| Title:
|
Controlled convergence theorems for Henstock-Kurzweil-Pettis integral on $m$-dimensional compact intervals (English) |
| Author:
|
Kaliaj, Sokol B. |
| Author:
|
Tato, Agron D. |
| Author:
|
Gumeni, Fatmir D. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
62 |
| Issue:
|
1 |
| Year:
|
2012 |
| Pages:
|
243-255 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we use a generalized version of absolute continuity defined by J. Kurzweil, J. Jarník, Equiintegrability and controlled convergence of Perron-type integrable functions, Real Anal. Exch. 17 (1992), 110–139. By applying uniformly this generalized version of absolute continuity to the primitives of the Henstock-Kurzweil-Pettis integrable functions, we obtain controlled convergence theorems for the Henstock-Kurzweil-Pettis integral. First, we present a controlled convergence theorem for Henstock-Kurzweil-Pettis integral of functions defined on $m$-dimensional compact intervals of $\mathbb {R}^{m}$ and taking values in a Banach space. Then, we extend this theorem to complete locally convex topological vector spaces. (English) |
| Keyword:
|
Henstock-Kurzweil-Pettis integral |
| Keyword:
|
controlled convergence theorem |
| Keyword:
|
complete locally convex spaces |
| Keyword:
|
$m$-dimensional compact interval |
| MSC:
|
26A39 |
| MSC:
|
28B05 |
| MSC:
|
46G10 |
| idZBL:
|
Zbl 1249.28017 |
| idMR:
|
MR2899748 |
| DOI:
|
10.1007/s10587-012-0009-6 |
| . |
| Date available:
|
2012-03-05T07:29:11Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142054 |
| . |
| Reference:
|
[1] Cichoń, M.: Convergence theorems for the Henstock-Kurzweil-Pettis integral.Acta Math. Hung. 92 (2001), 75-82. Zbl 1001.26003, MR 1924251, 10.1023/A:1013756111769 |
| Reference:
|
[2] Piazza, L. Di, Musiał, K.: Characterizations of Kurzweil-Henstock-Pettis integrable functions.Stud. Math. 176 (2006), 159-176. MR 2264361, 10.4064/sm176-2-4 |
| Reference:
|
[3] Piazza, L. Di: Kurzweil-Henstock type integration on Banach spaces.Real Anal. Exch. 29 (2003-2004), 543-556. MR 2083796 |
| Reference:
|
[4] Fremlin, D. H.: Pointwise compact sets of measurable functions.Manuscr. Math. 15 (1975), 219-242. Zbl 0303.28006, MR 0372594, 10.1007/BF01168675 |
| Reference:
|
[5] Gordon, R. A.: The Integrals of Lebesgue, Denjoy, Perron, and Henstock.Graduate Studies in Mathematics. Vol. 4. Providence, AMS (1994), 395. Zbl 0807.26004, MR 1288751 |
| Reference:
|
[6] Guoju, Y., Tianqing, A.: On Henstock-Dunford and Henstock-Pettis integrals.Int. J. Math. Sci. 25 (2001), 467-478. MR 1823609, 10.1155/S0161171201002381 |
| Reference:
|
[7] Guoju, Y.: On the Henstock-Kurzweil-Dunford and Kurzweil-Henstock-Pettis integrals.Rocky Mt. J. Math. 39 (2009), 1233-1244. Zbl 1214.28009, MR 2524711, 10.1216/RMJ-2009-39-4-1233 |
| Reference:
|
[8] James, R.: Weak compactness and reflexivity.Isr. J. Math. 2 (1964), 101-119. Zbl 0127.32502, MR 0176310, 10.1007/BF02759950 |
| Reference:
|
[9] Kurzweil, J., Jarník, J.: Equiintegrability and controlled convergence of Perron-type integrable functions.Real Anal. Exch. 17 (1992), 110-139. Zbl 0754.26003, 10.2307/44152200 |
| Reference:
|
[10] Musiał, K.: Vitali and Lebesgue convergence theorems for Pettis integral in locally convex spaces.Atti Semin. Mat. Fis. Univ. Modena 35 (1987), 159-165. Zbl 0636.28005, MR 0922998 |
| Reference:
|
[11] Musiał, K.: Topics in the theory of Pettis integration.Rend. Ist. Math. Univ. Trieste 23 (1991), 177-262. Zbl 0798.46042, MR 1248654 |
| Reference:
|
[12] Musiał, K.: Pettis integral.Handbook of Measure Theory Vol. I and II E. Pap Amsterdam: North-Holland (2002), 531-586. Zbl 1043.28010, MR 1954622 |
| Reference:
|
[13] Schaefer, H. H.: Topological Vector Spaces.Graduate Texts in Mathematics. 3. 3rd printing corrected. New York-Heidelberg-Berlin: Springer-Verlag XI (1971), 294. Zbl 0217.16002, MR 0342978 |
| Reference:
|
[14] Schwabik, Š., Guoju, Y.: Topics in Banach Space Integration.Series in Real Analysis 10. Hackensack, NJ: World Scientific (2005), 312. MR 2167754 |
| . |
