Standard Alphabets

Abstract

It is desirable to develop a theory under the most general possible assumptions. Random process models with very general alphabets are useful because they include all conceivable cases of practical importance. On the other hand, considering only the abstract spaces of the previous chapter can result in both weaker properties and more complicated proofs. Restricting the alphabets to possess some structure is necessary for some results and convenient for others. Ideally, however, we can focus on a class of alphabets that both possesses useful structure and still is sufficiently general to well model all examples likely to be encountered in the real world. Standard spaces are a candidate for this goal and are the topic of this chapter and the next. In this chapter we focus on the definitions and properties of standard spaces, leaving the more complicated demonstration that specific spaces are standard to the next chapter. The reader in a hurry can skip the next chapter. The theory of standard spaces is usually somewhat hidden in theories of topological measure spaces. Standard spaces are related to or include as special cases standard Borel spaces, analytic spaces, Lusin spaces, Suslin spaces, and Radon spaces. Such spaces are usually defined by their relation via a mapping to a complete separable metric space, a topic to be introduced in Chapter 3.