Sigma-ideals and outer measures on the real line

• Autorzy: Garcia-Ferreira Salvador , Tomita Artur H. , Ortiz-Castillo Yasser Ferman

Identyfikatory

DOI

10.1515/jaa-2019-0003

• Języki publikacji: EN

Abstrakty

EN

A weak selection on ℝ is a function f : [ℝ]² → ℝ such that f({x, y}) ∈ {x, y} for each {x, y} ∈ [ℝ]². In this article, we continue with the study (which was initiated in [1]) of the outer measures λ_f on the real line ℝ defined by weak selections f . One of the main results is to show that CH is equivalent to the existence of a weak selection f for which λ_f (A) = 0 whenever |A| ≤ ω and λ_f (A) = ∞ otherwise. Some conditions are given for a σ-ideal of ℝ in order to be exactly the family N_f of λ_f -null subsets for some weak selection f. It is shown that there are 2^c pairwise distinct ideals on ℝ of the form N_f , where f is a weak selection. Also, we prove that the Martin axiom implies the existence of a weak selection f such that N_f is exactly the σ-ideal of meager subsets of ℝ. Finally, we shall study pairs of weak selections which are “almost equal” but they have different families of λ_f -measurable sets.

Słowa kluczowe

EN

Lebesgue outer measures; outer measures; weak selections; M-equivalence between weak selections; congruence between weak selections; null-ideals of outer measures; sigma-ideals

PL

miara zewnętrzna Lebesgue'a; miary zewnętrzne; słaby wybór; m-równorzędność pomiędzy słabymi wyborami; kongruencja pomiędzy słabymi wyboram

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2019
• Tom: Vol. 25, nr 1
• Strony: 25--36
• Opis fizyczny: Bibliogr. 9 poz.

Twórcy

autor

Garcia-Ferreira Salvador

• Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, Campus Morelia, Apartado Postal 61-3, Santa Maria, 58089, Morelia, Michoacán, Mexico

autor

Tomita Artur H.

• Instituto de Matemática e Estatística, Universidade de São Paulo Rua do Matão, 1010, CEP 05508-090, São Paulo, Brazil

autor

Ortiz-Castillo Yasser Ferman

• Instituto de Matemática e Estatística, Universidade de São Paulo Rua do Matão, 1010, CEP 05508-090, São Paulo, Brazil

Bibliografia

• [1] J. A. Astorga-Moreno and S. Garcia-Ferreira, Outer measures on the real line by weak selections, Real Anal. Exchange 39 (2014), 101-116.
• [2] C. Costantini, Weak orderability of some spaces which admit a weak selection, Comment. Math. Univ. Carolin. 47 (2006), 609-615.
• [3] S. Garcia-Ferreira, K. Miyazaki and T. Nogura, Continuous weak selections for products, Topology Appl. 160 (2013), 2465-2472.
• [4] S. Garcia-Ferreira, K. Miyazaki, T. Nogura and A. H. Tomita, Topologies generated by weak selection topologies, Houston J. Math. 39 (2013), 1385-1399.
• [5] S. Garcia-Ferreira and A. H. Tomita, A non-normal topology generated by a two-point selection, Topology Appl. 155 (2008), 1105-1110.
• [6] V. Gutev, Selections and hyperspaces, in: Recent Progress in General Topology. III, Atlantis Press, Amsterdam (2014), 535-580.
• [7] V. Gutev and T. Nogura, Selections and order-like selections, Appl. Gen. Topol. 2 (2001), 205-218.
• [8] V. Gutev and T. Nogura, A topology generated by selections, Topology Appl. 153 (2004), 900-911.
• [9] M. Nagao and D. Shakhmatov, On the existence of kings in continuous tournaments, Topology Appl. 159 (2012), 3089-3096.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).