Perfect Measurable and Interpolation Theorems for Variable Exponent Lebesgue Spaces with Relatively Compact Sets

Abstract

We give a general treatment of the union problem for not necessarily perfect or weakly perfect ,measurable spaces. We introduce variable exponent Lebesgue spaces on metric measure spaces and consider a central tool in geometric analysis and the Hardy –Littlewood maximal operator. We study relatively compact sets in variable Lebesgue spaces. The full characterization of such sets is given in the case of variable Lebesgue space on metric measure spaces .