On Birkhoff integrability for scalar functions and vector measures (Q1027750)

\textit{R. G. Bartle, N. Dunford} and \textit{J. Schwartz} [Can. J. Math. 7, 289--305 (1955; Zbl 0068.09301)] developed an integral for scalar functions with respect to vector measures as an analogue of the Riesz Representation Theorem for compact operators on Banach spaces of continuous functions on a compact Hausdorff space. In this paper this integral is related to the S*-integral studied intensively by \textit{I. Dobrakov} [Czech. Math. J. 38(113), No. 3, 434--449 (1988; Zbl 0674.28003)]. It is shown that a scalar function \(f\) is B-D-S integrable with respect to a vector measure \(m\) iff it is S*-integrable, which in turn is also equivalent to saying that \(f\) is measurable and there is a partition of the domain into countably many measurable sets such that, for every finer, countable, partition \(\{A_n\}_n\) and any choice of points \(\omega_n \in A_n\), the series \(\sum_n f(\omega_n)m\{A_n\}\) is unconditionally convergent.