The space of integrable functions with respect to a vector measure which is already interesting by itself, finds applications in important problems as the integral representation and the study of the optimal domain of linear operators or the representation of abstract Banach lattices as spaces of functions. Classical vector measures m are considered to be defined on a sigma-algebra and with values in a Banach space, and the corresponding spaces L1(m) and L1w(m) of integrable and weakly integrable functions respectively have been studied in depth by many authors being their behavior well understood.
However, this framework is not enough, for instance, for applications to operators on spaces which do not contain the characteristic functions of sets or Banach lattices without weak unit. These cases require m to be defined on a weaker structure than sigma-algebra, namely, a delta-ring. Furthermore, integration with respect to vector measures defined on delta-rings is the natural vector valued generalization of the case of integration with respect to positive sigma-finite measures, which is not included in the frame of vector measures on sigma-algebras if such a measure is non finite.
Consequently, vector measures defined on a delta-ring also play an important role and deserve to be studied together with their spaces of integrable functions. The integration theory with respect to these vector measures m is due to Lewis and Masani and Niemi.
In this work we are mainly interested in providing the properties which guarantee the representation of a Banach lattice by means of an space of integrable functions. Chapter 4 is devoted to this aim and contains our main result (Theorem 4.1.7).
Some interesting questions appeared when we tried to solve this abstract representation problem. The analytic properties of a vector measure m defined on a delta-ring are directly related to the lattice properties of the space L1(m). It will be also the aim of this work to study the effect of certain properties of m on the lattice properties of the space L1w(m) and Chapter 2 is devoted to develop our results in this context. Concretely, we analyze order continuity, order density and Fatou type properties for L1w(m). We will see that the behavior of L1w(m) differs from the case in which m is defined on a sigma-algebra whenever m does not satisfies certain local sigma-finiteness property.
In Chapter 3 we study the lattice properties of the Banach lattices Lp(m) and Lpw(m) for a vector measure m defined on a delta-ring. The relation between these two spaces, the study of the continuity and some kind of compactness properties of certain multiplication operators between different spaces Lp(m) and/or Lqw(m) play a fundamental role.