Probability and Statistical Inference. Robert Bartoszynski

Probability and Statistical Inference

follows:

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

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

(2.8) equation

the events on the right‐hand side being disjoint. Since images (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 images from below. To prove continuity from above, we pass to the complements, and proceed as above.

Let us now assume that images is finitely additive and continuous from below, and let images be a sequence of mutually disjoint events. Put images so that images is a monotone increasing sequence with images . 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 images is countably additive.

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

(2.9) equation

Since (2.9) holds for every images , we can write

Again, by the definition of series and the assumption that images , images 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 images is a sequence of events such that

(2.10) equation

then

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

Paraphrasing the assertion of the lemma, if probabilities of events images decrease to zero fast enough to make the series converge, then with probability 1 only finitely many among events images 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 images 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 images is defined unambiguously at this event.

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