Banach-valued Henstock-Kurzweil integrable functions are McShane integrable on a portion. (English) Zbl 1112.28008

Lee, Tuo-Yeong

Math. Bohem. 130, No. 4, 349-354 (2005).

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
\[ (\text{P}) \int _0^x \bigg \{ (\text{P}) \int _0^yf(u,v)\, dv \bigg \}\, du = (\text{P} ) \int _0^y \bigg \{ (\text{P}) \int _0^xf(u,v)\, du \bigg \}\, dv = F(x,y) \]
for all \((x,y) \in [0,1]^2\).

MSC:

28B05	Vector-valued set functions, measures and integrals
26A39	Denjoy and Perron integrals, other special integrals

Keywords:

Henstock-Kurzweil integral; McShane integral