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		§9.4.1 Background Probability Theory

		A probability space consists of

		•  W , a set of possible outcomes of experiments;

		• ß, a collection of subsets of  W  (satisfying conditions of Definition 9.9 below), called events or measurable sets; and

		• pr, a function from ß to the real interval (satisfying the conditions of Definition 9.10 below), called a probability measure.

		Before discussing the conditions on ß and pt, we consider the familiar example of the uniform probability measure on a finite, nonempty set  W . In this case ß = the collection of all subsets of  W  and pr(E) = card(E)/card( W ) for . So if  W  = N2, then the probability of 0 (heads) = pr({0}) = 1/2 = pr({1}) = the probability of 1 (tails) and pr( W ) = 1. If , then the probability of an even number of 0's in three flips = pt(E) (where . If  W  is a finite or countable set, then one can usually take every subset of  W  to be an event. If  W  is uncountable (such as ), then set-theoretic difficulties typically force one to take ß to be a proper subset of the powerset of  W  that satisfies the properties given in the following definition.

		9.9 Definition is a Borel field (or  s -algebra) if and only if

		(a) ,

		(b) , and

		(c) ß is closed under countable unions, i.e., if is countable, then .

		The smallest Borel field that contains a particular collection, A, of subsets of  W  is called the Borel field generated by A.

		As , Borel fields are also closed under countable intersections.

		9.10 Definition A probability measure on a Borel field ß of subsets of  W  is a real-valued function pt: such that

		(a) pr( W ) = 1, and

		(b) pr is countably additive, i.e., if is a countable collection of pairwise disjoint sets, then .