A Survey on Valdivia Open Question on Nikodým Sets
[EN] Let A be an algebra of subsets of a set Ω and ba(A) the Banach space of bounded finitely additive scalar-valued measures on A endowed with the variation norm. A subset B of A is a Nikodým set for ba(A) if each countable B-pointwise bounded subset M of ba(A) is norm bounded. A subset B of A is a...
| Autores: | , , , |
|---|---|
| Tipo de recurso: | artículo |
| Fecha de publicación: | 2022 |
| País: | España |
| Institución: | Universitat Politècnica de València (UPV) |
| Repositorio: | RiuNet. Repositorio Institucional de la Universitat Politécnica de Valéncia |
| Idioma: | inglés |
| OAI Identifier: | oai:riunet.upv.es:10251/194479 |
| Acceso en línea: | https://riunet.upv.es/handle/10251/194479 |
| Access Level: | acceso abierto |
| Palabra clave: | Grothendieck set Nikodým set Strong Grothendieck set Strong Nikodým set Algebra of subsets Bounded scalar measure σ-algebra Variation norm MATEMATICA APLICADA CONSTRUCCIONES ARQUITECTONICAS |
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A Survey on Valdivia Open Question on Nikodým SetsLópez Alfonso, Salvador|||0000-0003-1655-2320López Pellicer, Manuel|||0000-0002-3918-1713Moll López, Santiago Emmanuel|||0000-0003-3388-5135Sánchez Ruiz, Luis Manuel|||0000-0001-7559-6724Grothendieck setNikodým setStrong Grothendieck setStrong Nikodým setAlgebra of subsetsBounded scalar measureσ-algebraVariation normMATEMATICA APLICADACONSTRUCCIONES ARQUITECTONICAS[EN] Let A be an algebra of subsets of a set Ω and ba(A) the Banach space of bounded finitely additive scalar-valued measures on A endowed with the variation norm. A subset B of A is a Nikodým set for ba(A) if each countable B-pointwise bounded subset M of ba(A) is norm bounded. A subset B of A is a Grothendieck set for ba(A) if for each bounded sequence {μn}∞ n=1 in ba(A) the B-pointwise convergence on ba(A) implies its ba(A)∗-pointwise convergence on ba(A). A subset B of an algebra A is a strong-Nikodým (Grothendieck) set for ba(A) if in each increasing covering {Bn : n ∈ N} of B there exists Bm which is a Nikodým (Grothendieck) set for ba(A). The answer of the following open question for an algebra A of subsets of a set Ω, proposed by Valdivia in 2013, has not yet been found: Is it true that if A is a Nikodým set for ba(A) then A is a strong Nikodým set for ba(A)? In this paper we surveyed some results related to this Valdivia’s open question, as well as the corresponding problem for strong Grothendieck sets. The new Propositions 1 and 3 provide more simplified proofs, particularly in their application to Theorems 1 and 2, which were the main results surveyed. Moreover, the proofs of almost all other propositions are wholly or partially original.This research was funded by grant PGC2018-094431-B-I00 of Ministry of Science, Innovation and Universities of Spain for the second named author.MDPI AGDepartamento de Matemática AplicadaEscuela Técnica Superior de Ingeniería Aeroespacial y Diseño IndustrialDepartamento de Construcciones ArquitectónicasEscuela Técnica Superior de ArquitecturaInstituto Universitario de Matemática Pura y AplicadaCentro de Investigación en Tecnologías GráficasMinisterio de Ciencia, Innovación y UniversidadesRepositorio Institucional de la Universitat Politècnica de València Riunet20222022-08-01journal articlehttp://purl.org/coar/resource_type/c_6501VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfhttps://riunet.upv.es/handle/10251/194479reponame:RiuNet. Repositorio Institucional de la Universitat Politécnica de Valénciainstname:Universitat Politècnica de València (UPV)InglésengAgencia Estatal de Investigación http://dx.doi.org/10.13039/501100011033 Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020 PGC2018-094431-B-I00 ESPACIOS DE FUNCIONES: FUNCIONES ANALITICAS Y OPERADORES DE COMPOSICION. RENORMAMIENTOS Y TOPOLOGIA DESCRIPTIVAopen accesshttp://purl.org/coar/access_right/c_abf2Reconocimiento (by)http://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccessoai:riunet.upv.es:10251/1944792026-06-13T07:49:27Z |
| dc.title.none.fl_str_mv |
A Survey on Valdivia Open Question on Nikodým Sets |
| title |
A Survey on Valdivia Open Question on Nikodým Sets |
| spellingShingle |
A Survey on Valdivia Open Question on Nikodým Sets López Alfonso, Salvador|||0000-0003-1655-2320 Grothendieck set Nikodým set Strong Grothendieck set Strong Nikodým set Algebra of subsets Bounded scalar measure σ-algebra Variation norm MATEMATICA APLICADA CONSTRUCCIONES ARQUITECTONICAS |
| title_short |
A Survey on Valdivia Open Question on Nikodým Sets |
| title_full |
A Survey on Valdivia Open Question on Nikodým Sets |
| title_fullStr |
A Survey on Valdivia Open Question on Nikodým Sets |
| title_full_unstemmed |
