Geodesic
Banach-valued Henstock-Kurzweil integrable functions are McShane integrable on a portion
Mathematica Bohemica, Tome 130 (2005) no. 4, pp. 349-354

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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 $f \: [0,1]^2 \longrightarrow {\mathbb{R}}$ 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 $(x,y) \in [0,1]^2$.
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 $(x,y) \in [0,1]^2$.
DOI : 10.21136/MB.2005.134207
Classification : 26A39, 28B05
Keywords: Henstock-Kurzweil integral; McShane integral 
Lee, Tuo-Yeong. Banach-valued Henstock-Kurzweil integrable functions are McShane integrable on a portion. Mathematica Bohemica, Tome 130 (2005) no. 4, pp. 349-354. doi: 10.21136/MB.2005.134207
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