Positive Representations of L1 of a Vector Measure

Abstract

We characterize the vector measures n on a Banach lattice such that the map |·|dn provides a quasi-norm which is equivalent to the canonical norm · n of the space L1(n) of integrable functions as an specific type of transformations of positive vector measures that we call cone-open transformations. We also prove that a vector measure m on a Banach space X constructed as a cone-open transformation of a positive vector measure can be considered in some sense as a positive vector measure by defining a new order on X.