On Grothendieck Sets

Ferrando, J. C.; López-Alfonso, Salvador; Pellicer, Manuel López

doi:10.3390/axioms9010034

Cited by 6 publications

(7 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is observed that the Grothendieck sets in the algebra of sets admit a pointwise convergent measure sequence of bounded scalar-valued measures and the measure sequence is weakly convergent in the corresponding Banach space [21]. Thus, a similar question can be asked in the null-extended σ − neighborhood system under finite measures.…”

Section: Topological and Group Algebraic Propertiesmentioning

confidence: 99%

Constructions and Properties of Quasi Sigma-Algebra in Topological Measure Space

Bagchi

2022

Axioms

View full text Add to dashboard Cite

The topological views of a measure space provide deep insights. In this paper, the sigma-set algebraic structure is extended in a Hausdorff topological space based on the locally compactable neighborhood systems without considering strictly (metrized) Borel variety. The null extension gives rise to a quasi sigma-semiring based on sigma-neighborhoods, which are rectifiable in view of Dieudonné measure in n-space. The concepts of symmetric signed measure, uniformly pushforward measure, and its interval-valued Lebesgue variety within a topological measure space are introduced. The symmetric signed measure preserves the total ordering on the real line; however, the collapse of symmetry admits Dieudonné measure within the topological space. The locally constant measures in compact supports in sigma-neighborhood systems are invariant under topological deformation retraction in a simply connected space where the sequence of deformation retractions induces a strongly convergent sequence of measures. Moreover, the extended sigma-structures in an automorphic and isomorphic topological space preserve the properties of uniformly pushforward measure. The Haar-measurable group algebraic structures equivalent to additive integer groups arise under the locally constant and signed measures as long as the topological space is non-compact and the null-extended sigma-neighborhood system admits compact groups. The comparative analyses of the proposed concepts with respect to existing results are outlined.

show abstract

Section: Topological and Group Algebraic Propertiesmentioning

confidence: 99%

Constructions and Properties of Quasi Sigma-Algebra in Topological Measure Space

Bagchi

2022

Axioms

View full text Add to dashboard Cite

show abstract

“…In the definition of the Grothendieck set given in [7] (Definition 1) the sentence "each bounded sequence" is replaced by "each sequence". Both definitions of Grothendieck sets agree when B is a Nikodým set for the Banach space ba(A), because then each sequence (µ n , n ∈ N) of ba(A) that B-pointwise converges is B-pointwise bounded, hence it is norm bounded.…”

Section: Strong Grothendieck Setsmentioning

confidence: 99%

A Survey on Valdivia Open Question on Nikodým Sets

et al. 2022

Self Cite

View full text Add to dashboard Cite

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 μnn=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.

show abstract

“…With the hypothesis of Proposition 3 the next theorem states that M and {χ A : A ∈ M} are strong Rainwater sets for E * and ba(A), respectively. Its argument is based in the proof of [8,Theorem 8].…”

Section: A Hereditary Property Of Rainwater Setsmentioning

confidence: 99%

“…Next lemma is a particular case of [8,Theorem 6], that is based on [7, Proposition 14]. We provide a brief proof for the sake of completeness.…”

Section: Author's Personal Copymentioning

confidence: 99%

Weak Sequential Convergence in Bounded Finitely Additive Measures

López-Alfonso

Pellicer

2020

Vietnam J. Math.

Self Cite

View full text Add to dashboard Cite

Your article is protected by copyright and all rights are held exclusively by Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd.. This e-offprint is for personal use only and shall not be self-archived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com".

show abstract

On Grothendieck Sets

Cited by 6 publications

References 15 publications

Constructions and Properties of Quasi Sigma-Algebra in Topological Measure Space

Constructions and Properties of Quasi Sigma-Algebra in Topological Measure Space

A Survey on Valdivia Open Question on Nikodým Sets

Weak Sequential Convergence in Bounded Finitely Additive Measures

Contact Info

Product

Resources

About