Some measurability results and applications to spaces with mixed family-norm
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 \(z(t)=\sup_{g \in B}\int_S | x(t,s)g(t,s)|\,ds\).
scientific article; zbMATH DE number 5125391