Lectures on the mean values of functionals -- An elementary introduction to infinite-dimensional probability

TL;DR

This paper provides an elementary introduction to infinite-dimensional probability, demonstrating how certain functionals on spaces like C[0,1] and L[0,1] have exact mean values with zero variance, illustrating the concentration of measure phenomenon.

Contribution

It introduces the computation of exact mean values of functionals in infinite-dimensional spaces and highlights the measure concentration phenomenon through zero variance results.

Findings

Exact mean values of functionals on C[0,1] and L[0,1]

Zero variance indicates measure concentration

Illustrates fundamental concepts in infinite-dimensional probability

Abstract

This is an elementary introduction to infinite-dimensional probability. In the lectures, we compute the exact mean values of some functionals on C[0,1] and L[0,1] by considering these functionals as infinite-dimensional random variables. The results show that there exist the complete concentration of measure phenomenon for these mean values since the variances are all zeroes.