Skip to main content
SpringerLink
Log in

Atomicity related to non-additive integrability

  • Published:
Rendiconti del Circolo Matematico di Palermo Series 2 Aims and scope Submit manuscript

Abstract

In this paper we present some results concerning Gould integrability of vector functions with respect to a monotone measure on finitely purely atomic measure spaces. As an application a Radon-Nikodym theorem in this setting is obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Boccuto, A., Candeloro, D., Sambucini, A.R.: Henstock multivalued integrability in Banach lattices with respect to pointwise non atomic measures. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26(4), 363–383 (2015). doi:10.4171/RLM/710. arXiv:1503.08285v1 [math. FA]

  2. Candeloro, D., Sambucini, A.R.: Order-type Henstock and McShane integrals in Banach lattice setting, Sisy. In: IEEE 12th International Symposium on Intelligent Systems and Informatics, Subotica–Serbia, pp. 55–59 (2014). doi:10.1109/SISY.2014.6923557. arXiv:1401.7818 [math.FA]

  3. Candeloro, D., Sambucini, A.R.: Comparison between some norm and order gauge integrals in Banach lattices. Pan Am. Math. J. 25(3), 1–16 (2015)

    MathSciNet  MATH  Google Scholar 

  4. Candeloro, D., Croitoru, A., Gavriluţ, A., Sambucini, A.R.: An extension of the Birkhoff integrability for multifunctions. Mediterr. J. Math. (2015). doi:10.1007/s00009-015-0639-7

  5. Candeloro, D., Croitoru, A., Gavrilut, A., Sambucini, A.R.: Radon Nikodym theorem for a pair of multimeasures with respect to the Gould integrability (2015). arXiv:1504.04110 [math.FA]

  6. Candeloro, D., Di Piazza, L., Musiał, K., Sambucini, A.R.: Gauge integrals and selections of weakly compact valued multifunctions (2016). arXiv:1602.00473 (submitted)

  7. Chiţescu, I.: Finitely purely atomic measures and \(L_{p}\)-spaces. An. Univ. Bucureşti Şt. Nat. 24, 23–29 (1975)

    Google Scholar 

  8. Chiţescu, I.: Finitely purely atomic measures: coincidence and rigidity properties. Rend. Circ. Mat. Palermo 50(3), 455–476 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Croitoru, A., Gavriluţ, A.: Comparison between Birkhoff integral and Gould integral. Mediterr. J. Math. 12, 329–347 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dobrakov, I.: On submeasures, I. Dissertationes Math. (1974)

  11. Drewnowski, L.: Topological rings of sets, continuous set functions. Integration. I, II, III. Bull. Acad. Polon. Sci. Sér. Math. Astron. Phys. 20, 269–286 (1972)

    MathSciNet  MATH  Google Scholar 

  12. Elton, J.H., Hill, P.T.: Ham Sandwich with Mayo: a stronger conclusion to the Classical Ham Sandwich Theorem. arXiv:0901.2589

  13. Gavriluţ, A.: On some properties of the Gould type integral with respect to a multisubmeasure. An. Şt. Univ. ”Al. I. Cuza” Iaşi LII(1), 177–194 (2006)

    MathSciNet  MATH  Google Scholar 

  14. Gavriluţ, A.: The Gould type integral with respect to a multisubmeasure. Math. Slovaca 58, 1–20 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gavriluţ, A., Croitoru, A.: Non-atomicity for fuzzy and non-fuzzy multivalued set functions. Fuzzy Sets Syst. 160(14), 2106–2116 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gavriluţ, A., Croitoru, A., Mastorakis, N.E., Gavriluţ, G.: Measurability and Gould integrability in finitely purely atomic multisubmeasure spaces. WSEAS Trans. Math. 8, 435–444 (2009)

    MathSciNet  Google Scholar 

  17. Gavriluţ, A.C., Iosif, A.E., Croitoru, A.: The Gould integral in Banach lattices. Positivity 19, 65–82 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gavriluţ, A., Petcu, A.: A Gould type integral with respect to a submeasure. An. Şt. Univ. Iaşi LIII(2), 351–368 (2007)

