The Henstock and McShane integrals of vector-valued functions
Let \(X\) be a Banach space and \(\phi: [0,1]\to X\) a function. Combining ideas from \textit{R. M. McLeod} [The generalized Riemann integral (1980; Zbl 0486.26005)] and \textit{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 \(E\subseteq [0,1]\).
scientific article; zbMATH DE number 599243