The average values of a kind of functionals in LP and concentration without measure

Abstract

This paper focuses on the average values of functionals like [Formula: see text] on the set [Formula: see text] in [Formula: see text]. The densities of coordinates of points in [Formula: see text] are derived out. The formula of average value [Formula: see text] of functional [Formula: see text] is obtained. The variance [Formula: see text] of [Formula: see text] is proven to be zero, which shows the phenomenon of concentration without measure, and then the nonlinear commutation identity [Formula: see text] is obtained for continuous function [Formula: see text]. Finally, particularly, it is proven that the average value depends on the discretization.