| Title:
|
The monotone convergence theorem for multidimensional abstract Kurzweil vector integrals (English) |
| Author:
|
Federson, Márcia |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
52 |
| Issue:
|
2 |
| Year:
|
2002 |
| Pages:
|
429-437 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We prove two versions of the Monotone Convergence Theorem for the vector integral of Kurzweil, $\int _R{\mathrm d}\alpha (t) f(t)$, where $R$ is a compact interval of $\mathbb{R}^n$, $\alpha $ and $f$ are functions with values on $L(Z,W)$ and $Z$ respectively, and $Z$ and $W$ are monotone ordered normed spaces. Analogous results can be obtained for the Kurzweil vector integral, $\int _R\alpha (t)\mathrm{d}f(t)$, as well as to unbounded intervals $R$. (English) |
| Keyword:
|
Monotone Convergence Theorem |
| Keyword:
|
Kurzweil vector integral |
| Keyword:
|
ordered normed spaces |
| MSC:
|
26A39 |
| MSC:
|
26A42 |
| MSC:
|
28B05 |
| idZBL:
|
Zbl 1022.28003 |
| idMR:
|
MR1905449 |
| . |
| Date available:
|
2009-09-24T10:52:36Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127730 |
| . |
| Reference:
|
[1] G. Birkhoff: Integration of functions on Banach spaces.Trans. Amer. Math. Soc. 38 (1935), 357–378. MR 1501815 |
| Reference:
|
[2] M. M. Day: Normed Linear Spaces.Springer-Verlag, 1973. Zbl 0268.46013, MR 0344849 |
| Reference:
|
[3] M. Federson: The Fundamental Theorem of Calculus for the multidimensional Banach space-valued Henstock vector integral.Real Anal. Exchange 25 (2000), 469–480. MR 1758903 |
| Reference:
|
[4] R. Henstock: A Riemann-type integral of Lebesgue power.Canad. J. Math. 20 (1968), 79–87. Zbl 0171.01804, MR 0219675, 10.4153/CJM-1968-010-5 |
| Reference:
|
[5] C. S. Hönig: On a remarkable differential characterization of the functions that are Kurzweil-Henstock integrals.Seminário Brasileiro de Análise 33 (1991), 331–341. |
| Reference:
|
[6] G. J. Murphy: $C^{*}$-Algebras and Operator Theory.Academic Press, 1990. Zbl 0714.46041, MR 1074574 |
| Reference:
|
[7] Š. Schwabik: Abstract Perron-Stieltjes integral.Math. Bohem. 121 (1996), 425–447. Zbl 0879.28021, MR 1428144 |
| . |
