By Robert Gray

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0, 1]3 is the unit cube in three-dimensional Euclidean space. Alternative notations for a Cartesian product space are k−1 Ai = i∈Zk Ai , i=0 where again the Ai are all replicas or copies of A, that is, where Ai = A, all i. Other notations for such a finite-dimensional Cartesian product are k ×i∈Zk Ai = ×k−1 i=0 Ai = A . This and other product spaces will prove to be a useful means of describing abstract spaces modeling sequences of elements from another abstract space. Observe that a finite-dimensional vector space constructed from a discrete space is also discrete since if one can count the number of possible values one coordinate can assume, then one can count the number of possible values that a finite number of coordinates can assume.

The first example is the simplest possible probability space and is commonly referred to as the trivial probability space. Although useless for application, the model does serve a purpose, however, by showing that a well-defined model need not be interesting. The second example is essentially the simplest nontrivial probability space, a slight generalization of the fair coin flip permitting an unfair coin. 0] Let Ω be any abstract space and let F = {Ω, ∅}; that is, F consists of exactly two sets — the sample space (everything) and the empty set (nothing).

K − 1}. 28 CHAPTER 2. 3] A countably infinite space Ω = {ak ; k = 0, 1, 2, . }, for some sequence {ak }. Specific examples are the space of all nonnegative integers {0, 1, 2, . }, which we denote by Z+ , and the space of all integers {. . , −2, −1, 0, 1, 2, . }, which we denote by Z. Other examples are the space of all rational numbers, the space of all even integers, and the space of all periodic sequences of integers. 3] are called discrete spaces. Spaces with finite or countably infinite numbers of elements are called discrete spaces.