A Survey on Valdivia Open Question on Nikodým Sets |
| title_sort |
A Survey on Valdivia Open Question on Nikodým Sets |
| dc.creator.none.fl_str_mv |
López Alfonso, Salvador|||0000-0003-1655-2320 López Pellicer, Manuel|||0000-0002-3918-1713 Moll López, Santiago Emmanuel|||0000-0003-3388-5135 Sánchez Ruiz, Luis Manuel|||0000-0001-7559-6724 |
| author |
López Alfonso, Salvador|||0000-0003-1655-2320 |
| author_facet |
López Alfonso, Salvador|||0000-0003-1655-2320 López Pellicer, Manuel|||0000-0002-3918-1713 Moll López, Santiago Emmanuel|||0000-0003-3388-5135 Sánchez Ruiz, Luis Manuel|||0000-0001-7559-6724 |
| author_role |
author |
| author2 |
López Pellicer, Manuel|||0000-0002-3918-1713 Moll López, Santiago Emmanuel|||0000-0003-3388-5135 Sánchez Ruiz, Luis Manuel|||0000-0001-7559-6724 |
| author2_role |
author author author |
| dc.contributor.none.fl_str_mv |
Departamento de Matemática Aplicada Escuela Técnica Superior de Ingeniería Aeroespacial y Diseño Industrial Departamento de Construcciones Arquitectónicas Escuela Técnica Superior de Arquitectura Instituto Universitario de Matemática Pura y Aplicada Centro de Investigación en Tecnologías Gráficas Ministerio de Ciencia, Innovación y Universidades Repositorio Institucional de la Universitat Politècnica de València Riunet |
| dc.subject.none.fl_str_mv |
Grothendieck set Nikodým set Strong Grothendieck set Strong Nikodým set Algebra of subsets Bounded scalar measure σ-algebra Variation norm MATEMATICA APLICADA CONSTRUCCIONES ARQUITECTONICAS |
| topic |
Grothendieck set Nikodým set Strong Grothendieck set Strong Nikodým set Algebra of subsets Bounded scalar measure σ-algebra Variation norm MATEMATICA APLICADA CONSTRUCCIONES ARQUITECTONICAS |
| description |
[EN] Let A be an algebra of subsets of a set Ω and ba(A) the Banach space of bounded finitely additive scalar-valued measures on A endowed with the variation norm. A subset B of A is a Nikodým set for ba(A) if each countable B-pointwise bounded subset M of ba(A) is norm bounded. A subset B of A is a Grothendieck set for ba(A) if for each bounded sequence {μn}∞ n=1 in ba(A) the B-pointwise convergence on ba(A) implies its ba(A)∗-pointwise convergence on ba(A). A subset B of an algebra A is a strong-Nikodým (Grothendieck) set for ba(A) if in each increasing covering {Bn : n ∈ N} of B there exists Bm which is a Nikodým (Grothendieck) set for ba(A). The answer of the following open question for an algebra A of subsets of a set Ω, proposed by Valdivia in 2013, has not yet been found: Is it true that if A is a Nikodým set for ba(A) then A is a strong Nikodým set for ba(A)? In this paper we surveyed some results related to this Valdivia’s open question, as well as the corresponding problem for strong Grothendieck sets. The new Propositions 1 and 3 provide more simplified proofs, particularly in their application to Theorems 1 and 2, which were the main results surveyed. Moreover, the proofs of almost all other propositions are wholly or partially original. |
| publishDate |
2022 |
| dc.date.none.fl_str_mv |
2022 2022-08-01 |
| dc.type.none.fl_str_mv |
journal article http://purl.org/coar/resource_type/c_6501 VoR http://purl.org/coar/version/c_970fb48d4fbd8a85 |
| dc.type.openaire.fl_str_mv |
info:eu-repo/semantics/article |
| format |
article |
| dc.identifier.none.fl_str_mv |
https://riunet.upv.es/handle/10251/194479 |
| url |
https://riunet.upv.es/handle/10251/194479 |
| dc.language.none.fl_str_mv |
Inglés eng |
| language_invalid_str_mv |
Inglés |
| language |
eng |
| dc.relation.none.fl_str_mv |
Agencia Estatal de Investigación http://dx.doi.org/10.13039/501100011033 Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020 PGC2018-094431-B-I00 ESPACIOS DE FUNCIONES: FUNCIONES ANALITICAS Y OPERADORES DE COMPOSICION. RENORMAMIENTOS Y TOPOLOGIA DESCRIPTIVA |
| dc.rights.none.fl_str_mv |
open access http://purl.org/coar/access_right/c_abf2 Reconocimiento (by) http://creativecommons.org/licenses/by/4.0/ |
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info:eu-repo/semantics/openAccess |
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open access http://purl.org/coar/access_right/c_abf2 Reconocimiento (by) http://creativecommons.org/licenses/by/4.0/ |
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openAccess |
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application/pdf |
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MDPI AG |
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MDPI AG |
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reponame:RiuNet. Repositorio Institucional de la Universitat Politécnica de Valéncia instname:Universitat Politècnica de València (UPV) |
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