A sufficient and necessary condition for the McShane integral (Q2720062)

In this paper, several versions of convergence theorems for Banach-valued multiple McShane integrals are given. The convergence theorems are in terms of uniform integrability. Let \(f_n: [a,b]\to X\), \(n= 1,2,\dots\), where \(X\) is a Banach space. Let \(X^*\) be its conjugate space. The sequence \(\{f_n\}\) is said to be uniformly integrable on \([a,b]\) if \(\{x^* f_n: x^*\in X^*, n= 1,2,\dots\}\) is uniformly absolutely continuous on \([a,b]\). The author has also pointed out that if \(f\) is McShane integrable, then there exists a sequence \(\{f_n\}\) of simple functions such that for each \(x^*\in X^*\), \(x^* f_n(t)\to x^* f(t)\) a.e. and \(\{f_n\}\) is uniformly integrable.