being, respectively, the event that “infinitely many 's occur” and the event that “all except finitely many 's occur.” Here the inner union in the event (1.7) is the event “at least one event with will occur”; call this event . The intersection over means that the event occurs for every . No matter how large we take, there will be at least one event with that will occur. But this is possible only if infinitely many s occur.
For the event , the argument is similar. The intersection occurs if all events with occur. The union means that at least one of the events will occur, and that means that all will occur, except possibly finitely many.
If all events (except possibly finitely many) occur, then infinitely many of them must occur, so that . If then (see the definition of equality of events) we say that the sequence converges, and .
The most important class of convergent sequences of events consists of monotone sequences, when (increasing sequence) or when (decreasing sequence). We have the following theorem:
Theorem 1.4.1 If the sequenceis increasing, then
and in case of a decreasing sequence, we have
Proof If the sequence is increasing, then the inner union () in remains the same independently of so that . On the other hand, the inner intersection in equals so that , which is the same as , as was to be shown. A similar argument holds for decreasing sequences.
The following two examples illustrate the concept of convergence of events.
Example 1.16
Let and be the sets of points on the plane satisfying the conditions and respectively. If , then is a decreasing sequence, and therefore . Since for all if and only if , we have . On the other hand, if , then is an increasing sequence, and . We leave a justification of the last equality to the reader.
Example 1.17
Let for odd and for even. The sequence is now so it is not monotone. We have here , since every point with belongs to infinitely many . On the other hand, . For
