Skip to main content
SpringerLink
Log in

Countable additivity of Henstock–Dunford integrable functions and Orlicz Space

  • Published:
Analysis and Mathematical Physics Aims and scope Submit manuscript

Abstract

Suppose \({\mathcal {X}}\) is a real Banach space and \((\varOmega , \varSigma , \mu )\) is a probability space. We characterize the countable additivity of Henstock–Dunford integrable functions taking values in \({\mathcal {X}}\) as those weakly measurable function \( g: \varOmega \rightarrow {\mathcal {X}}, \) for which \(\{y^*g: y^* \in B_{\mathcal {X}}^* \} \) is relatively weakly compact in some separable Henstock–Orlicz space (in briefly H–Orlicz space) \( H^{\theta }(\mu )\), where \(B_{\mathcal {X}}^*\) is the closed unit ball in \({\mathcal {X}}^{*}.\) We find relatively weakly compactness of some H–Orlicz space of Henstock–Gel’fand integrable functions.

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.

Similar content being viewed by others

Availability of data and materials

The article does not contain any data for analysis.

References

  1. Alexopoulos, J.: De la Vallèe Poussin’s theorem and weakly compact sets in Orlicz spaces. Quaest. Math. 17(2), 231–248 (1994)

    Article  MathSciNet  Google Scholar 

  2. Barcenas, D., Finol, C.E.: On vector measures, Uniformly integrable and Orlicz Spaces in vector measures. In: Integration and Related Topics: Operator Theory: Advances and Applications, vol. 201, pp 51–57. Birkhäuser, Basel (2010)

  3. Bhatnagar, S.: The radon Nikodym property and multipliers of HK-integrable functions. Real Anal. Exchange 44(2), 391–402 (2019)

    Article  MathSciNet  Google Scholar 

  4. Boccuto, A., Riećan, B.: On the Henstock–Kurzweil Integral for Riesz–Space-valued functions defined on unbounded intervals. Czechoslov. Math. J. 54, 591–607 (2004)

    Article  MathSciNet  Google Scholar 

  5. Boccuto, A., Minotti, A.M., Sambucini, A.R.: Set-valued Kurzweil–Henstock Integral in Riesz spaces. Panam. Math. J. 23(1), 57–74 (2013)

    MathSciNet  MATH  Google Scholar 

  6. Boccuto, A., Hazarika, B., Kalita, H.: Kuelbs–Steadman spaces for Banach space-valued measures. Mathematics 8(1005), 1–12 (2020)

    Google Scholar 

  7. Bongiorno, B., Di Piazza, L., Musiał, K.: Differentiation of additive interval measure with values in a conjugate Banach Space. Funct. Approx. 50(1), 169–180 (2014)

    MathSciNet  MATH  Google Scholar 

  8. Boonpogkrong, V.: Stokes theorem on manifolds: a Kurzweil–Henstock approach. Taiwan. J. Math. 17(4), 1183–1196 (2013)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. Candeloro, D., Di Piazza, L., Musiał, K., Sambucini, A.R.: Gauge integrals and selections of weakly compact valued multifunctions. J. Math. Anal. Appl. 441, 293–308 (2016)

    Article  MathSciNet  Google Scholar 

  11. Candeloro, D., Di Piazza, L., Musiał, K., Sambucini, A.R.: Some new results on integration for multifunctions. Ric. Math. 67, 361–372 (2018)

    Article  MathSciNet  Google Scholar 

  12. Candeloro, D., Di Piazza, L., Musiał, K., Sambucini, A.R.: Integration of multifunctions with closed convex valued in arbitrary Banach spaces. J. Convex Anal. 27(4), 1233–1246 (2020)

    MathSciNet  MATH  Google Scholar 

  13. Cascales, B., Kadets, V., Rodriguez, J.: The Gel’fand integral for multi-valued functions. J. Convex Anal. 18, 873–895 (2011)

    MathSciNet  MATH  Google Scholar 

  14. del Campo, R., Fernandez, A., Mayolal, F., Naranjo, F.: The de la Vallèe-Poussin theorem and Orlicz spaces associated to vector measure. J. Math. Anal. Appl. 470, 279–291 (2019)

    Article  MathSciNet  Google Scholar 

  15. Di Piazza, L., Musiał, K.: Henstock–Kurzweil–Pettis integrability of compact valued multifunctions with values in an arbitrary Banach space. J. Math. Anal. Appl. 408, 452–464 (2013)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. Di Piazza, L., Musiał, K.: Set valued Kurzweil–Henstock–Pettis integral. Set-Valued Anal. 13, 167–169 (2005)

    Article  MathSciNet  Google Scholar 

  18. Diestel, J., Joseph, J.U., Jr.: Vector measures. Math. Survey 15 AMS. Series Providence (1977)

  19. Dunford, N., Schwartz, J.T.: Linear Operators. Part I. Wiley, New York (1988)

    MATH  Google Scholar 

  20. Fong, C.K.: A continuous version of Orlicz–Pettis theorem via vector valued Henstock–Kurzweil integrals. Canad. Math. Bull. 24(2), 169–176 (1981)

