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Rearrangement and Convergence in Spaces of Measurable Functions
Journal of Inequalities and Applications volume 2007, Article number: 063439 (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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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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DOI: https://doi.org/10.1155/2007/63439