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

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 M are derived out. The formula of average value EY of functional Y is obtained. The variance DY of Y is proven to be zero, which shows the phenomenon of concentration without measure, and then the nonlinear commutation identity Eh(Y) = h(EY) is obtained for continuous function [Formula: see text]. Finally, particularly, it is proven that the average value depends on the discretization.