General integration and extensions. I

Štefan Schwabik

Czechoslovak Mathematical Journal (2010)

  • Volume: 60, Issue: 4, page 961-981
  • ISSN: 0011-4642

Abstract

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A general concept of integral is presented in the form given by S. Saks in his famous book Theory of the Integral. A special subclass of integrals is introduced in such a way that the classical integrals (Newton, Riemann, Lebesgue, Perron, Kurzweil-Henstock...) belong to it. A general approach to extensions is presented. The Cauchy and Harnack extensions are introduced for general integrals. The general results give, as a specimen, the Kurzweil-Henstock integration in the form of the extension of the Lebesgue integral.

How to cite

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Schwabik, Štefan. "General integration and extensions. I." Czechoslovak Mathematical Journal 60.4 (2010): 961-981. <http://eudml.org/doc/196779>.

@article{Schwabik2010,
abstract = {A general concept of integral is presented in the form given by S. Saks in his famous book Theory of the Integral. A special subclass of integrals is introduced in such a way that the classical integrals (Newton, Riemann, Lebesgue, Perron, Kurzweil-Henstock...) belong to it. A general approach to extensions is presented. The Cauchy and Harnack extensions are introduced for general integrals. The general results give, as a specimen, the Kurzweil-Henstock integration in the form of the extension of the Lebesgue integral.},
author = {Schwabik, Štefan},
journal = {Czechoslovak Mathematical Journal},
keywords = {abstract integration; extension of integral; Kurzweil-Henstock integration; abstract integration; extension of integral; Kurzweil-Henstock integration},
language = {eng},
number = {4},
pages = {961-981},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {General integration and extensions. I},
url = {http://eudml.org/doc/196779},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Schwabik, Štefan
TI - General integration and extensions. I
JO - Czechoslovak Mathematical Journal
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 4
SP - 961
EP - 981
AB - A general concept of integral is presented in the form given by S. Saks in his famous book Theory of the Integral. A special subclass of integrals is introduced in such a way that the classical integrals (Newton, Riemann, Lebesgue, Perron, Kurzweil-Henstock...) belong to it. A general approach to extensions is presented. The Cauchy and Harnack extensions are introduced for general integrals. The general results give, as a specimen, the Kurzweil-Henstock integration in the form of the extension of the Lebesgue integral.
LA - eng
KW - abstract integration; extension of integral; Kurzweil-Henstock integration; abstract integration; extension of integral; Kurzweil-Henstock integration
UR - http://eudml.org/doc/196779
ER -

References

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  1. Dunford, N., Schwartz, J. T., Linear Operators I, Interscience Publishers New York (1958). (1958) Zbl0084.10402MR0117523
  2. Foran, J., Fundamentals of Real Analysis, Marcel Dekker New York (1991). (1991) Zbl0744.26004MR1201817
  3. Gordon, R. A., The Integrals of Lebesgue, Denjoy, Perron and Henstock, American Mathematical Society (1994). (1994) Zbl0807.26004MR1288751
  4. Kubota, Y., 10.5036/mjiu.29.41, Math. J. Ibaraki Univ. 29 (1997), 41-54. (1997) Zbl0924.26005MR1601363DOI10.5036/mjiu.29.41
  5. Lee, P.-Y., Lanzhou Lectures on Henstock Integration, World Scientific Singapore (1989). (1989) Zbl0699.26004MR1050957
  6. Saks, S., Theory of the Integral, Hafner New York (1937). (1937) Zbl0017.30004
  7. Schwabik, Š., 10.2478/s12175-009-0160-1, Math. Slovaca 59 (2009), 731-752. (2009) MR2564330DOI10.2478/s12175-009-0160-1
  8. Thomson, B. S., Derivates of Interval Functions, Mem. Am. Math. Soc. 452, (1991). (1991) MR1078198

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