Compactness in quasi-Banach function spaces with applications to $$L^1$$ of the semivariation of a vector measure

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We characterize the relatively compact subsets of the order continuous part $$E_a$$ 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 $$E_a$$ as a compact subset of a smaller quasi-Banach Orlicz space $$E^\varPhi .$$ Our results generalize the previous known results for the Banach function spaces $$L^1(m)$$ and $$L^1_w(m)$$ associated to a vector measure m and moreover they can also be applied to the quasi-Banach function space $$L^1\left( \Vert m \Vert \right) $$ associated to the semivariation of m.