## The Henstock and McShane integrals of vector-valued functions.(English)Zbl 0797.28006

Let $$X$$ be a Banach space and $$\phi: [0,1]\to X$$ a function. Combining ideas from R. M. McLeod [The generalized Riemann integral (1980; Zbl 0486.26005)] and 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 $\left\| w- \sum^ n_{i=1} (a_ i- a_{i-1})\phi(t_ i)\right\|\leq \varepsilon,$ whenever $$0= a_ 0$$, $$a_ n= 1$$ and $$t_ i- \delta(t_ i)\leq a_{i-1}\leq t_ i\leq a_ i\leq t_ i+ \delta(t_ i)$$ for every $$i\leq n$$; 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 $\left\| w-\sum^ n_{i=1} (a_ i- a_{i-1})\phi(t_ i)\right\|\leq \varepsilon,$ whenever $$0= a_ 0$$, $$a_ n=1$$ and $$t_ 1,\dots,t_ n\in [0,1]$$ are such that $$t_ i- \delta(t_ i)\leq a_{i-1}\leq a_ i\leq t_ i+ \delta(t_ i)$$ for every $$i\leq n$$ – the difference being that in the latter case the $$t_ i$$ are no longer restricted to the intervals $$[a_{i-1},a_ i]$$. It has long been known that in the case $$X=\mathbb{R}$$ 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 $$E\subseteq [0,1]$$.

### MSC:

 28B05 Vector-valued set functions, measures and integrals

### Citations:

Zbl 0486.26005; Zbl 0714.28008