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

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
(\P) \int_0^x \bigg\{ (\P) \int_0^yf(u,v) \dd v \bigg\} \dd u = (\P) \int_0^y \bigg\{ (\P) \int_0^xf(u,v) \dd u \bigg\} \dd v = F(x,y)
for all $(x,y) \in[0,1]^2$.

Keywords: Henstock-Kurzweil integral, McShane integral

Classification (MSC2000): 28B05, 26A39

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