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

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.

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Authors’ Affiliations

(1)
Department of Mathematics, University of Palermo
(2)
Department of Mathematics, University of Calabria

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Copyright

© 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.