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Zbl 0797.28006
Fremlin, D.H.
The Henstock and McShane integrals of vector-valued functions. (English)
[J] Ill. J. Math. 38, No.3, 471-479 (1994). ISSN 0019-2082

Let $X$ be a Banach space and $\phi: [0,1]\to X$ a function. Combining ideas from {\it R. M. McLeod} [The generalized Riemann integral (1980; Zbl 0486.26005)] and {\it R. A. Gordon} [Ill. J. Math. 34, No. 3, 557-567 (1990; Zbl 0714.28008)] I say that $\phi$ is Henstock integrable, with integral $w\in X$, if for every $\varepsilon> 0$ there is a function $\delta: [0,1]\to ]0,\infty[$ such that $$\left\Vert w- \sum\sp n\sb{i=1} (a\sb i- a\sb{i-1})\phi(t\sb i)\right\Vert\le \varepsilon,$$ whenever $0= a\sb 0$, $a\sb n= 1$ and $t\sb i- \delta(t\sb i)\le a\sb{i-1}\le t\sb i\le a\sb i\le t\sb i+ \delta(t\sb i)$ for every $i\le n$; and that $\phi$ is McShane integrable, with integral $w$, if for every $\varepsilon> 0$ there is a function $\delta: [0,1]\to ]0,\infty[$ such that $$\left\Vert w-\sum\sp n\sb{i=1} (a\sb i- a\sb{i-1})\phi(t\sb i)\right\Vert\le \varepsilon,$$ whenever $0= a\sb 0$, $a\sb n=1$ and $t\sb 1,\dots,t\sb n\in [0,1]$ are such that $t\sb i- \delta(t\sb i)\le a\sb{i-1}\le a\sb i\le t\sb i+ \delta(t\sb i)$ for every $i\le n$ -- the difference being that in the latter case the $t\sb i$ are no longer restricted to the intervals $[a\sb{i-1},a\sb i]$. It has long been known that in the case $X=\bbfR$ the McShane integral agrees with the Lebesgue integral, while the Henstock integral is a proper extension. In this paper I show that, for a general Banach space $X$ and any function $\phi: [0,1]\to X$, the following are equivalent: (i) $\phi$ is McShane integrable; (ii) $\phi$ is Henstock integrable and Pettis integrable; (iii) $\phi\times \chi E$ is Henstock integrable for every measurable set $E\subseteq [0,1]$.
[D.H.Fremlin (Colchester)]

MSC 2000:: *28B05 Vector-valued set functions etc. (measure theory)

Keywords: Pettis integral; vector-valued functions; McShane integral; Henstock integral

Citations: Zbl 0486.26005; Zbl 0714.28008

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