    MathSciNet  MATH  Google Scholar 

  19. Gould, G.G.: On integration of vector-valued measures. Proc. Lond. Math. Soc. 15, 193–225 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  20. Klement, E.P., Li, J., Mesiar, R., Pap, E.: Integrals based on monotone set functions. Fuzzy Sets Syst. 281, 88–102 (2015). doi:10.1016/j.fss.2015.07.010

    Article  MathSciNet  Google Scholar 

  21. Ionascu, E.J., Stancu, A.A.: On independent sets in purely atomic probability spaces with geometric distribution. Acta Math. Univ. Comenianae 79(1), 31–38 (2010)

    MathSciNet  MATH  Google Scholar 

  22. Leung, D.H.: Purely non-atomic weak \(L^{p}\) spaces. Studia Math. 122, 55–66 (1997)

    MathSciNet  MATH  Google Scholar 

  23. Li, J., Mesiar, R., Pap, E.: Atoms of weakly null-additive monotone measures and integrals. Inf. Sci. 257, 183–192 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Li, J., Mesiar, R., Pap, E.: Atoms of weakly null-additive monotone measures and integrals. Inf. Sci. 257, 183–192 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Martellotti, A., Musial, K., Sambucini, A.R.: A Radon-Nikodym theorem for the Bartle-Dunford-Schwartz Integral with respect to finitely additive measures. Atti Sem. Mat. Fis. Univ. Modena XLII, 625–633 (1994)

    MathSciNet  MATH  Google Scholar 

  26. Matveev, M.: Rothberger property and purely atomic measures. Quest. Answ. Gen. Topol. 28, 157–160 (2010)

    MathSciNet  MATH  Google Scholar 

  27. Mesiar, R.: Fuzzy integrals and linearity. Int. J. Approx. Reason. 47, 352–358 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Narukawa, Y., Murofushi, T.: Regular null-additive measure and Choquet integral. Fuzzy Sets Syst. 143, 487–492 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. Pap, E.: The range of null-additive fuzzy and non-fuzzy measure. Fuzzy Sets Syst. 65, 105–115 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  30. Pap, E.: Null-additive set functions. Kluwer Academic Publishers, Dordrecht, Boston, London (1995)

    MATH  Google Scholar 

  31. Pap, E.: Regular null additive monotone set functions. Univ. Novom Sadu, Zb. Rad. Privod-Mat. Fak., Ser. Mat. 25(2), 93–101 (1995)

    MathSciNet  MATH  Google Scholar 

  32. Precupanu, A., Croitoru, A.: A Gould type integral with respect to a multimeasure. I. An. Şt. Univ. ”Al.I. Cuza” Iaşi 48, 165–200 (2002)

    MathSciNet  MATH  Google Scholar 

  33. Precupanu, A., Croitoru, A.: A Gould type integral with respect to a multimeasure. II. An. Şt. Univ. ”Al.I. Cuza” Iaşi 49, 183–207 (2003)

    MathSciNet  MATH  Google Scholar 

  34. Precupanu, A., Gavriluţ, A., Croitoru, A.: A fuzzy Gould type integral. Fuzzy Sets Syst. 161(5), 661–680 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. Satco, B.: A Vitali type theorem for the set-valued Gould integral. An. Şt. Univ. ”Al. I. Cuza” Iaşi. Mat. 51, 191–200 (2005)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

Domenico Candeloro and Anna Rita Sambucini have been supported by University of Perugia—Department of Mathematics and Computer Sciences—Grant No. 2010.011.0403 and, respectively, by Prin “Metodi logici per il trattamento dell’informazione”, “Descartes” and by the Grant prot. U2014/000237 of GNAMPA-INDAM (Italy).

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Sciences, University of Perugia, 1, Via Vanvitelli, 06123, Perugia, Italy

    Domenico Candeloro & Anna Rita Sambucini

  2. Faculty of Mathematics, “Al. I. Cuza” University, 700506, Iaşi, Romania

    Anca Croitoru & Alina Gavriluţ

Authors
  1. Domenico Candeloro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anca Croitoru
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alina Gavriluţ
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Anna Rita Sambucini
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Domenico Candeloro.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Candeloro, D., Croitoru, A., Gavriluţ, A. et al. Atomicity related to non-additive integrability. Rend. Circ. Mat. Palermo, II. Ser 65, 435–449 (2016). https://doi.org/10.1007/s12215-016-0244-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12215-016-0244-z

Keywords

Mathematics Subject Classification

Search

Navigation