Average values of functionals and concentration without measure

TL;DR

This paper explores the concept of concentration without measure in infinite-dimensional spaces, defining average values and variances of functionals through limiting procedures, and reveals that variance can be zero, indicating functional concentration.

Contribution

It introduces a measure-free approach to define and analyze average values and variances of functionals in infinite-dimensional spaces, including explicit formulas and the concept of concentration without measure.

Findings

Variance of certain functionals is zero, indicating concentration.

Probability densities of coordinates exist despite lack of measure.

Explicit finite-dimensional integral formulas for functional expectations.

Abstract

Although there doesn't exist the Lebesgue measure in the ball $M$ of $C[0,1]$ with $p-$norm, the average values (expectation) $EY$ and variance $DY$ of some functionals $Y$ on $M$ can still be defined through the procedure of limitation from finite dimension to infinite dimension. In particular, the probability densities of coordinates of points in the ball $M$ exist and are derived out even though the density of points in $M$ doesn't exist. These densities include high order normal distribution, high order exponent distribution. This also can be considered as the geometrical origins of these probability distributions. Further, the exact values (which is represented in terms of finite dimensional integral) of a kind of infinite-dimensional functional integrals are obtained, and specially the variance $DY$ is proven to be zero, and then the nonlinear exchange formulas of average values of functionals are also given. Instead of measure, the variance is used to measure the deviation of functional from its average value. $DY=0$ means that a functional takes its average on a ball with probability 1 by using the language of probability theory, and this is just the concentration without measure. In addition, we prove that the average value depends on the discretization.