By Grau R.
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Extra resources for Probability,Random Processes and Ergodic Properties
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Use this basis to enumerate all of the elements of F in order, that is, first list the elements of F0 , then the elements of F1 , and so on. This produces a sequence {Fn , n = 0, 1, . } = F. Now construct a corresponding binary indexed asymptotically generating sequence of fields, say Gn , n = 0, 1, . 1. 4). First observe that for every n there is an m ≥ n such that Fn = Gm . That is, eventually the binary indexed fields Gm match the original fields, they just take longer to get all of the elements since they include them one new one at a time.
The latter sequence cannot be empty since the Gn form a basis, hence neither can the former sequence since the mappings are one-to-one. Reversing the argument shows that if either space is standard, then so is the other, completing the proof. 1. Indexing the atoms by binary vectors will prove a simple and useful representation. 4 If a field F is standard, then there exists a binary indexed basis and a corresponding canonical binary sequence function. Proof: The proof consists simply of relabeling of a basis in binary vectors.
Since F does not possess a basis, we know that for any such sequence Fn there must exist a decreasing sequence of atoms Gn of Fn such that Gn ↓ ∅. Define set functions Pn on Fn as follows: If Gn ⊂ F , then Pn (F ) = 1, if F ∩ Gn = ∅, then Pn (F ) = 0. Since F ∈ Fn , F either wholly contains the atom Gn or F and Gn are disjoint, hence the Pn are well defined. , if the Fn are constructed as before as the field generated by the first n elements of F, then eventually every element of the countable field F must appear in one of the Fn ).