Rearrangement and Convergence in Spaces of Measurable Functions

Abstract

We prove that the convergence of a sequence of functions in the space L 0 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 L N with respect to a finitely additive extended real-valued set function. In the space L ∞ and in the space E Φ , of finite elements of an Orlicz space L Φ 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 L Φ , to the set of rearrangements.

Effects of Thermal Treatment on the Structure of Eu:YAG Nanopowder

“…In order to obtain a deep knowledge of all steps involved in the preparation of doped YAG nanopowders, the degree of solubility of neodymium atoms in the yttrium aluminium based oxide matrix was investigated. Additionally, the effect of thermal treatment at high temperature on the phase stability of crystallization products of the coprecipitated hydroxides were studied systematically [11,12]. When the dopant amount was lower than 3.2 at.%, only the YAG phase was obtained by means of thermal treatment at 900°C for 1 hour.…”