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  • Research Article
  • Open Access

Rearrangement and Convergence in Spaces of Measurable Functions

Journal of Inequalities and Applications20072007:063439

https://doi.org/10.1155/2007/63439

  • Received: 3 November 2006
  • Accepted: 25 February 2007
  • Published:

Abstract

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.

Keywords

  • Measurable Function
  • Lebesgue Measure
  • Orlicz Space
  • Hausdorff Measure

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Authors’ Affiliations

(1)
Department of Mathematics, University of Palermo, Palermo, 90123, Italy
(2)
Department of Mathematics, University of Calabria, Rende (CS), 87036, Italy

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