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Concrete It may seem strange that the axioms of probability deal with apparently abstract ideas of measures instead of corresponding physical intuition that the probability tells you something about the fraction of times specific events will occur in a sequence of trials, such as the relative frequency of a pair of dice summing to seven in a sequence of many roles, or a decision algorithm correctly detecting a single binary symbol in the presence of noise in a transmitted data file. Such real world behavior can be quantified by the idea of a relative frequency, that is, suppose the output of the nth trial of a sequence of trials is xn and we wish to know the relative frequency that xn takes on a particular value, say a.

Let Ω be the space of all one-sided binary sequences. Suppose that you consider the smallest field formed by all finite set-theoretic operations on the individual one-sided binary sequences, that is, on singleton sets in the sequence space. Then many countably infinite sets of binary sequences (say the set of all periodic sequences) are not events since they cannot be expressed as finite sequences of set-theoretic operations on the singleton sets. Obviously, the sigma-field formed by including countable set-theoretic operations does not have this defect.

5] A space consisting of k−dimensional vectors with coordinates taking values in one of the previously described spaces. A useful notation for such vector spaces is a product space. Let A denote one of the abstract spaces previously considered. Define the Cartesian product Ak by Ak = { all vectors a = (a0 , a1 , . . , ak−1 ) with ai ∈ A} . Here for the moment we follow a somewhat dated convention of denoting a vector in boldface. At the beginning, however, this can help keep bookkeeping straight and this notation will be used heavily in the earlier chapters of the book.