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Compactness in quasi-Banach function spaces with applications to \(L^1\) of the semivariation of a vector measure. (English) Zbl 1447.46029

Relatively compact subsets of the order continuous part \(E_a\) of a quasi-Banach function space \(E\) are studied. The authors use the general representation theory that allows one to write a broad class of quasi-Banach function spaces as spaces of integrable functions with respect to a vector measure on a sigma-algebra. In particular, the space \(L^1(\|m\|)\) for a vector meaure \(m\) is the quasi-Banach space of the functions that are integrable (in the sense of Choquet) with respect to its semivariation \(\|m\|,\) and can be used for representing more general classes of quasi-Banach function spaces. Relative compactness of subsets is studied in these spaces, providing similar characterizations as in the case of classical Lebesgue spaces, in relation to the usual properties related to compactness, such as uniform absolute continuity, uniform integrability, almost order boundedness, and \(L\)-weak compactness. In this sense, Corollary 3.3 gives a complete characterization, establishing that for a vector measure \(m,\) a subset \(H\) of a space \(L^1(\|m\|)\) is relatively compact if and only if it satisfies any of these properties and is also relatively compact in the space of measurable functions \(L^0(m)\).
With the aim of locating relatively compact subsets, the authors also prove a de la Vallée-Poussin-type theorem (Theorem 5.1), which allows them to locate each compact subset of \(E_a\) as a compact subset of a smaller quasi-Banach Orlicz space \(E^\phi\). This result is given in:
Corollary 5.3. Let \(m : \Sigma \to X\) be a vector measure and \(H \subset L^0(m).\)
(1)
\(H\) is relatively compact in \(L^1(\|m\|)\) if and only if there exists an \(N\)-function \(\phi \in \Delta_2\) such that \(H\) is relatively compact in \(L^\phi(\|m\|)\).
(2)
\(H\) is \(L\)-weakly compact in \(L^1(\|m\|)\) if and only if there exists an \(N\)-function \(\phi \in \Delta_2\) such that \(H\) is \(L\)-weakly compact in \(L^\phi(\|m\|)\).
Theorem 5.1 is interesting in itself. It seems to be the first completely general de la Vallée-Poussin-type result for quasi-Banach function spaces. Other interesting results on Orlicz-type spaces constructed from a quasi-Banach function space by an Orlicz function – inclusions, characterization of the order continuous part, etc. – are also shown in the paper.

MSC:

46G10 Vector-valued measures and integration
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B50 Compactness in Banach (or normed) spaces
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