ABSTRACT
In this chapter, the authors begin to focus on infinite dimensional spaces by examining RN and some of its subspaces. They examine Sigma-algebras and probability measures constructed on the dual of a nuclear space. Some common results related to probability theory are presented. The authors provide a brief discussion of Sigma-algebras and probability measures along with some key theorems. They focus on the collections of sets probability measures typically act on and their properties. The Dynkin Class Theorem is an important result that will prove useful as the authors progress to extend functions on collections of sets to probability measures on Sigma-algebras. There is a strong connection between the probability measures and certain linear functionals. The result that connects these two seemingly separate concepts is the Riesz–Markov Theorem or Riesz Representation Theorem.
