Home 𝜎-ideals and outer measures on the real line
Article
Licensed
Unlicensed Requires Authentication

𝜎-ideals and outer measures on the real line

  • Salvador Garcia-Ferreira EMAIL logo , Artur H. Tomita and Yasser Ferman Ortiz-Castillo
Published/Copyright: May 21, 2019
Journal of Applied Analysis
From the journal Journal of Applied Analysis Volume 25 Issue 1

Abstract

A weak selection on ℝ is a function f:[ℝ]2→ℝ such that f⁹({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 λf 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 λf⁹(A)=0 whenever |A|≀ω and λf⁹(A)=∞ otherwise. Some conditions are given for a σ-ideal of ℝ in order to be exactly the family đ’©f of λf-null subsets for some weak selection f. It is shown that there are 2𝔠 pairwise distinct ideals on ℝ of the form đ’©f, where f is a weak selection. Also, we prove that the Martin axiom implies the existence of a weak selection f such that đ’©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 λf-measurable sets.

Keywords: Lebesgue outer measures; outer measures; weak selections; congruence between weak selections; null-ideals of outer measures
MSC 2010: 28A12; 28A99; 28B15

Funding source: Consejo Nacional de Ciencia y TecnologĂ­a

Award Identifier / Grant number: 176202

Funding source: Universidad Nacional AutĂłnoma de MĂ©xico

Award Identifier / Grant number: IN105318

Funding source: Conselho Nacional de Desenvolvimento CientĂ­fico e TecnolĂłgico

Award Identifier / Grant number: 307130/2013-4

Funding source: Fundação de Amparo å Pesquisa do Estado de São Paulo

Award Identifier / Grant number: 2014/16955-2

Funding statement: Research of the first-named author was supported by CONACYT grant no. 176202 and PAPIIT grant no. IN105318. The second author has support from CNPq (Brazil) – “Bolsa de Produtividade em Pesquisa, processo 307130/2013-4”. The third author was supported by FAPESP (Brazil) Proc. 2014/16955-2.

Acknowledgements

The authors would like to thank the anonymous referee for careful reading and very useful suggestions and comments that help to improve the presentation of the paper. The research leading to this paper was essentially performed while the first-listed author was visiting the Instituto de Matemática and Estatística of University of São Paulo. He would like to gratefully acknowledge the hospitality received from this institution and the financial support received from the project ”Grupos topológicos fortemente pseudocompactos ”, Proc. FAPESP 2016/23440-4 (Auxílio professor visitante).

References

[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. 10.14321/realanalexch.39.1.0101Search in Google Scholar

[2] C. Costantini, Weak orderability of some spaces which admit a weak selection, Comment. Math. Univ. Carolin. 47 (2006), 609–615. Search in Google Scholar

[3] S. Garcia-Ferreira, K. Miyazaki and T. Nogura, Continuous weak selections for products, Topology Appl. 160 (2013), 2465–2472. 10.1016/j.topol.2013.07.040Search in Google Scholar

[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. Search in Google Scholar

[5] S. Garcia-Ferreira and A. H. Tomita, A non-normal topology generated by a two-point selection, Topology Appl. 155 (2008), 1105–1110. 10.1016/j.topol.2008.01.013Search in Google Scholar

[6] V. Gutev, Selections and hyperspaces, Recent Progress in General Topology. III, Atlantis Press, Amsterdam (2014), 535–580. 10.2991/978-94-6239-024-9_12Search in Google Scholar

[7] V. Gutev and T. Nogura, Selections and order-like selections, Appl. Gen. Topol. 2 (2001), 205–218. 10.4995/agt.2001.2150Search in Google Scholar

[8] V. Gutev and T. Nogura, A topology generated by selections, Topology Appl. 153 (2004), 900–911. 10.1016/j.topol.2005.01.019Search in Google Scholar

[9] M. Nagao and D. Shakhmatov, On the existence of kings in continuous tournaments, Topology Appl. 159 (2012), 3089–3096. 10.1016/j.topol.2012.05.021Search in Google Scholar

Received: 2017-11-10
Revised: 2018-07-03
Accepted: 2018-07-06
Published Online: 2019-05-21
Published in Print: 2019-06-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Two-step collocation methods for two-dimensional Volterra integral equations of the second kind
  3. Riesz basis of exponential family for a hyperbolic system
  4. 𝜎-ideals and outer measures on the real line
  5. Some fractional differential equations involving generalized hypergeometric functions
  6. Global existence of solutions to nonlinear Volterra integral equations
  7. Second-order characterization of convex functions and its applications
  8. On some k-fractional integral inequalities of~Hermite--Hadamard type for twice differentiable generalized beta (r, g)-preinvex functions
  9. Refinements of the Hermite–Hadamard inequality for co-ordinated convex mappings
  10. Fractional Ostrowski type inequalities for functions whose certain power of modulus of the first derivatives are pre-quasi-invex via power mean inequality
  11. Some families of sublinear correspondences
  12. Decay rates for a coupled quasilinear system of nonlinear viscoelastic equations
  13. A note on the regularity of weak solutions to the coupled 2D Allen–Cahn–Navier–Stokes system
https://doi.org/10.1515/jaa-2019-0003
Keywords for this article
Lebesgue outer measures; outer measures; weak selections; congruence between weak selections; null-ideals of outer measures

Articles in the same Issue

  1. Frontmatter
  2. Two-step collocation methods for two-dimensional Volterra integral equations of the second kind
  3. Riesz basis of exponential family for a hyperbolic system
  4. 𝜎-ideals and outer measures on the real line
  5. Some fractional differential equations involving generalized hypergeometric functions
  6. Global existence of solutions to nonlinear Volterra integral equations
  7. Second-order characterization of convex functions and its applications
  8. On some k-fractional integral inequalities of~Hermite--Hadamard type for twice differentiable generalized beta (r, g)-preinvex functions
  9. Refinements of the Hermite–Hadamard inequality for co-ordinated convex mappings
  10. Fractional Ostrowski type inequalities for functions whose certain power of modulus of the first derivatives are pre-quasi-invex via power mean inequality
  11. Some families of sublinear correspondences
  12. Decay rates for a coupled quasilinear system of nonlinear viscoelastic equations
  13. A note on the regularity of weak solutions to the coupled 2D Allen–Cahn–Navier–Stokes system
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 10.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/jaa-2019-0003/html
Scroll to top button