Rearrangement and convergence in spaces of measurable functions

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 convergenceμ-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.