Skip to main content
SpringerLink
Log in

Multi-integrals of finite variation

  • Published:
Bollettino dell'Unione Matematica Italiana Aims and scope Submit manuscript

Abstract

The aim of this paper is to investigate different types of multi-integrals of finite variation and to obtain decomposition results.

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. Beer, G.: Topologies on Closed and Closed Convex Sets, Mathematics and its Applications, vol. 268. Kluwer Academic Publishers Group, Dordrecht (1993)

    Book  Google Scholar 

  2. Bongiorno, B., Di Piazza, L., Musiał, K.: A variational Henstock integral characterization of the Radon–Nikodym property. Ill. J. Math. 53(1), 87–99 (2009)

    Article  MathSciNet  Google Scholar 

  3. Candeloro, D., Croitoru, A., Gavrilut, A., Sambucini, A.R.: An extension of the Birkhoff integrability for multifunctions. Mediterr. J. Math. 13(5), 2551–2575 (2016). https://doi.org/10.1007/s00009-015-0639-7

    Article  MathSciNet  MATH  Google Scholar 

  4. Candeloro, D., Croitoru, A., Gavrilut, A., Iosif, A., Sambucini, A.R.: Properties of the Riemann–Lebesgue integrability in the non-additive case. Rend. Circ. Mat. Palermo II Ser. (2019). https://doi.org/10.1007/s12215-019-00419-y

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Candeloro, D., Di Piazza, L., Musiał, K., Sambucini, A.R.: Some new results on integration for multifunction. Ricerche Mat. 67(2), 361–372 (2018). https://doi.org/10.1007/s11587-018-0376-x

    Article  MathSciNet  MATH  Google Scholar 

  8. Candeloro, D., Di Piazza, L., Musiał, K., Sambucini, A.R.: Integration of multifunctions with closed convex values in arbitrary Banach spaces. J. Convex Anal. 27(4) (2020) (to appear)

  9. Candeloro, D., Di Piazza, L., Musiał, K., Sambucini, A.R.: Multifunctions determined by integrable functions. Int. J. Approx. Reason. 112, 140–148 (2019). https://doi.org/10.1016/j.ijar.2019.06.002

    Article  MathSciNet  MATH  Google Scholar 

  10. Candeloro, D., Sambucini, A.R.: Order-type Henstock and McShane integrals in Banach lattices setting, Sisy 20014- IEEE 12th International Symposium on Intelligent Systems and Informatics, Subotica-Serbia. https://doi.org/10.1109/SISY.2014.6923557 (2014)

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

    Article  MathSciNet  Google Scholar 

  12. Cascales, C., Kadets, V., Rodriguez, J.: Birkhoff integral for multi-valued functions. J. Math. Anal. Appl. 297, 540–560 (2004)

    Article  MathSciNet  Google Scholar 

  13. Cascales, B., Kadets, V., Rodriguez, J.: Measurable selectors and set-valued Pettis integral in non-separable Banach spaces. J. Funct. Anal. 256, 673–699 (2009)

    Article  MathSciNet  Google Scholar 

  14. Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions. Lecture Notes Math, vol. 580. Springer, Berlin (1977)

    Book  Google Scholar 

  15. Diestel, J., Uhl, J.J.: Vector Measures. Mathematical Surveys and Monographs, vol. 15. American Mathematical Society, Providence, RI (1977)

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

    Article  MathSciNet  Google Scholar 

  17. Di Piazza, L., Marraffa, V., Satco, B.: Set valued integrability and measurability in non separable Fréchet spaces and applications. Math. Slovaca 66(5), 1119–1138 (2016)

    Article  MathSciNet  Google Scholar 

  18. Di Piazza, L., Musiał, K.: A characterization of variationally McShane integrable Banach-space valued function. Ill. J. Math. 45(1), 279–289 (2001)

    Article  MathSciNet  Google Scholar 

  19. Di Piazza, L., Musiał, K.: A decomposition of Denjoy–Khintchine–Pettis and Henstock–Kurzweil–Pettis integrable multifunctions, vector measures, integration and related topics. In: Curbera, G.P., Mockenhaupt, G., Ricker, W.J. (eds.) Operator Theory: Advances and Applications, vol. 201, pp. 171–182. Birkhauser Verlag, Basel (2010)

