Liouville's theorem and the restricted mean value property in the plane

Abstract

Let U be a domain in R 2 such that U c is polar and let τ be a real function on U such that 0 < r ≤ ∥ · ∥ + M. A positive numerical function f on U is called r-supermedian if, for every x ∈ U, the average of f on the disk of center x and radius r(x) is at most f(x). The purpose of this note is to give a short proof for the fact that every l.s.c. r-median function is constant.