Standard Alphabets

Abstract

Standard spaces provide a model for random process outputs that is arguably as general as needed in almost all applications and has sufficient structure to yield several fundamental results for random processes that do not necessarily hold for more abstract models. The structure eases the construction of probability measures on product spaces and from sample sequences which is essential to the Kolmogorov extension theorem, the existence of regular conditional probability measures, and the ergodic decomposition, all results encountered in this book. In this chapter the basic definitions and properties are developed for standard measurable spaces and random processes with standard alphabets and simple examples are given. The properties are used to prove the Kolmogorov extension theorem for processes with standard alphabets. As an example of the Kolmogorov theorem, simple Bernoulli processes (discrete IID processes) are constructed and used to generate a fairly general class of discrete random processes known as B-processes. In the next chapter more complicated and general examples of standard spaces are treated.