A Riemann sums method in the theory of vector integration (Q2906224)

Consider the following vector-valued integral extending Riemann-type integration to Banach-space-valued functions. Fix the interval \([0,1]\). A partition \(\mathcal P\) is a finite collection of non-overlapping compact intervals covering \([0,1]\). Then \(\operatorname{mesh}(\mathcal P)\) is the supremum of the lengths of the intervals in \(\mathcal P\).NEWLINENEWLINEA sequence \(\overline{t}=\{t_n\}\) in \([0,1]\) is a trajectory if it is a sequence of distinct points in \([0,1]\). If \(J \subseteq [0,1]\) is an interval, \(n(J)\) is the first index such that \(t_{n(J)} \in J\).NEWLINENEWLINEA Banach-space-valued function \(f:[0,1] \to X\) is Darji-Evans integrable with respect to \(\overline{t}\) if there exists a vector \(A \in X\) such that, for each \(\varepsilon >0\), there is a constant \(\delta >0\) with NEWLINE\[NEWLINE \big\| \sum_{J \in \mathcal P} f(t_{n(J)}) |J| - A \big\| < \varepsilon NEWLINE\]NEWLINE for each partition \(\mathcal P\) with \(\operatorname{mesh}(\mathcal P) < \delta\).NEWLINENEWLINEIn this paper, the extension of the Darji-Evans integration to functions taking values in an infinite-dimensional Banach space is considered. The author proves that this extension is possible for the case of Bochner integrable functions. Theorem 3 establishes that, if \(f:[0,1] \to X\) is Bochner integrable, then there is a trajectory such that \(f\) is Darji-Evans integrable with respect to \(\overline t\) and the Bochner integral coincides with the Darji-Evans integral associated to \(\overline t\). However, as it is also shown in this paper, this is not possible for the other usual integration procedures for vector-valued functions, the Pettis integral, the Birkhoff integral and the McShane integral: a counterexample involving an \(\ell^2\)-valued function is presented.