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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 \(z(t)=\sup_{g \in B}\int_S | x(t,s)g(t,s)|\,ds\).

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 \(z(t)=\sup_{g \in B}\int_S | x(t,s)g(t,s)|\,ds\).

Reviewer: Vladimir Kadets (Murcia)

### 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 |

### Keywords:

Banach function space; space with mixed norm; space with mixed family-norm; Luxemburg-Gribanov theorem
Full Text:
DOI

### References:

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