| Title:
|
Variational Henstock integrability of Banach space valued functions (English) |
| Author:
|
Di Piazza, Luisa |
| Author:
|
Marraffa, Valeria |
| Author:
|
Musiał, Kazimierz |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
141 |
| Issue:
|
2 |
| Year:
|
2016 |
| Pages:
|
287-296 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study the integrability of Banach space valued strongly measurable functions defined on $[0,1]$. In the case of functions $f$ given by $\sum \nolimits _{n=1}^{\infty } x_n\chi _{E_n}$, where $x_n $ are points of a Banach space and the sets $E_n$ are Lebesgue measurable and pairwise disjoint subsets of $[0,1]$, there are well known characterizations for Bochner and Pettis integrability of $f$. The function $f$ is Bochner integrable if and only if the series $\sum \nolimits _{n=1}^{\infty }x_n|E_n|$ is absolutely convergent. Unconditional convergence of the series is equivalent to Pettis integrability of $f$. In this paper we give some conditions for variational Henstock integrability of a certain class of such functions. (English) |
| Keyword:
|
Kurzweil-Henstock integral |
| Keyword:
|
variational Henstock integral |
| Keyword:
|
Pettis integral |
| MSC:
|
26A39 |
| idZBL:
|
Zbl 06587866 |
| idMR:
|
MR3499788 |
| DOI:
|
10.21136/MB.2016.19 |
| . |
| Date available:
|
2016-05-19T09:11:08Z |
| Last updated:
|
2020-07-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145716 |
| . |
| Reference:
|
[1] Bongiorno, B., Piazza, L. Di, Musiał, K.: Kurzweil-Henstock and Kurzweil-Henstock-Pettis integrability of strongly measurable functions.Math. Bohem. 131 (2006), 211-223. Zbl 1112.26015, MR 2242846 |
| Reference:
|
[2] J. Diestel, J. J. Uhl, Jr.: Vector Measures.Mathematical Surveys 15 American Mathematical Society 13, Providence (1977). Zbl 0369.46039, MR 0453964 |
| Reference:
|
[3] Marraffa, V.: A characterization of strongly measurable Kurzweil-Henstock integrable functions and weakly continuous operators.J. Math. Anal. Appl. 340 (2008), 1171-1179. Zbl 1141.46021, MR 2390920, 10.1016/j.jmaa.2007.09.033 |
| Reference:
|
[4] Marraffa, V.: Strongly measurable Kurzweil-Henstock type integrable functions and series.Quaest. Math. 31 (2008), 379-386. Zbl 1177.28030, MR 2527448, 10.2989/QM.2008.31.4.6.610 |
| Reference:
|
[5] Musia{ł}, K.: Topics in the theory of Pettis integration.School on Measure Theory and Real Analysis, Grado, 1991 Rend. Ist. Mat. Univ. Trieste 23 (1993), 177-262. MR 1248654 |
| . |
