A Survey on Valdivia Open Question on Nikodým Sets

Salvador López-Alfonso, Manuel López-Pellicer, Santiago Moll-López and Luis M. Sánchez-Ruiz
Additional contact information
Salvador López-Alfonso: Departamento de Construcciones Arquitectónicas, Universitat Politècnica de València, 46022 Valencia, Spain
Manuel López-Pellicer: Instituto Universitario de Matemática Pura y Aplicada (IUMPA), Universitat Politècnica de València, 46022 Valencia, Spain
Santiago Moll-López: Departamento de Matemática Aplicada, Universitat Politècnica de València, 46022 Valencia, Spain
Luis M. Sánchez-Ruiz: Departamento de Matemática Aplicada, Universitat Politècnica de València, 46022 Valencia, Spain

Mathematics, 2022, vol. 10, issue 15, 1-11

Abstract: Let A be an algebra of subsets of a set Ω and b a ( 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 b a ( A ) if each countable B -pointwise bounded subset M of b a ( A ) is norm bounded. A subset B of A is a Grothendieck set for b a ( A ) if for each bounded sequence μ n n = 1 ∞ in b a ( A ) the B -pointwise convergence on b a ( A ) implies its b a ( A ) * -pointwise convergence on b a ( A ) . A subset B of an algebra A is a strong-Nikodým (Grothendieck) set for b a ( A ) if in each increasing covering { B n : n ∈ N } of B there exists B m which is a Nikodým (Grothendieck) set for b a ( 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 b a ( A ) then A is a strong Nikodým set for b a ( 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.

Keywords: Grothendieck set; Nikodým set; strong Grothendieck set; strong Nikodým set; algebra of subsets; bounded scalar measure; ? -algebra; variation norm (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/10/15/2660/pdf (application/pdf)
https://www.mdpi.com/2227-7390/10/15/2660/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:10:y:2022:i:15:p:2660-:d:874537

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().