Teubner-Texte zur Mathematik, Bd. 26. BSB B. G. Teubner Verlagsgesellschaft, Leipzig (1980), German. (1980) Zbl0441.28001MR0597703DOI10.1112/blms/16.4.432
  27. Lusin, N., Sur les propriétés de l'intégrale de M. Denjoy, C. R. Acad. Sci. Paris Sér. I Math. 155 (1912), 1475-1477 French. (1912) Zbl43.0360.03MR0067182
  28. Malý, J., 10.1007/s10231-013-0338-6, Ann. Mat. Pura Appl. (4) 193 (2014), 1457-1484. (2014) Zbl1304.26006MR3262642DOI10.1007/s10231-013-0338-6
  29. Malý, J., Pfeffer, W. F., 10.21136/MB.2016.16, Math. Bohem. 141 (2016), 217-237. (2016) Zbl06587863MR3499785DOI10.21136/MB.2016.16
  30. Mawhin, J., Generalized multiple Perron integrals and the Green-Goursat theorem for differentiable vector fields, Czech. Math. J. 31 (1981), 614-632. (1981) Zbl0562.26004MR0631606
  31. Mawhin, J., 10.1007/978-3-0348-5452-8_55, E. B. Christoffel: The Influence of his Work on Mathematics and The Physical Sciences. Int. Christoffel Sym., Aachen and Monschau, 1979 Birkhäuser, Basel (1981), 704-714 P. L. Butzer et al. (1981) Zbl0562.26003MR0661109DOI10.1007/978-3-0348-5452-8_55
  32. Monteiro, G. A., Slavík, A., Tvrdý, M., 10.1142/9432, Series in Real Analysis 15. World Scientific Publishing, Hackensack (2018). (2018) Zbl06758513MR3839599DOI10.1142/9432
  33. Novikov, A., Pfeffer, W. F., 10.1090/S0002-9939-1994-1182703-4, Proc. Am. Math. Soc. 120 (1994), 849-853. (1994) Zbl0808.26006MR1182703DOI10.1090/S0002-9939-1994-1182703-4
  34. Perron, O., Über den Integralbegriff, Heidelb. Ak. Sitzungsber 14 (1914), 1-16 German. (1914) Zbl45.0445.01
  35. Pfeffer, W. F., Une intégrale Riemannienne et le théorème de divergence, C. R. Acad. Sci. Paris, Sér. I Math. 299 (1984), 299-301 French. (1984) Zbl0574.26009MR0761251
  36. Pfeffer, W. F., 10.1017/s1446788700029293, J. Aust. Math. Soc., Ser. A 43 (1987), 143-170. (1987) Zbl0638.26011MR0896622DOI10.1017/s1446788700029293
  37. Pfeffer, W. F., 10.1016/0001-8708(91)90063-D, Adv. Math. 87 (1991), 93-147. (1991) Zbl0732.26013MR1102966DOI10.1016/0001-8708(91)90063-D
  38. Pfeffer, W. F., 10.1017/CBO9780511574764, Cambridge Tracts in Mathematics 140. Cambridge University Press, Cambridge (2001). (2001) Zbl0980.26008MR1816996DOI10.1017/CBO9780511574764
  39. Pfeffer, W. F., Derivatives and primitives, Sci. Math. Jpn. 55 (2002), 399-425. (2002) Zbl1010.26012MR1887074
  40. Preiss, D., Thomson, B. S., 10.4153/CJM-1989-023-8, Can. J. Math. 41 (1989), 508-555. (1989) Zbl0696.26004MR1013466DOI10.4153/CJM-1989-023-8
  41. Saks, S., Theory of the Integral, With two Additional Notes by Stefan Banach. Dover Publications, Mineola (1964). (1964) Zbl1196.28001MR0167578

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.