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By Gray Robert

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Such spaces can usually be handled by treating the discrete and continuous components separately. 5] A space consisting of k−dimensional vectors with coordinates taking values in one of the previously described spaces. 3. PROBABILITY SPACES 29 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} . Thus, for example, k is k−dimensional Euclidean space.

N }) = 0 (since the integral of 1 over a finite set of points is zero) and therefore the limit as N → ∞ must also be zero, a “continuity of probability” idea that we shall later make rigorous. A Single Coin Flip The original example of a spinning wheel is continuous in that the sample space consists of a continuum of possible outcomes, all points in the unit interval. Sample spaces can also be discrete, as is the case of modeling a single flip of a “fair” coin with heads labeled “1” and tails labeled “0”, 20 CHAPTER 2.

While we will not describe such a subset, we can guarantee that these “unmeasurable” sets have no physical importance, that they are very hard to construct, and that an engineer will never encounter such a subset in practice. It may, however, be necessary to demonstrate that some weird subset is in fact an event in this sigma-field. This is typically accomplished by showing that it is the limit of simple Borel sets. In some cases we wish to deal not with a sample space that is the entire real line, but one that is some subset of the real line.

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