    Google Scholar 

  20. Di Piazza, L., Musiał, K.: Relations among Henstock, McShane and Pettis integrals for multifunctions with compact convex values. Monatshefte Math. 173, 459–470 (2014)

    Article  MathSciNet  Google Scholar 

  21. El Amri, K., Hess, C.: On the Pettis integral of closed valued multifunctions. Set Valued Anal. 8, 329–360 (2000)

    Article  MathSciNet  Google Scholar 

  22. Fremlin, D.H., Mendoza, J.: On the integration of vector-valued functions. Ill. J. Math. 38, 127–147 (1994)

    Article  MathSciNet  Google Scholar 

  23. Fremlin, D.H.: The Henstock and McShane integrals of vector-valued functions. Ill. J. Math. 38(3), 471–479 (1994)

    Article  MathSciNet  Google Scholar 

  24. Fremlin, D., Talagrand, M.: A decomposition theorem for additive set functions and applications to Pettis integral and ergodic means. Math. Z. 168, 117–142 (1979)

    Article  MathSciNet  Google Scholar 

  25. Gordon, R.A.: The Denjoy extension of the Bochner, Pettis and Dunford integrals. Stud. Math. XCII, 73–91 (1989)

    Article  MathSciNet  Google Scholar 

  26. Gordon, R.A.: The Integrals of Lebesgue, Denjoy, Perron and Henstock, Grad. Stud. Math., vol. 4. AMS, Providence (1994)

    Google Scholar 

  27. Katz, M.P.: On extension of vector measures. Sibirskij Mat. J. 13(5), 1158–1168 (1972). in Russian

    MathSciNet  Google Scholar 

  28. Labuschagne, C.C.A., Pinchuck, A.L., van Alten, C.J.: A vector lattice version of Rådström’s embedding theorem. Quaest. Math. 30(3), 285–308 (2007)

    Article  MathSciNet  Google Scholar 

  29. Musiał, K.: Topics in the theory of Pettis integration. Rend. Istit. Mat. Univ. Trieste 23, 177–262 (1991)

    MathSciNet  MATH  Google Scholar 

  30. Musiał, K.: Approximation of Pettis integrable multifunctions with values in arbitrary Banach spaces. J. Convex Anal. 20(3), 833–870 (2013)

    MathSciNet  MATH  Google Scholar 

  31. Naralenkov, K.M.: A Lusin type measurability property for vector-valued functions. J. Math. Anal. Appl. 417(1), 293–307 (2014)

    Article  MathSciNet  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. Department of Mathematics and Computer Sciences, University of Palermo, Via Archirafi 34, 90123, Palermo, Italy

    Luisa Di Piazza

  3. Institute of Mathematics, Wrocław University, Pl. Grunwaldzki 2/4, 50-384, Wrocław, Poland

    Kazimierz Musiał

Authors
  1. Domenico Candeloro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luisa Di Piazza
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kazimierz Musiał
    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 Luisa Di Piazza.

Additional information

This work concludes a research cycle, but not the friendship that has tied us. You left us, dear Mimmo, too soon. The disease has won, but your memories will always be with us.

Publisher's Note

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

This research was partially supported by Grant “Metodi di analisi reale per l’approssimazione attraverso operatori discreti e applicazioni” (2019) of GNAMPA – INDAM (Italy), by University of Perugia – Fondo Ricerca di Base 2019 “Integrazione, Approssimazione, Analisi Nonlineare e loro Applicazioni”, by University of Palermo – Fondo Ricerca di Base 2019 and by Progetto Fondazione Cassa di Risparmio cod. nr. 2018.0419.021 (title: Metodi e Processi di Intelligenza artificiale per lo sviluppo di una banca di immagini mediche per fini diagnostici (B.I.M.)).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Candeloro, D., Di Piazza, L., Musiał, K. et al. Multi-integrals of finite variation. Boll Unione Mat Ital 13, 459–468 (2020). https://doi.org/10.1007/s40574-020-00217-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40574-020-00217-w

Keywords

Mathematics Subject Classification

Search

Navigation