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.