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Rearrangement and Convergence in Spaces of Measurable Functions

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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Correspondence to D Caponetti.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Caponetti, D., Trombetta, A. & Trombetta, G. Rearrangement and Convergence in Spaces of Measurable Functions. J Inequal Appl 2007, 063439 (2007). https://doi.org/10.1155/2007/63439

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Keywords

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