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

top
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

top

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

top
  1. 10.1090/S0002-9939-1991-1034883-6, Proc. Amer. Math. Soc. 111 (1991), 127–129. (1991) Zbl0732.26011MR1034883DOI10.1090/S0002-9939-1991-1034883-6
  2. The Henstock and McShane integrals of vector-valued functions, Illinois J. Math. 38 (1994), 471–479. (1994) Zbl0797.28006MR1269699
  3. The integration of vector-valued functions, Illinois J. Math. 38 (1994), 127–147. (1994) MR1245838
  4. The McShane integral of Banach-valued functions, Illinois J. Math. 34 (1990), 557–567. (1990) Zbl0685.28003MR1053562
  5. The Integrals of Lebesgue, Denjoy, Perron, and Henstock, Graduate Studies in Mathematics Volume 4, AMS, 1994. (1994) Zbl0807.26004MR1288751
  6. Zur Theorie des mehrdimensionalen Integrals, Časopis Pěst. Mat. 80 (1955), 400–414. (Czech) (1955) MR0089249
  7. Equi-integrability and controlled convergence of Perron-type integrable functions, Real Anal. Exchange 17 (1991/92), 110–139. (1991/92) MR1147361
  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. The McShane and the Pettis integral of Banach space-valued functions defined on m , Illinois J. Math. 46 (2002), 1125–1144. (2002) MR1988254
  12. On the curvilinear and iterated integral, Trudy Mat. Inst. Steklov. 35 (1950), 102 pp. (Russian) (1950) MR0044612

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.