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Keywords:
Henstock-Kurzweil integral; McShane integral
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 $(x,y) \in [0,1]^2$.
References:
[1] Z. Buczolich: Henstock integrable functions are Lebesgue integrable on a portion. Proc. Amer. Math. Soc. 111 (1991), 127–129. DOI 10.1090/S0002-9939-1991-1034883-6 | MR 1034883 | Zbl 0732.26011
[2] D. H. Fremlin: The Henstock and McShane integrals of vector-valued functions. Illinois J. Math. 38 (1994), 471–479. MR 1269699 | Zbl 0797.28006
[3] D. H. Fremlin, J. Mendoza: The integration of vector-valued functions. Illinois J. Math. 38 (1994), 127–147. MR 1245838
[4] R. A. Gordon: The McShane integral of Banach-valued functions. Illinois J. Math. 34 (1990), 557–567. MR 1053562 | Zbl 0685.28003
[5] R. A. Gordon: The Integrals of Lebesgue, Denjoy, Perron, and Henstock. Graduate Studies in Mathematics Volume 4, AMS, 1994. MR 1288751 | Zbl 0807.26004
[6] K. Karták: Zur Theorie des mehrdimensionalen Integrals. Časopis Pěst. Mat. 80 (1955), 400–414. (Czech) MR 0089249
[7] J. Kurzweil, J. Jarník: Equi-integrability and controlled convergence of Perron-type integrable functions. Real Anal. Exchange 17 (1991/92), 110–139. MR 1147361
[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
[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. MR 2005879 | Zbl 1047.26006
[10] Tuo-Yeong Lee: Some full characterizations of the strong McShane integral. Math. Bohem. 129 (2004), 305–312. MR 2092716 | Zbl 1080.26006
[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
[12] G. P. Tolstov: On the curvilinear and iterated integral. Trudy Mat. Inst. Steklov. 35 (1950), 102 pp. (Russian) MR 0044612
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