Orlicz spaces associated to a quasi-Banach function space. Applications to vector measures and interpolation

Abstract

We characterize the relatively compact subsets of $L^1\left(\| m \| \right),$ the quasi-Banach function space associated to the semivariation of a given vector measure $m$ showing that the strong connection between compactness, uniform absolute continuity, uniform integrability, almost order boundedness and L-weak compactness that appears in the classic setting of Lebesgue spaces remains almost invariant in this new context of the Choquet integration. Also we present a de la Vall\'ee-Poussin type theorem in the context of these spaces $L^1\left(\|m\|\right)$ that allows us to locate each compact subset of $L^1\left(\|m\|\right)$ as a compact subset of a smaller quasi-Banach Orlicz space $L^\Phi\left(\|m\|\right)$ associated to the semivariation of the measure $m.$