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A Komlós theorem for abstract Banach lattices of measurable functions. (English) Zbl 1228.46024

In this interesting paper, the authors consider a \(\sigma\)-ring \(\mathcal R\) and a vector measure \(\nu:\mathcal R \to X,\) where \(X\) is a Banach space. Let \(E\) be an ideal of \(L^{1}(\nu).\) Then they prove that \(E\) has the weak \(\sigma\)-Fatou property if and only if it has the Komlós property, that is, if for every bounded sequence \((f_{n})_{n}\) in \(E,\) there exists a subsequence \((f_{n_{k}})_{k}\) and a function \(f\in E\) such that for any further subsequence \((h_{j})_{j}\) of \((f_{n_{k}})_{k},\) the series \(\frac{1}{n}\sum_{j=1}^{n}h_{j}\) converges \(\mu\)-a.e. to \(f\).
Moreover, they give an example of a Banach lattice of measurable functions that is Fatou but does not satisfy the Komlós theorem.

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B42 Banach lattices
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References:

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