Probability and Statistical Inference. Robert Bartoszynski

follows:

      

      Theorem 2.6.1 If the probability satisfies Axiom 3 of countable additivity, then is continuous from above and from below. Conversely, if a function satisfies Axioms 1 and 2, is finitely additive, and is either continuous from below or continuous from above at the empty set , then is countably additive.

      Proof: Assume that satisfies Axiom 3, and let be a monotone increasing sequence. We have

(2.8)

the events on the right‐hand side being disjoint. Since (see Section 1.5), using (2.8), and the assumption of countable additivity, we obtain

      (passing from the first to the second line, we used the fact that the infinite series is defined as the limit of its partial sums). This proves continuity of from below. To prove continuity from above, we pass to the complements, and proceed as above.

      Let us now assume that is finitely additive and continuous from below, and let be a sequence of mutually disjoint events. Put so that is a monotone increasing sequence with . We have then, using continuity from below and finite additivity,

      again by definition of a numerical series being the limit of its partial sums. This shows that is countably additive.

      Finally, let us assume that is finitely additive and continuous from above at the empty set (impossible event). Taking again a sequence of disjoint events let . We have and . By finite additivity, we obtain

(2.9)

Since (2.9) holds for every , we can write

      Again, by the definition of series and the assumption that , is countably additive, and the proof is complete.

      As an illustration, we now prove the following theorem:

      

      Theorem 2.6.2 (First Borel–Cantelli Lemma) If is a sequence of events such that

(2.10)

       then

Proof: Recall (1.7) from Chapter 1, where = “infinitely many events occur” = (because the unions form a decreasing sequence). Consequently, using the continuity of , subadditivity property (2.2), and assumption (2.10), we have

      Paraphrasing the assertion of the lemma, if probabilities of events decrease to zero fast enough to make the series converge, then with probability 1 only finitely many among events will occur. We will prove the converse (under an additional assumption), known as the second Borel–Cantelli lemma, in Chapter 4.

      In the remainder of this section, we will discuss some theoretical issues related to defining probability in practical situations. Let us start with the observation that the analysis above should leave some more perceptive readers disturbed. Clearly, one should not write a formula without being certain that it is well defined. In particular, when writing two things ought to be certain: (1) that what appears in the parentheses is a legitimate object of probability, that is, an event and (2) that the function is defined unambiguously at this event.

      With regard to the first point, the situation is rather simple. All reasonable questions concern events such as