Robert Bartoszynski - Probability and Statistical Inference

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 sequence is increasing, then

and in case of a decreasing sequence, we have

ProofIf 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.

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.

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 we have if is large enough (and also for all ). However, if , then does not belong to any with even . Thus, , and the sequence does not converge.