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Some measurability results and applications to spaces with mixed family-norm. (English) Zbl 1120.46013

Let \(T, S\) be measure spaces, \(U(t)\), \(t\in T\), be a family of normed function spaces on \(S\), and \(V\) be a normed function space on \(T\). One can define the mixed family-normed space \([U(\cdot) \to V]\) as the space of those measurable functions \(x\) on \(T \times S\) that \(x(t, \cdot) \in U(t)\) for all \(t \in T\) such that the corresponding function \(y(t):= \| x(t, \cdot)\| _{U(t)}\) belongs to \(V\). \([U(\cdot) \to V]\) can be equipped with the norm \(\| x\| := \| y\| _V\). To make this definition applicable, one needs some measurability restriction on the family \(U\) to ensure that the corresponding \(y\) is measurable for every measurable \(x\).
The author proves that a sufficient condition for this is the “uniform measurability” of the family \(\{ \| u\| _{U(t)}: u \in L_0(S)\}\). The Köthe dual to \([U(\cdot) \to V]\) is studied as well. A large part of the paper is devoted to the definition and properties of “uniform measurable” families of functions and to the proof of a Luxemburg-Gribanov type theorem which shows, for some uncountable families \(B\) of measurable functions, the measurability of \(z(t)=\sup_{g \in B}\int_S | x(t,s)g(t,s)|\,ds\).

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
28A35 Measures and integrals in product spaces
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