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

Lee, Tuo-Yeong

Mathematica Bohemica, Tome 130 (2005) no. 4, pp. 349-354.

Résumé

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 $(x,y) \in [0,1]^2$.

Détail

DOI : 10.21136/MB.2005.134207

Classification : 26A39, 28B05
Keywords: Henstock-Kurzweil integral; McShane integral