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A general diagonal process and compactness in spaces of measurable functions. (English) Zbl 0876.04002

Let \(\Omega\) be the set of all strictly increasing functions from \({\mathbb N}\) to \({\mathbb N}\). Let \(\Omega^{\alpha}=\{\beta\in\Omega:\alpha\leq\beta\}\), where \(\alpha\leq\beta\), with \(\alpha,\beta\in\Omega\), means that \(\beta\) is eventually a subsequence of \(\alpha\). Let \(U\) be a fixed set and \(\perp\), \(*\) be binary relations on \(U\). A set \(V\subseteq U\) is a \(\perp\)-set if \(v\perp w\) for all \(v\neq w\) in \(V\). We suppose that all \(\perp\)-sets are countable and \(*\) is \(\perp\)-compatible, i.e.if \(V\) is a \(\perp\)-set, \(u\in V\), and \(v*u\), then \(v\notin V\setminus\{u\}\) and \((V\setminus\{u\})\cup\{v\}\) is a \(\perp\)-set. The author proves a subsequence selection principle which generalizes the diagonal principle (often used in analysis): Assume that \({\mathcal A}\) is a property on \(U\times\Omega\) such that for every \((u,\alpha)\in U\times\Omega\) there are \(\beta\in\Omega^\alpha\) and \(v\in U\) with \(v*u\) for which \({\mathcal A}(v,\beta)\) holds true, and whenever \({\mathcal A}(w,\gamma)\) holds true, then \({\mathcal A}(w,\delta)\) holds true for all \(\delta\in\Omega^\gamma\). Then there exists a maximal \(\perp\)-set \(V\) and some \(\psi\in\Omega\) such that \({\mathcal A}(v,\psi)\) holds true for all \(v\in V\). In measure theory this principle is used to obtain some criteria for relative compactness and for sequential convergence with respect to the topology of convergence in measure in spaces of vector-valued measurable functions.

MSC:

03E99 Set theory
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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