Rearrangement and Convergence in Spaces of Measurable Functions

• Main Authors: Trombetta A, Trombetta G, Caponetti D
• Format: Article
• Language: English
• Published: SpringerOpen 2007-01-01
• Series: Journal of Inequalities and Applications
• Online Access: http://www.journalofinequalitiesandapplications.com/content/2007/063439
• ISSN: 1025-5834; 1029-242X

<p/> <p>We prove that the convergence of a sequence of functions in the space <inline-formula><graphic file="1029-242X-2007-063439-i1.gif"/></inline-formula> of measurable functions, with respect to the topology of convergence in measure, implies the convergence <inline-formula><graphic file="1029-242X-2007-063439-i2.gif"/></inline-formula>-almost everywhere ( <inline-formula><graphic file="1029-242X-2007-063439-i3.gif"/></inline-formula> denotes the Lebesgue measure) of the sequence of rearrangements. We obtain nonexpansivity of rearrangement on the space <inline-formula><graphic file="1029-242X-2007-063439-i4.gif"/></inline-formula>, and also on Orlicz spaces <inline-formula><graphic file="1029-242X-2007-063439-i5.gif"/></inline-formula> with respect to a finitely additive extended real-valued set function. In the space <inline-formula><graphic file="1029-242X-2007-063439-i6.gif"/></inline-formula> and in the space <inline-formula><graphic file="1029-242X-2007-063439-i7.gif"/></inline-formula>, of finite elements of an Orlicz space <inline-formula><graphic file="1029-242X-2007-063439-i8.gif"/></inline-formula> of a <inline-formula><graphic file="1029-242X-2007-063439-i9.gif"/></inline-formula>-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 <inline-formula><graphic file="1029-242X-2007-063439-i10.gif"/></inline-formula>, or <inline-formula><graphic file="1029-242X-2007-063439-i11.gif"/></inline-formula>, to the set of rearrangements.</p>