    Article  MathSciNet  Google Scholar 

  21. Fremlin, D.H.: The Henstock and McShane integrals of vector valued function. Ilinois J. Math. 38(3), 471–479 (1994)

    MathSciNet  MATH  Google Scholar 

  22. Gamez, J.L., Mendoza, J.: Denjoy–Dunford and Denjoy—Pettis integral. Stud. Math. 130(2), 115–133 (1998)

    MathSciNet  MATH  Google Scholar 

  23. Gill, T.L., Zachary, W.W.: A new class of Banach spaces. J. Phys. A Math. Theor. 41(49), 1–15 (2008)

    Article  MathSciNet  Google Scholar 

  24. Gordon, R.A.: The integrals of lebesgue, Denjoy, Perron and Henstock, Grad. Stud. Math 4. Amer. Math Soc., Providence (1994)

  25. Gordon, R.A.: Denjoy extension of Bochner, Pettis and Dunford integral. Stud. Math. 92, 73–91 (1989)

    Article  MathSciNet  Google Scholar 

  26. Guoju, Y.: On the Henstock–Kurzweil–Dunford and Kurzweil–Henstock–Pettis. Rocky Mount. J. Math. 39(4), 1233–1244 (2009)

    Article  MathSciNet  Google Scholar 

  27. Guoju, Y., Tianguing, A.N.: On Henstock–Dunford and Henstock–Pettis integral. Int. J. Math. Math. Sci. 25(7), 467–478 (2001)

    Article  MathSciNet  Google Scholar 

  28. Guoju, Y., Schwabik, S.: The McShane and weak McShane integrals of Banach space-valued functions defined on \(\mathbb{R}^m\), Miskolc Math. Notes 2(2), 127–136 (2001)

    MathSciNet  MATH  Google Scholar 

  29. Hazarika, B., Kalita, H.: Henstock–Orlicz space and its dense space. Asian Eur. J. Math. (2021). https://doi.org/10.1142/S179355712150114X

    Article  Google Scholar 

  30. Hazarika, B.: Some Lacunary difference sequence spaces defined by Musielak–Orlicz functions. Asia-Eur. J. Math. 4(4), 613–626 (2011)

    Article  MathSciNet  Google Scholar 

  31. Krasnoselskii, M.A., Rutickii, V.B.: Convex Functions and Orlicz Spaces. Noordhoff, Groningen (1961)

    Google Scholar 

  32. Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces—I, Sequence Spaces (1977)

  33. Morrison, T.J.: A note on Denjoy integrability of abstractly valued functions. Proc. Am. Math. Soc. 61(2), 385–386 (1976)

    MathSciNet  MATH  Google Scholar 

  34. Musiał, K.: Topics in the theory of Pettis integration. Rend. Istit. mat. Univ. Trieste 23, 177–262 (1992)

    MathSciNet  MATH  Google Scholar 

  35. Musiał, K.: Pettis integral. In: Pap, E. (ed.) Hand Book of Measure Theory, pp. 531–568. North Holland, Amsterdan (2002)

    Google Scholar 

  36. Pérez-Becerra, T., Sánchez-Perales, S., Oliveros-Oliveros, J.J.: The HK–Sobolev space and applications to one-dimensional boundary value problems. J. King Saud Univ. Sci. 32(6), 2790–2796 (2020)

    Article  Google Scholar 

  37. Rao, M.M., Ren, Z.: The Theory of Orlicz Spaces. Marcel Dekker, Inc., Newwork (1991)

    MATH  Google Scholar 

  38. Saks, S.: Theory of the Integral, 2 revised Hafner, New York (1937)

    MATH  Google Scholar 

  39. Sánchez-Perales, S.: The initial value problem for the Schrödinger equation involving the Henstock–Kurzweil integral. Revista de la Unión Matemática Argentina 58(2), 297–306 (2017)

    MathSciNet  MATH  Google Scholar 

  40. Sánchez-Perales, S., Mendoza-Torres, F.J.: Boundary value problems for the Schrödinger equation involving the Henstock–Kurzweil integral. Czechoslov. Math. J. 70(145), 519–537 (2020)

    Article  Google Scholar 

  41. Schwabik, S., Guoju, Y.: Topics in Banach Spaces Integration, Series in Real Analysis, vol. 10. World Scientific, Singapore (2005)

    Book  Google Scholar 

  42. Stefansson, A.G.: Pettis intrgrability. Trans. Am. Math. Soc. 330(1), 401–418 (1992)

    Article  MathSciNet  Google Scholar 

  43. Talvila, E.: Limits and Henstock integrals of products. Real Anal. Exchange 25, 907–918 (1999)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We are very grateful to the referee for useful comments and suggestions which have led us to improve the readability of the article.

Funding

Not applicable, the research is not supported by any funding agency.

Author information

Authors and Affiliations

  1. Department of Mathematics, Patkai Christian College (Autonomous), Dimapur, Patkai, Nagaland, 797103, India

    Hemanta Kalita

  2. Department of Mathematics, Gauhati University, Guwahati, Assam, 781014, India

    Bipan Hazarika

Authors
  1. Hemanta Kalita
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bipan Hazarika
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All the authors have equal contribution for the preparation of the article.

Corresponding author

Correspondence to Bipan Hazarika.

Ethics declarations

Conflict of interest

The authors declare that there are no conflict of interest.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kalita, H., Hazarika, B. Countable additivity of Henstock–Dunford integrable functions and Orlicz Space. Anal.Math.Phys. 11, 96 (2021). https://doi.org/10.1007/s13324-021-00533-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13324-021-00533-0

Keywords

Mathematics Subject Classification

Search

Navigation