Some measurability results and applications to spaces with mixed family-norm
type: Item
instance of: scholarly article
title: Some measurability results and applications to spaces with mixed family-norm
zbMATH Open document ID: 1120.46013
author: Martin Väth
published in: Positivity
publication date: 2007-02-13T00:00:00Z
review text: 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 \(z(t)=\sup_{g \in B}\int_S | x(t,s)g(t,s)|\,ds\).
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 28A20
Mathematics Subject Classification ID: 28A35
zbMATH DE Number: 5125391
zbMATH Keywords: Banach function space
zbMATH Keywords: space with mixed norm
zbMATH Keywords: space with mixed family-norm
zbMATH Keywords: Luxemburg-Gribanov theorem
reviewed by: Vladimir Kadets
Wikidata QID: Q57363446
MaRDI profile type: Publication
full work available at URL: s11117-005-0035-9
OpenAlex ID: W1973177049
cites work: The spaces \(L^ p\), with mixed norm
cites work: Q3943431
cites work: Q3281180
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cites work: Q3669960
DOI: 10.1007/S11117-005-0035-9
description: scientific article; zbMATH DE number 5125391