Abstract
Given a vector measure ν with values in a Banach space X, we consider the space L 1(ν) of real functions which are integrable with respect to ν. We prove that every order continuous Banach function space Y continuously contained in L 1(ν) is generated via a certain positive map ϱ related to ν and defined on X* x M, where X* is the dual space of X and M the space of measurable functions. This procedure provides a way of defining Orlicz spaces with respect to the vector measure ν.
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