Compactness in quasi-Banach function spaces with applications to L1 of the semivariation of a vector measure

We characterize the relatively compact subsets of the order continuous part Ea of a quasi-Banach function space E showing that the strong connection between compactness, uniform absolute continuity, uniform integrability, almost order boundedness and L-weak compactness that appears in the classical setting of Lebesgue spaces remains almost invariant in this new context under mild assumptions. We also present a de la Vallee-Poussin type theorem in this context that allows us to locate each compact subset of Ea as a compact subset of a smaller quasi-Banach Orlicz space E phi. Our results generalize the previous known results for the Banach function spaces L-1(m)and Lw1(m) associated to a vector measure m and moreover they can also be applied to the quasi-Banach function space L-1 associated to the semivariation of m.