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For each  and t > 1, let  and, for each  define  . So ( ,  ,  ) is just the uniform probability space on  . Easy exercise: Show that, for each k and t,  and  satisfy the conditions for a Borel field and probability measure, respectively. |
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Now let us consider the problem of constructing for  an appropriate probability space ( ,  ,  ) where, as usual, t is an integer greater that 1. Somehow, this space should be the "limit" of the ( ,  ,  )'s. Moreover, we want the probability that an infinite sequence of coin flips starts with a 0 (respectively, 1, . . ., t - 1) to be 1/t, and, in general, we want the probability that an infinite sequence of coin flips has r (a particular finite t-ary sequence) as an initial subsequence to be  . So, defining, for each  , |
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we know that all of the  should be in  with  . Thus, we take  to be the Borel field generated by  , and, less obviously, for each  , define |
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It can be shown (see Dudley [56]) that this  is a probability measure for  . This probability space on  will suffice for our purposes. Henceforth, we will say that an  is measurable if and only if  . Many natural sets turn out to be measurable, as shown by the next lemma. First, we introduce a bit of terminology. For  and  , we say a relation  is arithmetic if and only if P can be expressed in the language of first-order arithmetic augmented by a 1-ary function variable O. So for example, the relation "there are infinitely many n such that O(n) = 0" is expressible by:  . (Rogers discusses and characterizes such arithmetic relations in Section 15.2 of [158].) |
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9.11 Lemma Suppose  ,  , and  is arithmetic. Then, for each x1, . . ., xk, the set  is measurable. |
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Proof: Without loss of generality, suppose that for all O, x1, . . . xk, |
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