Search results for "Convergence in measure"
showing 2 items of 2 documents
When a convergence of filters is measure-theoretic
2022
Abstract Convergence almost everywhere cannot be induced by a topology, and if measure is finite, it coincides with almost uniform convergence and is finer than convergence in measure, which is induced by a metrizable topology. Measures are assumed to be finite. It is proved that convergence in measure is the Urysohn modification of convergence almost everywhere, which is pseudotopological. Extensions of these convergences from sequences to arbitrary filters are discussed, and a concept of measure-theoretic convergence is introduced. A natural extension of convergence almost everywhere is neither measure-theoretic, nor finer than a natural extension of convergence in measure. A straightforw…
Rearrangement and convergence in spaces of measurable functions
2007
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 th…