Borel Spaces and Polish Alphabets

Abstract

We have seen that standard measurable spaces are the only measurable spaces for which all finitely additive candidate probability measures are also countably additive, and we have developed several properties and some important simple examples. In particular, sequence spaces drawn from countable alphabets and certain subspaces thereof are standard. In this chapter we develop the most important (and, in a sense, the most general) class of standard spaces-Borel spaces formed from complete separable metric spaces. We will accomplish this by showing that such spaces are isomorphic to a standard subspace of a countable alphabet sequence space and hence are themselves standard. The proof will involve a form of coding or quantization.