Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Rearrangement and Convergence in Spaces of Measurable Functions | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 63439 17 pages doi 2007 63439 Research Article Rearrangement and Convergence in Spaces of Measurable Functions D. Caponetti A. Trombetta and G. Trombetta Received 3 November 2006 Accepted 25 February 2007 Recommended by Nikolaos S. Papageorgiou 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 n-almost everywhere n denotes the Lebesgue measure of the sequence of rearrangements. We obtain nonexpansivity of rearrangement on the space Lx 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 Eq of finite elements of an Orlicz space Lq of a Ơ-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 Lx or Lq to the set of rearrangements. 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. 1. Introduction The notion of rearrangement of a real-valued n-measurable function was introduced by Hardy et al. in 1 . It has been studied by many authors and leads to interesting results in Lebesgue spaces and more generally in Orlicz spaces see . 2-5 . The space L0 is a space of real-valued measurable functions defined on a nonempty set o in which we can give a natural generalization of the topology of convergence in measure using a group pseudonorm which depends on a submeasure defined on the power set Q of o see 6 7 and the references given there . In the second section of this note we study rearrangements of functions of the space L0. The .
