Journal of Inequalities and Applications 
Volume 2007 (2007), Article ID 63439, 17 pages
doi:10.1155/2007/63439
Research Article

Rearrangement and Convergence in Spaces of Measurable Functions

D. Caponetti,1 A. Trombetta,2 and G. Trombetta2

 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

Recommended by Nikolaos S. Papageorgiou

Abstract

We prove that the convergence of a sequence of functions in the space L0 of measurable functions, with respect to the topology of convergence in measure, implies the convergence μ-almost everywhere (μ denotes the Lebesgue measure) of the sequence of rearrangements. We obtain nonexpansivity of rearrangement on the space L∞, and also on Orlicz spaces LN with respect to a finitely additive extended real-valued set function. In the space L∞ and in the space EΦ, of finite elements of an Orlicz space LΦ of a σ-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 L∞, or LΦ, to the set of rearrangements.