J. Aust. Math. Soc.  82 (2007), 85-109
Spaces of vector functions that are integrable with respect to vector measures

José Rodríguez
  Departamento de Matemáticas
  Universidad de Murcia
  30.100 Espinardo

We study the normed spaces of (equivalence classes of) Banach space-valued functions that are Dobrakov, S* or McShane integrable with respect to a Banach space-valued measure, where the norm is the natural one given by the total semivariation of the indefinite integral. We show that simple functions are dense in these spaces. As a consequence we characterize when the corresponding indefinite integrals have norm relatively compact range. On the other hand, we also determine when these spaces are ultrabornological. Our results apply to conclude, for instance, that the spaces of Birkhoff (respectively McShane) integrable functions defined on a complete (respectively quasi-Radon) probability space, endowed with the Pettis norm, are ultrabornological.
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