A characterization of the existence of statistical limit of real-valued measurable functions

Abstract

We introduce the concept of the statistical limit (at ∞) of a measurable function in several variables and recall the concept of the statistical convergence of a multiple sequence. Then we extend a classical theorem of Schoenberg (which characterizes statistical convergence) from single to multiple sequences, and prove an analogous theorem on statistical limit. These theorems even may be extended to vector-valued sequences or functions, respectively.