Banach-valued Henstock-Kurzweil integrable functions are McShane integrable on a portion

Tuo-Yeong Lee

Mathematica Bohemica (2005)

  • Volume: 130, Issue: 4, page 349-354
  • ISSN: 0862-7959

Abstract

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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 and a continuous function F [ 0 , 1 ] 2 such that ( ) 0 x ( ) 0 y f ( u , v ) d v d u = ( ) 0 y ( ) 0 x f ( u , v ) d u d v = F ( x , y ) for all ( x , y ) [ 0 , 1 ] 2 .

How to cite

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Lee, Tuo-Yeong. "Banach-valued Henstock-Kurzweil integrable functions are McShane integrable on a portion." Mathematica Bohemica 130.4 (2005): 349-354. <http://eudml.org/doc/249574>.

@article{Lee2005,
abstract = {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 $(x,y) \in [0,1]^2$.},
author = {Lee, Tuo-Yeong},
journal = {Mathematica Bohemica},
keywords = {Henstock-Kurzweil integral; McShane integral; Henstock-Kurzweil integral; McShane integral},
language = {eng},
number = {4},
pages = {349-354},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Banach-valued Henstock-Kurzweil integrable functions are McShane integrable on a portion},
url = {http://eudml.org/doc/249574},
volume = {130},
year = {2005},
}

TY - JOUR
AU - Lee, Tuo-Yeong
TI - Banach-valued Henstock-Kurzweil integrable functions are McShane integrable on a portion
JO - Mathematica Bohemica
PY - 2005
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 130
IS - 4
SP - 349
EP - 354
AB - 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 $(x,y) \in [0,1]^2$.
LA - eng
KW - Henstock-Kurzweil integral; McShane integral; Henstock-Kurzweil integral; McShane integral
UR - http://eudml.org/doc/249574
ER -

References

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  8. The integral, An Easy Approach after Kurzweil and Henstock, Australian Mathematical Society Lecture Series 14 (Cambridge University Press, 2000). MR1756319
  9. A full descriptive definition of the Henstock-Kurzweil integral in the Euclidean space, Proc. London Math. Soc. 87 (2003), 677–700. (2003) Zbl1047.26006MR2005879
  10. Some full characterizations of the strong McShane integral, Math. Bohem. 129 (2004), 305–312. (2004) Zbl1080.26006MR2092716
  11. 10.1215/ijm/1258138470, Illinois J. Math. 46 (2002), 1125–1144. (2002) MR1988254DOI10.1215/ijm/1258138470
  12. On the curvilinear and iterated integral, Trudy Mat. Inst. Steklov. 35 (1950), 102 pp. (Russian) (1950) MR0044612

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