Skip to main content
SpringerLink
Log in

Properties of the Riemann–Lebesgue integrability in the non-additive case

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

Abstract

We study Riemann–Lebesgue integrability of a vector function relative to an arbitrary non-negative set function. We obtain some classical integral properties. Results regarding the continuity properties of the integral and relationships among Riemann–Lebesgue, Birkhoff simple and Gould integrabilities are also established.

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. Birkhoff, G.: Integration of functions with values in a Banach space. Trans. Am. Math. Soc. 38(2), 357–378 (1935)

    MathSciNet  MATH  Google Scholar 

  2. Balcerzak, M., Musial, K.: A convergence theorem for the Birkhoff integral. Funct. Approx. Comment. Math. 50(1), 161–168 (2014)

    Article  MathSciNet  Google Scholar 

  3. 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). https://doi.org/10.4171/RLM/710

    Article  MathSciNet  MATH  Google Scholar 

  4. Boccuto, A., Minotti, A.M., Sambucini, A.R.: Set-valued Kurzweil-Henstock integral in Riesz space setting. PanAm. Math. J. 23(1), 57–74 (2013)

    MathSciNet  MATH  Google Scholar 

  5. Boccuto, A., Sambucini, A.R.: A note on comparison between Birkhoff and Mc Shane integrals for multifunctions. Real Anal. Exchange 37(2), 3–15 (2012)

    Google Scholar 

  6. Candeloro, D., Di Piazza, L., Musial, K., Sambucini, A.R.: Gauge integrals and selections of weakly compact valued multifunctions. J.M.A.A. 441(1), 293–308 (2016). https://doi.org/10.1016/j.jmaa.2016.04.009

    Article  MathSciNet  MATH  Google Scholar 

  7. Candeloro, D., Di Piazza, L., Musial, K., Sambucini, A.R.: Relations among gauge and Pettis integrals for multifunctions with weakly compact convex values. Ann. Mat. 197(1), 171–183 (2018). https://doi.org/10.1007/s10231-017-0674-z

    Article  MathSciNet  MATH  Google Scholar 

  8. 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 

  9. Candeloro, D., Croitoru, A., Gavriluţ, A., Sambucini, A.R.: Atomicity related to non-additive integrability. Rend. Circolo Matem. Palermo 65(3), 435–449 (2016)

    Article  MathSciNet  Google Scholar 

  10. Candeloro, D., Croitoru, A., Gavriluţ, A., Sambucini, A.R.: An extension of the Birkhoff integrability for multifunctions. Mediterr. J. Math. 13(5), 2551–2575 (2016)

    Article  MathSciNet  Google Scholar 

  11. Candeloro, D., Croitoru, A., Gavriluţ, A., Sambucini, A.R.: A multivalued version of the Radon-Nikodym theorem, via the single-valued Gould integral. Aust. J. Math. Anal. Appl. 15(2), art. 9, 1–16 (2018)

  12. Caponetti, D., Marraffa, V., Naralenkov, K.: On the integration of Riemann-measurable vector-valued functions. Monatsh. Math. 182(3), 513–536 (2017)

    Article  MathSciNet  Google Scholar 

  13. Cascales, B., Rodriguez, J.: Birkhoff integral for multi-valued functions. J. Math. Anal. Appl. 297(2), 540–560 (2004)

    Article  MathSciNet  Google Scholar 

  14. Cascales, B., Rodriguez, J.: Birkhoff integral and the property of Bourgain. Math. Ann. 331(2), 259–279 (2005)

    Article  MathSciNet  Google Scholar 

  15. Croitoru, A., Gavriluţ, A.: Comparison between Birkhoff integral and Gould integral. Mediterr. J. Math. 12, 329–347 (2015). https://doi.org/10.1007/s00009-014-0410-5

    Article  MathSciNet  MATH  Google Scholar 

  16. Di Piazza, L., Marraffa, V.: The Mc Shane, PU and Henstock integrals of Banach valued functions. Czechoslov. Math. J. 52(3), 609–633 (2002)

    Article  Google Scholar 

  17. Di Piazza, L., Preiss, D.: When do McShane and Pettis integrals coincide? Ill. J. Math. 47(4), 1177–1187 (2003)

    Article  MathSciNet  Google Scholar 

  18. Di Piazza, L., Marraffa, V., Musial, K.: Variational Henstock integrability of Banach space valued functions. Math. Bohem. 141(2), 287–296 (2016)

    Article  MathSciNet  Google Scholar 

  19. Fernandez, A., Mayoral, F., Naranjo, F., Rodriguez, J.: On Birkhoff integrability for scalar functions and vector measures. Monatsh. Math. 157, 131–142 (2009)

    Article  MathSciNet  Google Scholar 

  20. Fremlin, D.H.: The McShane and Birkhoff integrals of vector-valued functions. University of Essex Mathematics Department Research Report 92-10, version of 13.10.04. http://www.essex.ac.uk/maths/staff/fremlin/preprints.htm

  21. Gavriluţ, A., Petcu, A.: A Gould type integral with respect to a submeasure. An. Şt. Univ. Al. I. Cuza Iaşi 53(2), 351–368 (2007)

    MathSciNet  MATH  Google Scholar 

  22. Gavriluţ, A.: A Gould type integral with respect to a multisubmeasure. Math. Slovaca 58, 43–62 (2008)

    Article  MathSciNet  Google Scholar 

  23. Gavriluţ, A.: A generalized Gould type integral with respect to a multisubmeasure. Math. Slovaca 60, 289–318 (2010)

    Article  MathSciNet  Google Scholar 

  24. Gavriluţ, A., Iosif, A., Croitoru, A.: The Gould integral in Banach lattices. Positivity 19(1), 65–82 (2015). https://doi.org/10.1007/s11117-014-0283-7

