Skip to main content
SpringerLink
Log in

The space of probabilistic 1-Lipschitz maps

  • Published:
Aequationes mathematicae Aims and scope Submit manuscript

Abstract

We introduce and study the natural notion of probabilistic 1-Lipschitz maps. We use the space of all probabilistic 1-Lipschitz maps to give a new method for the construction of probabilistic metric completion (respectively of probabilistic invariant metric group completion). Our construction is of independent interest. We prove that the space of all probabilistic 1-Lipschitz maps defined on a probabilistic invariant metric group can be endowed with a monoid structure. Next, we explicit the set of all invertible elements of this monoid and characterize probabilistic invariant complete Menger groups by the space of all probabilistic 1-Lipschitz maps in the spirit of the classical Banach–Stone theorem.

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. Alsina, C., Schweizer, B., Sklar, A.: On the definition of a probabilistic normed space. Aequ. Math. 46(1–2), 91–98 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bachir, M.: Well-posedness and inf-convolution. J. Optim. Theory Appl. 177(2), 271–290 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bachir, M.: The inf-convolution as a law of monoid. An analogue to the Banach–Stone theorem. J. Math. Anal. Appl. 420(1), 145–166 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bachir, M.: A Banach–Stone type theorem for invariant metric groups. Topol. Appl. 209, 189–197 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bachir, M.: Representation of isometric isomorphisms between monoids of Lipschitz functions. Methods Funct. Anal. Topol 23(4), 309–319 (2017)

    MathSciNet  MATH  Google Scholar 

  6. Chu, H.Y.: On the Mazur–Ulam problem in linear 2-normed spaces. J. Math. Anal. Appl. 327, 1041–1045 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cobzas, S.: A Mazur–Ulam theorem for probabilistic normed spaces. Aequ. Math. 77, 197–205 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cobzas, S.: Completeness with respect to the probabilistic Pompeiy–Hausdorff metric. Stud. Univ. “Babes-Bolyai” Math. LII(3), 43–65 (2007)

    MATH  Google Scholar 

  9. Hadžić, O., Pap, E.: Fixed Point Theory in Probabilistic Metric Space, Volume 536 of Mathematics and Its Applications. Kluwer, Dordrecht (2001)

    Google Scholar 

  10. Hadžić, O., Pap, E.: On the Local Uniqueness of the Fixed Point of the Probabilistic q-Contraction in Fuzzy Metric Spaces, pp. 3453–3458. Published by Faculty of Sciences and Mathematics, University of Niš, Niš (2017). https://doi.org/10.2298/FIL1711453H

    Google Scholar 

  11. Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer, Dordrecht (2000)

    Book  MATH  Google Scholar 

  12. Klement, E.P., Mesiar, R., Pap, E.: Triangular norms I: basic analytical and algebraic properties. Fuzzy Sets Syst. 143, 5–26 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Klee, V.L.: Invariant metrics in groups (solution of a problem of Banach). Proc. Am. Math. Soc. 3, 484–487 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lafuerza Guillén, B., Rodriguez Lallena, J.A., Sempi, C.: Completion of probabilistic normed spaces. Int. J. Math. Math. Sci. 18, 649–652 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  15. Mac Shane, E.: Extension of range of functions. Bull. Am. Math. Soc. 40(12), 837–842 (1934)

    Article  MathSciNet  Google Scholar 

  16. Menger, K.: Statistical metrics. Proc. Nat. Acad. Sci. USA 28, 535–537 (1942)

    Article  MathSciNet  MATH  Google Scholar 

  17. Menger, K.: Probabilistic geometry. Proc. Nat. Acad. Sci. USA 37, 226–229 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  18. Menger, K.: Géométrie générale (Chap. VII), Memorial des Sciences Mathematiques, No. 124, Paris (1954)

  19. Nourouzi, K., Pourmoslemi, A.R.: Probabilistic normed groups. Iran. J. Fuzzy Syst. 14(1), 99–113 (2017)

    MathSciNet  MATH  Google Scholar 

  20. Pap, E.: Pseudo-additive measures and their applications. In: Pap, E. (ed.) Handbook of Measure Theory, vol. II, pp. 1403–1468. Elsevier, Amsterdam (2002)

    Chapter  Google Scholar 

  21. Pap, E., Štajner, I.: Generalized pseudo-convolution in the theory of probabilistic metric spaces, information, fuzzy numbers, optimization, system theory. Fuzzy Sets Syst. 102, 393–415 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  22. Schweizer, B., Sklar, A.: Probabilistic Metric Spaces, North-Holland Series in Probability and Applied Mathematics. North-Holland Publishing Co., New York (1983)

    Google Scholar 

  23. Schweizer, B., Sklar, A.: Espaces metriques aleatoires. C. R. Acad. Sci. Paris 247, 2092–2094 (1958)

    MathSciNet  MATH  Google Scholar 

  24. Schweizer, B., Sklar, A.: Statistical metric spaces. Pac. J. Math. 10, 313–334 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  25. Schweizer, B., Sklar, A.: Triangle inequalities in a class of statistical metric spaces. J. Lond. Math. Soc. 38, 401–406 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  26. Schweizer, B., Sklar, A., Thorp, E.: The metrization of statistical metric spaces. Pac. J. Math. 10, 673–675 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  27. Shaabani, M.H., Pourmemar, L.: On the definition of a probabilistic normed group. J. Interdiscip. Math. 20, 383–395 (2017)

    Article  Google Scholar 

  28. Sherwood, H.: On E-spaces and their relation to other classes of probabilistic metric spaces. J. Lond. Math. Soc. s1–44, 441–448 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  29. Sherwood, H.: Complete probabilistic metric spaces. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 20, 117–128 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  30. Sibley, D.A.: A metric for weak convergence of distribution functions. Rocky Mt. J. Math. 1, 427–430 (1971)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire SAMM 4543, Université Paris 1 Panthéon-Sorbonne, Centre P.M.F. 90 rue Tolbiac, 75634, Paris Cedex 13, France

    Mohammed Bachir

Authors
  1. Mohammed Bachir
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohammed Bachir.

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

Bachir, M. The space of probabilistic 1-Lipschitz maps. Aequat. Math. 93, 955–983 (2019). https://doi.org/10.1007/s00010-019-00641-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00010-019-00641-0

Keywords

Mathematics Subject Classification

Search

Navigation