By R. Gray

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**Additional resources for Probability, Random Processes, and Ergodic Properties**

**Sample text**

Thus any countable measurable space is standard. CHAPTER 2. STANDARD ALPHABETS 34 The above example points out that it is easy to find sequences of asymptotically generating finite fields for a standard space that are not bases and hence which cannot be used to extend arbitrary finitely additive set functions. In this case, however, we were able to rig the field so that it did provide a basis. 1 any countable product of standard spaces must be standard. Thus products of finite or countable alphabets give standard spaces, a conclusion summarized in the following lemma.

N − 1}, Bi ⊂ {0, 1} all i. 10) c(bn ) = {x : xi = bi ; i = 0, 1, . . , n − 1} for bn ∈ {0, 1}n . These sets are called thin cylinders and are the atoms of Fn . The thin cylinders generate the full σ-field and any decreasing sequence of such rectangles converges down to a single binary sequence. Hence it is easy to show that this space is standard. Observe that this also implies that individual binary sequences x = {xi ; i = 0, 1, . } are themselves events since they are a countable intersection of thin cylinders.

Since it is an element of F, it is the union of disjoint intersection sets. There can be only one nonempty intersection set in the union, however, or G would not be an atom. Hence every atom is an intersection set. Conversely, if G is an intersection set, then it must also be an atom since otherwise it would contain more than one atom and hence contain more than one intersection set, contradicting the disjointness of the intersection sets. In summary, given any finite field F of a space Ω we can find a unique collection of atoms A of the field such that the sets in A are disjoint, nonempty, and have the entire space Ω as their union (since Ω is a field element and hence can be written as a union of atoms).