property (1) it is enough to show that there exists at least one field containing . We may take here the class of all subsets of : it is a field (as well as a ‐field and monotone class), and it contains all sets in . Property (2) follows from the fact that the intersection of fields containing is a field containing . Property (3) (i.e., uniqueness of ) follows from the fact that the result of the operation of intersection is unique.
Finally, suppose that there exists a field containing such that . Then must appear as one of the factors in the intersection defining so that . Consequently, . This completes the proof for the case of fields. The proofs for ‐fields and monotone classes are exactly the same, since an intersection of ‐fields (or monotone classes) containing is again a ‐field (monotone class) containing .
One may find it disturbing that Theorem 1.4.3 asserts the existence and uniqueness of some objects without giving a clue as to how to practically find them. In fact, the nonconstructive character of the theorem, combined with its generality, is instead a great help. As we will see in Chapter 2, the natural objects of our interest (the domains of definition of probability) will be ‐fields of events. Beyond the trivial situations of finite or countably infinite sample spaces , where one can always consider the maximal ‐field consisting of all subsets of , one is forced to restrict consideration to classes of events that form ‐fields generated by some “simple” events. The events in these ‐fields are typically of a very rich structure, and one seldom has useful criteria for distinguishing events (elements of the ‐field in question) from “nonevents,” that is, subsets of to which probabilities are not assigned. However, as shown by the two examples below, the smallest ‐field generated by some class is richer than the smallest field generated by the same class.
Example 1.21
A point moves randomly on the plane, and its location is recorded at some time . The outcome of this experiment is the pair of coordinates of the observed location of the point (e.g., imagine here the location of a particle of dust in a liquid, tossed about by random hits from molecules of the medium, and performing Brownian motion; or imagine a location of a previously marked bird at the time of its capture in a bird migration study or the ages of both husband and wife at the time one of them dies).
In any study of this kind (regardless of its ultimate purpose), the “natural” sample space is a plane or part of the plane, (the positive quadrant, etc.). The “simple” events here are of the form , that is, rectangles with sides parallel to the axes. The reason for distinguishing these events as “simple” is that, as will be explained in later chapters, it is often easy to assign probabilities to these events. The reason for the particular configuration of strict and nonstrict inequalities (i.e., north and east side included, south and west side excluded) will also become apparent from the analysis below. To simplify the language, we will call such events Rectangles, and use a capital letter to signify the specific assumption about which sides are included and which are not. Naturally, we will allow for infinite Rectangles, such as .
It is easy to determine the field generated by all Rectangles: These are events that result from finite operations on Rectangles. Clearly, the complement of a Rectangle is a union of at most eight disjoint (infinite) Rectangles (see Figure 1.7), whereas the intersection of Rectangles is again a Rectangle (or is empty). Since unions are reduced to intersections of complements by De Morgan's laws, every element of the smallest field containing all Rectangles is the union of a finite number of disjoint Rectangles. On the other hand, there exist events that do not belong to this field of events. As a simple example, one might be interested in the event that the point lies within distance from some fixed point (from the initial location of the particle, the point of release of the bird, etc.). This event is a circle on the plane, and hence a subset of which is not decomposable into a finite number of Rectangles. On the other hand, a circle does belong to the ‐field spanned by Rectangles: it is representable as a countable union of Rectangles, or equivalently, as an infinite intersection of sets built up of Rectangles.
