| Title:
|
Banach-valued Henstock-Kurzweil integrable functions are McShane integrable on a portion (English) |
| Author:
|
Lee, Tuo-Yeong |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
130 |
| Issue:
|
4 |
| Year:
|
2005 |
| Pages:
|
349-354 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
It is shown that a Banach-valued Henstock-Kurzweil integrable function on an $m$-dimensional compact interval is McShane integrable on a portion of the interval. As a consequence, there exist a non-Perron integrable function $f \: [0,1]^2 \longrightarrow {\mathbb{R}}$ and a continuous function $F \: [0,1]^2 \longrightarrow {\mathbb{R}}$ such that \[ (¶) \int _0^x \bigg \lbrace (¶) \int _0^yf(u,v) \mathrm{d}v \bigg \rbrace \mathrm{d}u = (¶) \int _0^y \bigg \lbrace (¶) \int _0^xf(u,v) \mathrm{d}u \bigg \rbrace \mathrm{d}v = F(x,y) \] for all $(x,y) \in [0,1]^2$. (English) |
| Keyword:
|
Henstock-Kurzweil integral |
| Keyword:
|
McShane integral |
| MSC:
|
26A39 |
| MSC:
|
28B05 |
| idZBL:
|
Zbl 1112.28008 |
| idMR:
|
MR2182381 |
| DOI:
|
10.21136/MB.2005.134207 |
| . |
| Date available:
|
2009-09-24T22:22:12Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134207 |
| . |
| Reference:
|
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| Reference:
|
[2] D. H. Fremlin: The Henstock and McShane integrals of vector-valued functions.Illinois J. Math. 38 (1994), 471–479. Zbl 0797.28006, MR 1269699, 10.1215/ijm/1255986726 |
| Reference:
|
[3] D. H. Fremlin, J. Mendoza: The integration of vector-valued functions.Illinois J. Math. 38 (1994), 127–147. MR 1245838, 10.1215/ijm/1255986891 |
| Reference:
|
[4] R. A. Gordon: The McShane integral of Banach-valued functions.Illinois J. Math. 34 (1990), 557–567. Zbl 0685.28003, MR 1053562, 10.1215/ijm/1255988170 |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
[8] Peng-Yee Lee, R. Výborný: The integral, An Easy Approach after Kurzweil and Henstock.Australian Mathematical Society Lecture Series 14 (Cambridge University Press, 2000). MR 1756319 |
| Reference:
|
[9] Tuo-Yeong Lee: A full descriptive definition of the Henstock-Kurzweil integral in the Euclidean space.Proc. London Math. Soc. 87 (2003), 677–700. Zbl 1047.26006, MR 2005879 |
| Reference:
|
[10] Tuo-Yeong Lee: Some full characterizations of the strong McShane integral.Math. Bohem. 129 (2004), 305–312. Zbl 1080.26006, MR 2092716 |
| Reference:
|
[11] Š. Schwabik, Guoju Ye: The McShane and the Pettis integral of Banach space-valued functions defined on ${\mathbb{R}}^m$.Illinois J. Math. 46 (2002), 1125–1144. MR 1988254, 10.1215/ijm/1258138470 |
| Reference:
|
[12] G. P. Tolstov: On the curvilinear and iterated integral.Trudy Mat. Inst. Steklov. 35 (1950), 102 pp. (Russian) MR 0044612 |
| . |
