Journal of Inequalities and Applications
Volume 2007 (2007), Article ID 63439, 17 pages
Rearrangement and Convergence in Spaces of Measurable Functions
1Department of Mathematics, University of Palermo, Palermo 90123, Italy
2Department of Mathematics, University of Calabria, Rende (CS) 87036, Italy
Received 3 November 2006; Accepted 25 February 2007
Academic Editor: Nikolaos S. Papageorgiou
Copyright © 2007 D. Caponetti et al. 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.
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.