The space of scalarly integrable functions for a Fréchet-space-valued measure
type: Item
instance of: scholarly article
title: The space of scalarly integrable functions for a Fréchet-space-valued measure
zbMATH Open document ID: 1187.28007
author: J. Martínez
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-05-19T00:00:00Z
review text: Let \(X\) be a Fréchet space, \(\nu:\Sigma\to X\) a vector measure and \((\Omega,\Sigma)\) be a measurable space. Consider the space \(L_w^1(\nu)\) of all equivalence classes of scalarly integrable functions \(f:\Omega\to\mathbb R\). Clearly the usual space \(L^1(\nu)\) is a closed subspace of \(L_w^1(\nu)\). When \(X\) is a Banach space, it is known that \(L_w^1(\nu)\) can be completed to a Banach space. In this interesting paper, the authors prove that \(L_w^1(\nu)\) is a complete space, when \(X\) is a metrizable locally convex Hausdorff space (and a Fréchet space when \(X\) is a Fréchet space). This result is important, since \(L^1(\nu)\) fails, in general, to be complete, when \(X\) is simply a metrizable locally convex Hausdorff space. The authors also prove that every Fréchet lattice with the \(\sigma\)-Fatou property and with a weak unit in its \(\sigma\)-order continuous part is isomorphic to \(L_w^1(\nu)\) for a suitable measure \(\nu\).
reviewed by: M. Federson
Mathematics Subject Classification ID: 28A25
Mathematics Subject Classification ID: 46G10
zbMATH DE Number: 5555221
zbMATH Keywords: Fréchet spaces (lattice)
zbMATH Keywords: vector measure
zbMATH Keywords: Fatou property
zbMATH Keywords: Lebesgue topology
zbMATH Keywords: scalarly integrable function
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.01.036
OpenAlex ID: W2046712888
cites work: Q4186829
cites work: Operators into \(L^ 1\) of a vector measure and applications to Banach lattices
cites work: Banach lattices with the Fatou property and optimal domains of kernel operators
cites work: Q5448134
cites work: Vector measures: Where are their integrals?
cites work: Q4148368
cites work: Reflexivity and Order Properties of Scalar-Type Spectral Operators in Locally Convex Spaces
cites work: Weakly sequentially complete Fréchet spaces of integrable functions
cites work: Q4235662
cites work: Completeness of \(L^1\)-spaces for measures with values in complex vector spaces
cites work: AL- and AM-spaces of integrable scalar functions with respect to a Fréchet-valued measure
cites work: Spaces of \(p\)-integrable functions with respect to a vector measure
cites work: Q5182141
cites work: Q4076072
cites work: Q5569793
cites work: Integration with respect to vector measures
cites work: On integrability and summability in vector spaces
cites work: Q5638775
cites work: Q4340900
cites work: Banach lattices
cites work: Vector measures and integration in non-complete spaces
cites work: Compact integration operators for Fréchet-space-valued measures.
cites work: Q4652149
cites work: Optimal domain and integral extension of operators, acting in function spaces
cites work: Q4400388
cites work: Spaces of complex functions and vector measures in incomplete spaces
cites work: Q4834940
cites work: Q5563474
cites work: Q3669960
DOI: 10.1016/J.JMAA.2009.01.036
Recommended article: Q4235662
Recommended article: AL- and AM-spaces of integrable scalar functions with respect to a Fréchet-valued measure
Recommended article: Spaces of \(p\)-integrable functions with respect to a vector measure
Recommended article: Spaces of vector functions that are integrable with respect to vector measures
Recommended article: Compact integration operators for Fréchet-space-valued measures.
Recommended article: Q5405573
Recommended article: Q3145211
Recommended article: Orlicz spaces of integrable functions with respect to vector-valued measures
Recommended article: Spaces of integrable functions with respect to a vector measure and factorizations through \(L^{p}\) and Hilbert spaces
Recommended article: The space of vector‐valued integrable functions under certain locally convex topologies
description: scientific article; zbMATH DE number 5555221