Compactness in quasi-Banach function spaces with applications to L 1 of the semivariation of a vector measure Ricardo del Campo 1 · Antonio Fernández 2 · Fernando Mayoral 2 · Francisco Naranjo 2 Abstract 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 compact- ness 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 Vallée–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 Φ . 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 (m) associated to the semivariation of m. Keywords Orlicz spaces · Vector measure · Semivariation · Uniform integrability · Uniform absolute continuity · Compactness · De la Vallée–Poussin’s theorem Mathematics Subject Classification 46G10 · 46E30 This research has been partially supported by La Junta de Andalucía (Spain) under the Grant FQM-133. B Antonio Fernández afcarrion@us.es Ricardo del Campo rcampo@us.es Fernando Mayoral mayoral@us.es Francisco Naranjo naranjo@us.es 1 Dpto. Matemática Aplicada I, Universidad de Sevilla, EUITA, Ctra. de Utrera Km. 1, 41013 Sevilla, Spain 2 Dpto. Matemática Aplicada II, Escuela Técnica Superior de Ingeniería, Camino de los Descubrimientos, s/n, 41092 Sevilla, Spain