Skip to main content

Rearrangement and Convergence in Spaces of Measurable Functions


We prove that the convergence of a sequence of functions in the space of measurable functions, with respect to the topology of convergence in measure, implies the convergence-almost everywhere ( denotes the Lebesgue measure) of the sequence of rearrangements. We obtain nonexpansivity of rearrangement on the space, and also on Orlicz spaces with respect to a finitely additive extended real-valued set function. In the space and in the space, of finite elements of an Orlicz space of a-additive set function, we introduce some parameters which estimate the Hausdorff measure of noncompactness. We obtain some relations involving these parameters when passing from a bounded set of, or, to the set of rearrangements.



  1. Hardy GH, Littlewood JE, Polya G: Inequalities. Cambridge University Press, Cambridge, UK; 1934.

    Google Scholar 

  2. Bennett C, Sharpley R: Interpolation of Operators, Pure and Applied Mathematics. Volume 129. Academic Press, Boston, Mass, USA; 1988:xiv+469.

    Google Scholar 

  3. Chiti G: Rearrangements of functions and convergence in Orlicz spaces. Applicable Analysis 1979,9(1):23–27. 10.1080/00036817908839248

    Article  MathSciNet  MATH  Google Scholar 

  4. Crandall MG, Tartar L: Some relations between nonexpansive and order preserving mappings. Proceedings of the American Mathematical Society 1980,78(3):385–390. 10.1090/S0002-9939-1980-0553381-X

    Article  MathSciNet  MATH  Google Scholar 

  5. Kolyada VI: Rearrangements of functions, and embedding theorems. Russian Mathematical Surveys 1989,44(5):73–117. 10.1070/RM1989v044n05ABEH002287

    Article  MathSciNet  MATH  Google Scholar 

  6. de Lucia P, Weber H: Completeness of function spaces. Ricerche di Matematica 1990,39(1):81–97.

    MathSciNet  MATH  Google Scholar 

  7. Trombetta G, Weber H: The Hausdorff measure of noncompactness for balls of-normed linear spaces and for subsets of. Bollettino dell'Unione Matematica Italiana. Serie VI. C. 1986,5(1):213–232.

    MathSciNet  MATH  Google Scholar 

  8. Banaś J, Goebel K: Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics. Volume 60. Marcel Dekker, New York, NY, USA; 1980:vi+97.

    Google Scholar 

  9. Dunford N, Schwartz JT: Linear Operators—Part I. John Wiley & Sons, New York, NY, USA; 1958.

    Google Scholar 

  10. Bhaskara Rao KPS, Bhaskara Rao M: Theory of Charges, Pure and Applied Mathematics. Volume 109. Academic Press, New York, NY, USA; 1983:x+315.

    Google Scholar 

  11. Jarchow H: Locally Convex Spaces. B. G. Teubner, Stuttgart, Germany; 1981:548.

    Book  MATH  Google Scholar 

  12. Avallone A, Trombetta G: Measures of noncompactness in the spaceand a generalization of the Arzelà-Ascoli theorem. Bollettino dell'Unione Matematica Italiana. Serie VII. B 1991,5(3):573–587.

    MathSciNet  MATH  Google Scholar 

  13. Weber H: Generalized Orlicz spaces. Locally solid group topologies. Mathematische Nachrichten 1990, 145: 201–215. 10.1002/mana.19901450115

    Article  MathSciNet  MATH  Google Scholar 

  14. Aliprantis CD, Burkinshaw O: Locally Solid Riesz Spaces. Academic Press, New York, NY, USA; 1978:xii+198.

    MATH  Google Scholar 

  15. Kufner A, John O, Fučík S: Function Spaces. Noordhoff, Leyden, The Netherlands; 1977:xv+454.

    MATH  Google Scholar 

  16. Musielak J: Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics. Volume 1034. Springer, Berlin, Germany; 1983:iii+222.

    Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to D Caponetti.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Caponetti, D., Trombetta, A. & Trombetta, G. Rearrangement and Convergence in Spaces of Measurable Functions. J Inequal Appl 2007, 063439 (2007).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: