| Title:
|
General integration and extensions. I (English) |
| Author:
|
Schwabik, Štefan |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
60 |
| Issue:
|
4 |
| Year:
|
2010 |
| Pages:
|
961-981 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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\dots ) belong to it. \endgraf 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. (English) |
| Keyword:
|
abstract integration |
| Keyword:
|
extension of integral |
| Keyword:
|
Kurzweil-Henstock integration |
| MSC:
|
26A39 |
| MSC:
|
26A42 |
| idZBL:
|
Zbl 1224.26030 |
| idMR:
|
MR2738960 |
| . |
| Date available:
|
2010-11-20T13:54:47Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140797 |
| . |
| Related article:
|
http://dml.cz/handle/10338.dmlcz/140798 |
| . |
| Reference:
|
[1] Dunford, N., Schwartz, J. T.: Linear Operators I.Interscience Publishers New York (1958). Zbl 0084.10402, MR 0117523 |
| Reference:
|
[2] Foran, J.: Fundamentals of Real Analysis.Marcel Dekker New York (1991). Zbl 0744.26004, MR 1201817 |
| Reference:
|
[3] Gordon, R. A.: The Integrals of Lebesgue, Denjoy, Perron and Henstock.American Mathematical Society (1994). Zbl 0807.26004, MR 1288751 |
| Reference:
|
[4] Kubota, Y.: Abstract treatment of integration.Math. J. Ibaraki Univ. 29 (1997), 41-54. Zbl 0924.26005, MR 1601363, 10.5036/mjiu.29.41 |
| Reference:
|
[5] Lee, P.-Y.: Lanzhou Lectures on Henstock Integration.World Scientific Singapore (1989). Zbl 0699.26004, MR 1050957 |
| Reference:
|
[6] Saks, S.: Theory of the Integral.Hafner New York (1937). Zbl 0017.30004 |
| Reference:
|
[7] Schwabik, Š.: Variational measures and the Kurzweil-Henstock integral.Math. Slovaca 59 (2009), 731-752. MR 2564330, 10.2478/s12175-009-0160-1 |
| Reference:
|
[8] Thomson, B. S.: Derivates of Interval Functions, Mem. Am. Math. Soc. 452.(1991). MR 1078198 |
| . |
