Integration with respect to orthogonally scattered measures

The aim of this paper is to compare two classical definitions of integral with respect to a vector measure with values in a Hilbert space, namely the Bochner integral, which is defined as a limit of integrals of a defining sequence of simple functions, and the De Giorgi-Letta integral (monotone integral). The use of an orthogonally scattered dilation provides a better insight of these two kinds of approach, in comparison to what known in the literature for Banach or LCTVS-valued finitely additive measures, and it also yields a sufficient condition for the equivalence between these two types of integration.