| Title:
|
On Kurzweil-Henstock equiintegrable sequences (English) |
| Author:
|
Schwabik, Štefan |
| Author:
|
Vrkoč, Ivo |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
121 |
| Issue:
|
2 |
| Year:
|
1996 |
| Pages:
|
189-207 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
For the Kurzweil-Henstock integral the equiintegrability of a pointwise convergent sequence of integrable functions implies the integrability of the limit function and the relation
\lim_{m \to\infty}\int_a^bf_m(s)\dd s = \int_a^b\lim_{m \to\infty}f_m(s)\dd s.
Conditions for the equiintegrability of a sequence of functions pointwise convergent to an integrable function are presented. These conditions are given in terms of convergence of some sequences of integrals. (English) |
| Keyword:
|
equiintegrable sequence |
| Keyword:
|
Kurzweil-Henstock integral |
| MSC:
|
26A39 |
| idZBL:
|
Zbl 0863.26009 |
| idMR:
|
MR1400612 |
| DOI:
|
10.21136/MB.1996.126102 |
| . |
| Date available:
|
2009-09-24T21:18:23Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/126102 |
| . |
| Reference:
|
[1] Gordon R. A.: Another look at a convergence theorem foг the Henstock integгal.Real Analysis Exchange 15 (1989-90), 724-728. MR 1059433, 10.2307/44152048 |
| Reference:
|
[2] Gordon R. A.: A general convergence theorem for non-absolute integгals.J. London Math. Soc. 44 (1991), 301-309. MR 1136442, 10.1112/jlms/s2-44.2.301 |
| Reference:
|
[3] Gordon R. A.: The Integrals of Lebesgue, Denjoy, Perron and Henstock.Graduate Studies in Math., Vol. 4, American Mathematical Society, 1994. Zbl 0807.26004, MR 1288751, 10.1090/gsm/004/09 |
| Reference:
|
[4] Henstock R.: Lectures on the Theory of Integration.Series in Real Analysis, Vol. 1, World Scientific, Singapore, 1988. Zbl 0668.28001, MR 0963249 |
| Reference:
|
[5] Kurzweil J.: Nichtabsolut konvergente Integrale.Teubner-Texte zur Mathematik, Band 26, Teubner, Leipzig, 1980. Zbl 0441.28001, MR 0597703 |
| Reference:
|
[6] Kurzweil J., Jarník J.: Equiintegrability and controlled convergence of Perron-type integrable functions.Real Anal. Exchange П (1991-92), 110-139. |
| Reference:
|
[7] Lee Peng Yee: Lanzhou Lectures on Henstock Integration.Series in Real Analysis, Vol. 2, World Scientific, Singapore, 1989. Zbl 0699.26004, MR 1050957 |
| Reference:
|
[8] McLeod R. M.: The Generalized Riemann Integral.Caгus Mathematical Monographs, No. 20, Mathematical Association of America, 1980. Zbl 0486.26005, MR 0588510 |
| Reference:
|
[9] Schwabik Š.: Generalized Ordinary Differential Equations.Series in Real Analysis, Vol. 5, World Scientific, Singapore, 1992. Zbl 0781.34003, MR 1200241 |
| Reference:
|
[10] Schwabik Š.: Convergence theorems for the Perron integral and Sklyarenko's condition.Comment. Math. Univ. Carolin. 33,2 (1992), 237-244. Zbl 0774.26004, MR 1189654 |
| . |
