| Title:
|
McShane equi-integrability and Vitali’s convergence theorem (English) |
| Author:
|
Kurzweil, Jaroslav |
| Author:
|
Schwabik, Štefan |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
129 |
| Issue:
|
2 |
| Year:
|
2004 |
| Pages:
|
141-157 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The McShane integral of functions $f\:I\rightarrow \mathbb{R}$ defined on an $m$-dimensional interval $I$ is considered in the paper. This integral is known to be equivalent to the Lebesgue integral for which the Vitali convergence theorem holds. For McShane integrable sequences of functions a convergence theorem based on the concept of equi-integrability is proved and it is shown that this theorem is equivalent to the Vitali convergence theorem. (English) |
| Keyword:
|
McShane integral |
| Keyword:
|
Vitali convergence theorem |
| Keyword:
|
equi-integrability |
| MSC:
|
26A39 |
| MSC:
|
26B99 |
| idZBL:
|
Zbl 1051.26012 |
| idMR:
|
MR2073511 |
| DOI:
|
10.21136/MB.2004.133903 |
| . |
| Date available:
|
2009-09-24T22:13:45Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/133903 |
| . |
| Reference:
|
[1] R. A. Gordon: The integrals of Lebesgue, Denjoy, Perron, and Henstock.American Mathematical Society, Providence, RI, 1994. Zbl 0807.26004, MR 1288751 |
| Reference:
|
[2] E. J. McShane: A Riemann-type integral that includes Lebesgue-Stieltjes, Bochner and stochastic integrals.Mem. Am. Math. Soc. 88 (1969). Zbl 0188.35702, MR 0265527 |
| Reference:
|
[3] I. P. Natanson: Theory of Functions of a Real Variable.Frederick Ungar, New York, 1955, 1960. MR 0067952 |
| Reference:
|
[4] Š. Schwabik, Ye Guoju: On the strong McShane integral of functions with values in a Banach space.Czechoslovak Math. J. 51 (2001), 819–828. MR 1864044, 10.1023/A:1013721114330 |
| Reference:
|
[5] J. Kurzweil, Š. Schwabik: On McShane integrability of Banach space-valued functions.(to appear). MR 2083811 |
| . |
