Advertisement

Equivalence Between Limit Theorems for Lattice Group-Valued k-Triangular Set Functions

  • Antonio Boccuto
  • Xenofon Dimitriou
Article
  • 15 Downloads

Abstract

We investigate some of the main properties of lattice group-valued k-triangular set functions and we prove some Brooks–Jewett, Nikodým, Vitali–Hahn–Saks and Schur-type theorems and their equivalence. A Drewnowski-type theorem on existence of continuous restrictions of (s)-bounded set functions is given.

Keywords

Lattice group k-triangular set function continuous set function (s)-bounded set function Fremlin lemma limit theorem 

Mathematics Subject Classification

26E50 28A12 28A33 28B10 28B15 40A35 46G10 

Notes

Acknowledgements

(a) Our thanks to the anonymous referee for his/her remarks which improved the exposition of the paper.

(b) This research was partially supported by the Italian project “Ricerca di Base 2015, Teoria della Misura e dell’Approssimazione e applicazioni a svariati campi della matematica e alla ricostruzione di immagini”.

References

  1. 1.
    Aviles Lopez, A., Cascales Salinas, B., Kadets, V., Leonov, A.: The Schur \(l_1\) theorem for filters. J. Math. Phys., Anal. Geom. 3(4), 383–398 (2007)zbMATHGoogle Scholar
  2. 2.
    Ban, A.I.: Intuitionistic fuzzy sets: theory and applications. Nova Science Publ. Inc, New York (2006)zbMATHGoogle Scholar
  3. 3.
    Boccuto, A.: Egorov property and weak \(\sigma \)-distributivity in Riesz spaces. Acta Math. (Nitra) 6, 61–66 (2003)Google Scholar
  4. 4.
    Boccuto, A., Candeloro, D.: Uniform \((s)\)-boundedness and convergence results for measures with values in complete \((\ell )\)-groups. J. Math. Anal. Appl. 265, 170–194 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Boccuto, A., Candeloro, D.: Convergence and decompositions for \((\ell )\)-group-valued set functions. Comment. Math. 44(1), 11–37 (2004)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Boccuto, A., Candeloro, D., Sambucini, A.R.: \(L^p\) spaces in vector lattices and applications. Math. Slovaca 67(6), 1409–1426 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Boccuto, A., Dimitriou, X.: Ideal limit theorems and their equivalence in \((\ell )\)-group setting. J. Math. Res. 5(2), 43–60 (2013)CrossRefGoogle Scholar
  8. 8.
    Boccuto, A., Dimitriou, X.: Limit theorems for \(k\)-subadditive lattice group-valued capacities in the filter convergence setting. Tatra Mt. Math. Publ. 65, 1–21 (2016)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Boccuto, A., Dimitriou, X.: Convergence theorems for Lattice group-valued measures. Bentham Science Publ, U. A. E (2015)CrossRefzbMATHGoogle Scholar
  10. 10.
    Boccuto, A., Dimitriou, X.: Matrix theorems and interchange for lattice group-valued series in the filter convergence setting, 2015. Bull. Hellenic Math. Soc. 59, 39–55 (2016)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Boccuto, A., Dimitriou, X.: Limit theorems for lattice group-valued \(k\)-triangular set functions. In: Proceedings of the 33rd PanHellenic Conference on Mathematical Education, Chania, Greece, 4–6 November 2016. pp. 1–10 (2016)Google Scholar
  12. 12.
    Boccuto, A., Dimitriou, X.: Schur-type theorems for \(k\)-triangular Lattice group-valued set functions with respect to filter convergence. Appl. Math. Sci. 11(57), 2825–2833 (2017)Google Scholar
  13. 13.
    Boccuto, A., Dimitriou, X.: Non-additive lattice group-valued set functions and limit theorems. Lambert Acad. Publ., (2017). ISBN: 978-613-4-91335-5Google Scholar
  14. 14.
    Boccuto, A., Dimitriou, X., Papanastassiou, N.: Countably additive restrictions and limit theorems in \((\ell )\)-groups. Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 57, 121–134 (2010); Addendum to: “Countably additive restrictions and limit theorems in \((\ell )\)-groups”, ibidem 58, 3–10 (2011)Google Scholar
  15. 15.
    Boccuto, A., Dimitriou, X., Papanastassiou, N.: Brooks–Jewett-type theorems for the pointwise ideal convergence of measures with values in \((\ell )\)-groups. Tatra Mt. Math. Publ. 49, 17–26 (2011)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Boccuto, A., Dimitriou, X., Papanastassiou, N.: Schur lemma and limit theorems in lattice groups with respect to filters. Math. Slovaca 62(6), 1145–1166 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Boccuto, A., Riečan ,B., Sambucini, A. R.: On the product of \(M\)-measures in \((\ell )\)-groups. Aust. J. Math. Anal. Appl. 7(1) Paper N. 9, 8 p (2010)Google Scholar
  18. 18.
    Boccuto, A., Riečan, B., Vrábelová, M.: Kurzweil–Henstock Integral in Riesz Spaces. Bentham Science Publ, U. A. E (2009)Google Scholar
  19. 19.
    Candeloro, D.: On the Vitali–Hahn–Saks, Dieudonné and Nikodým theorems (Italian). Rend. Circ. Mat. Palermo Ser. II(Suppl. 8), 439–445 (1985)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Candeloro, D., Letta, G.: On Vitali–Hahn–Saks and Dieudonné theorems (Italian), Rend. Accad. Naz. Sci. XL Mem. Mat. 9(1), 203–213 (1985)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Candeloro, D., Sambucini, A.R.: Filter convergence and decompositions for vector lattice-valued measures. Mediterranean J. Math. 12, 621–637 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Dobrakov, I.: On submeasures I. Dissertationes Math. 112, 5–35 (1974)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Drewnowski, L.: Equivalence of Brooks–Jewett, Vitali–Hahn–Saks and Nikodym theorems. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 20, 725–731 (1972)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Fremlin, D.H.: A direct proof of the Matthes–Wright integral extension theorem. J. Lond. Math. Soc. 11(2), 276–284 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Guariglia, E.: \(k\)-triangular functions on an orthomodular lattice and the Brooks–Jewett theorem. Radovi Mat. 6, 241–251 (1990)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Pap, E.: The Vitali–Hahn–Saks theorems for \(k\)-triangular set functions. Atti Sem. Mat. Fis. Univ. Modena 35, 21–32 (1987)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Pap, E.: Null-Additive Set Functions. Kluwer Acad. Publishers/Ister Science, Bratislava (1995)zbMATHGoogle Scholar
  28. 28.
    Riečan, B., Neubrunn, T.: Integral, Measure and Ordering. Kluwer Acad. Publ./Ister Science, Dordrecht/Bratislava (1997)zbMATHGoogle Scholar
  29. 29.
    Saeki, S.: The Vitali–Hahn–Saks theorem and measuroids. Proc. Am. Math. Soc. 114(3), 775–782 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Schachermayer, W.: On some classical measure-theoretic theorems for non-sigma-complete Boolean algebras. Dissertationes Math. 214, 1–33 (1982)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Ventriglia, F.: Cafiero theorem for \(k\)-triangular functions on an orthomodular lattice. Rend. Accad. Sci. Fis. Mat. Napoli 75, 45–52 (2008)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Wang, Z., Klir, G.J.: Generalized Measure theory. Springer, Berlin (2009)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversita degli Studi di PerugiaPerugiaItaly
  2. 2.Department of MathematicsUniversity of AthensAthensGreece

Personalised recommendations