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.

中文翻译：

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拟巴拿赫函数空间中的紧致性以及向量测度半变分的 $$L^1$$L1 的应用

我们刻画了准 Banach 函数空间 E 的阶连续部分 $$E_a$$ 的相对紧子集，表明紧致性、一致绝对连续性、一致可积性、几乎有序有界性和 L-弱紧致性之间的强联系出现在在温和的假设下，勒贝格空间的经典设置在这种新环境中几乎保持不变。在此上下文中，我们还提出了 de la Vallee–Poussin 类型定理，它允许我们将 $$E_a$$ 的每个紧子集定位为较小的准 Banach Orlicz 空间 $$E^\varPhi 的紧子集。