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  26. Kadets, V.M., Tseytlin, L.M.: On integration of non-integrable vector-valued functions. Mat. Fiz. Anal. Geom. 7, 49–65 (2000)

    MathSciNet  MATH  Google Scholar 

  27. Kadets, V.M., Shumyatskiy, B., Shvidkoy, R., Tseytlin, L.M., Zheltukhin, K.: Some remarks on vector-valued integration. Mat. Fiz. Anal. Geom. 9, 48–65 (2002)

    MathSciNet  MATH  Google Scholar 

  28. Maraffa, V.: A Birkhoff type integral and the Bourgain property in a locally convex space. Real. Anal. Exchange 32(2), 409–428 (2006–2007)

  29. Memetaj, S.B.: Some convergence theorems for bk-integral in locally convex spaces. Tatra Mt. Math. Publ. 46(1), 29–40 (2010)

    MathSciNet  MATH  Google Scholar 

  30. Pap, E.: An integral generated by decomposable measure. Univ. u Novom Sadu Zh. Rad. Prirod. Mat. Fak. Ser. Mat. 20, 135–144 (1990)

    MathSciNet  MATH  Google Scholar 

  31. Pap, E.: Pseudo-additive measures and their applications. In: Pap, E. (ed.) Handbook of measure theory, II, pp. 1403–1465. Elsevier, Amsterdam (2002)

    Chapter  Google Scholar 

  32. Popov, M.M.: On integrability in \(F\)-spaces. Stud. Math. 110(3), 205–220 (1994)

    Article  MathSciNet  Google Scholar 

  33. Potyrala, M.: Some remarks about Birkhoff and Riemann–Lebesgue integrability of vector valued functions. Tatra Mt. Math. Publ. 35, 97–106 (2007)

    MathSciNet  MATH  Google Scholar 

  34. Potyrala, M.: The Birkhoff and variational McShane integral of vector valued functions. Folia Math. Acta Univ. Lodziensis 13, 31–40 (2006)

    MathSciNet  MATH  Google Scholar 

  35. Precupanu, A., Croitoru, A.: A Gould type integral with respect to a multimeasure I/II. An. Şt. Univ. “Al.I. Cuza” Iaşi 48, 165–200 (2002)/49, 183–207 (2003)

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

    Article  MathSciNet  Google Scholar 

  37. Precupanu, A., Satco, B.: The Aumann-Gould integral. Mediterr. J. Math. 5, 429–441 (2008)

    Article  MathSciNet  Google Scholar 

  38. Rodriguez, J.: Convergence theorems for the Birkhoff integral. Houston J. Math. 35, 541–551 (2009)

    MathSciNet  MATH  Google Scholar 

  39. Rodriguez, J.: Pointwise limits of Birkhoff integrable functions. Proc. Am. Math. Soc. 137, 235–245 (2009)

    Article  MathSciNet  Google Scholar 

  40. Rodriguez, J.: Some examples in vector integration. Bull. Aust. Math. Soc. 80(3), 384–392 (2009)

    Article  MathSciNet  Google Scholar 

  41. Sipos, J.: Integral with respect to a pre-measure. Math. Slovaca 29, 141–155 (1979)

    MathSciNet  MATH  Google Scholar 

  42. Sofian-Boca, F.N.: Another Gould type integral with respect to a multisubmeasure. An. Ştiinţ. Univ. ”Al.I. Cuza” Iaşi 57, 13–30 (2011)

    MathSciNet  MATH  Google Scholar 

  43. Spaltenstein, N.: A definition of integrals. J. Math. Anal. Appl. 195, 835–871 (1995)

    Article  MathSciNet  Google Scholar 

  44. Torra, V.: Use and applications of non-additive measures and integrals. In: Torra, V., Narukawa, Y., Sugeno, M. (eds.) Non-additive measures, theory and applications. Studies in fuzziness and soft computing, vol. 310, pp. 1–33. Springer, Berlin (2014). https://doi.org/10.1007/978-3-319-03155-2_2

    Chapter  Google Scholar 

Download references

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, University “Alexandru Ioan Cuza”, Bd. Carol I, No. 11, 700506, Iasi, Romania

    Anca Croitoru & Alina Gavriluţ

  3. Department of Computer Science, Information Technology, Mathematics and Physics, Petroleum-Gas University of Ploieşti, Bd. Bucureşti, No. 39, 100680, Ploiesti, Romania

    Alina Iosif

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. Alina Iosif
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Anna Rita Sambucini
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anna Rita Sambucini.

Additional information

Dedicated to Mimmo: a Master, a Colleague, a dear Friend and a Gentleman who passed away, but lives in our hearts.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The first and the last authors have been partially supported by University of Perugia – Department of Mathematics and Computer Sciences – Ricerca di Base 2018 “Metodi di Teoria dell’Approssimazione, Analisi Reale, Analisi Nonlineare e loro applicazioni” and the last author by GNAMPA - INDAM (Italy) “Metodi di Analisi Reale per l’Approssimazione attraverso operatori discreti e applicazioni” (2019).

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. Properties of the Riemann–Lebesgue integrability in the non-additive case. Rend. Circ. Mat. Palermo, II. Ser 69, 577–589 (2020). https://doi.org/10.1007/s12215-019-00419-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12215-019-00419-y

Keywords

Mathematics Subject Classification

Search

Navigation