Representation Theorems of Abstract Banach Lattices 12+

📓 The space of integrable functions with respect to a vector measure 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 are defined on a σ-algebra and with values in a Banach space. However, this framework is not enough for applications to operators on spaces which do not contain the characteristic functions of sets or Banach lattices without weak unit. These cases require the vector measure to be defined on a δ-ring. 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 with respect to a vector measure on a δ-ring. The analytic properties of a vector measure are directly related to the lattice properties of the space L1. It will be also the aim of this work to study the effect of certain properties of the vector measure on the lattice structure of the space L1w. We also study the spaces Lp, Lpw for a vector measure on a δ-ring and the corresponding representation theorems by means of these spaces.