Open Access

Rearrangement and Convergence in Spaces of Measurable Functions

Journal of Inequalities and Applications20072007:063439

DOI: 10.1155/2007/63439

Received: 3 November 2006

Accepted: 25 February 2007

Published: 19 April 2007


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.


Authors’ Affiliations

Department of Mathematics, University of Palermo
Department of Mathematics, University of Calabria


  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. 10.1080/00036817908839248MathSciNetView ArticleMATHGoogle 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-XMathSciNetView ArticleMATHGoogle Scholar
  5. Kolyada VI: Rearrangements of functions, and embedding theorems. Russian Mathematical Surveys 1989,44(5):73–117. 10.1070/RM1989v044n05ABEH002287MathSciNetView ArticleMATHGoogle Scholar
  6. de Lucia P, Weber H: Completeness of function spaces. Ricerche di Matematica 1990,39(1):81–97.MathSciNetMATHGoogle 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.MathSciNetMATHGoogle 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.View ArticleMATHGoogle 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.MathSciNetMATHGoogle Scholar
  13. Weber H: Generalized Orlicz spaces. Locally solid group topologies. Mathematische Nachrichten 1990, 145: 201–215. 10.1002/mana.19901450115MathSciNetView ArticleMATHGoogle Scholar
  14. Aliprantis CD, Burkinshaw O: Locally Solid Riesz Spaces. Academic Press, New York, NY, USA; 1978:xii+198.MATHGoogle Scholar
  15. Kufner A, John O, Fučík S: Function Spaces. Noordhoff, Leyden, The Netherlands; 1977:xv+454.MATHGoogle Scholar
  16. Musielak J: Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics. Volume 1034. Springer, Berlin, Germany; 1983:iii+222.Google Scholar


© D. Caponetti et al. 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.