Approximation of the Bochner integral by means of Riemann sums (Q2706889)

Let \((T,\mu)\) be a measure space with finite measure \(\mu\) and \(E\) a Banach space. Suppose \(f: T\to E\). An integral of \(f\) is defined by Riemann sums, which are generated by partitions of \(T\). The integral of \(f\) is the Moore-Smith limit of Riemann sums. The directed set is induced by partitions. The language used in this paper is different from that used by Henstock, Kurzweil and their school. The author has studied the relation between the above integral and the Bochner integral in detail. The integral defined is more general than the Bochner integral. Sufficient conditions are given under which two integrals are equivalent. Most of the results are proved without topology imposed on the measure space \(T\).