MATH INTRODUCTION TO PROBABILITY FALL 2011

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1 MATH INTRODUCTION TO PROBABILITY FALL 2011 Lecture 3. Some theorems about probabilities. Facts belonging to the settheoretic introduction to both probability theory and measure theory. The modern theory of probability deals not so much with events and their probabilities, as with random variables. But before we get to random variables, I have to speak a little more about events and probabilities. I ll formulate and prove some simple theorems. Theorem 3.1. P (A c = 1 P (A. Theorem 3.2. If A B for events A and B, then P (A P (B. These theorems are so simple that I won t give their proofs: they are immediate consequences of Axiom 2 (also Axiom 1 band the equalities Ω = A A c, B = A (B A for A B (disjoint unions. What I want to prove now is that the probability of the limit of a sequence of events is equal to the limit of their probabilities. Except that we don t know what a limit of a sequence of sets is. The limit of a sequence of numbers is not always defined; so it is not surprising that the limit of a sequence of sets also will be defined only in some cases. I ll define it in the following two cases. If B 1 B 2 B 3... B n... is a non-decreasing sequence of sets, I take lim B n = B i (3.1 (please draw a picture illustrating this; and if B 1 B 2 B 3... B n... is a non-increasing sequence of sets, I take lim B n = B i. (3.2 Theorem 3.3. For both a non-decreasing sequence of events B n and for a nonincreasing one P ( lim n = lim n. (3.3 Proof. It is clear that we need to prove the theorem only for, say, non-increasing sequences (if it is proved, the non-decreasing ones are handled by introducing C i = Bi c: this sequence is non-increasing. We have: B 1 = (B 1 B 2 (B 2 B 3... B i ; (3.4 these sets are mutually disjoint (make a picture. So by Axiom 3 P (B 1 = P (B i B i+1 + P ( B i = lim 1 P (B i B i+1 + P ( lim B n. (3.5

2 Using finite additivity of P we get: P (B i B i+1 = P ( n (B i B i+1 = P (B 1 B n+1 = P (B 1 P (B n+1, (3.6 and we get lim P n+1 = P (lim B n, i. e. formula (3.3. We know that a function f : R 1 R 1 is continuous if and only if it follows from lim x n = x 0 that f(lim x n = lim f(x n. So Theorem 3.3 means, in fact, that the probability function P is continuous. Theorem 3.3 is true also for arbitrary measures m if the m-measures of all sets considered are finite. Theorem 3.4. Let A be an algebra of sets in a space X; let m be a finite nonnegative finitely additive function on A (i. e. 0 m(a <, and m ( n A n i = m(a i for disjoint A 1, A 2,..., A n A such that n A i A. Then the function m is countably additive (i. e. satisfies (1 2.3 for disjoint A 1, A 2,..., A n,... A such that A i A if and only if it is continuous at zero : B 1 B 2... B n B n+1..., B n = n=1 lim m(b n = 0. (3.7 Proof. That (3.7 follows from countable additivity is, in fact, Theorem 3.3 for finite measures. Let us prove that countable additivity follows from (3.7. Let A 1, A 2,..., A n,... A, and A i A. We have to prove that m ( A i = m(a i. The B n = n A i is a non-decreasing sequence, the sequence C n = A i n A i is non-increasing, and lim C n = n=1 C n =. So by for7 (continuity at zero we have: lim m( A i n A i = 0. (3.8 By finite additivity the probability under the limit sign is equal to m ( n A i, and we have: [ ( ( n ] ( lim m A i m A i = m A i lim which was to be proved. m ( A i m(a i = 0, (3.9 Now back to probabilities (though the results that follow are true for all measures. The next theorem is a little more complicated than Theorems 3.1, 3.2: Theorem 3.5. For any events A 1, A 2,..., A n,... (disjoint or not we have: P ( A i P (A i. (3.10 2

3 Proof. We represent the non-disjoint union A i as one of disjoint sets: A i = A 1 (A 2 A 1 ( A 3 (A 1 A 2... ( A n (A 1... A n (3.11 So P ( A i = P (A1 + P (A 2 A 1 + P ( A 3 (A 1 A P ( A n (A 1... A n (3.12 By A i (A 1... A i 1 A i and Theorem 3.2, we have P ( A i (A 1... A i 1 P (A i ; from which (3.10 follows. Theorem 3.6 (Borel Cantelli s Lemma. If A 1, A 2,..., A n,... are events such that P (A i <, then almost surely only finitely many of the events A i occur. The statement of the theorem requires a translation. According to our meaning of the expression almost surely, this statement means that P {infinitely many of A i occur} = 0. (3.13 Again this requires a translation into the language of sets and operations on them. Infinitely many means that however large a natural number n we take, for at least one i > n the event A i occurs. So here is the translation: {infinitely many of A i occur} = n=1 A i. (3.14 By Theorem 3.5 we have: P ( A i P (A i; the events B n = A i clearly form a non-increasing sequence, so we have: P ( n=1 A i = lim P ( A i lim P (A i. (3.15 But we know that for every convergent series its tail after the n-th summand goes to 0 as n : which proves (3.13. lim P (A i = lim [ P (A i P (A i ] = 0; (3.16 Still we have a little to do before we start on random variables; but what follows is not probability or measure theory, but something in the common set-theoretic introduction to them. 3

4 Theorem 3.7. For any class A of subsets of a set X there exists the smallest σalgebra C in X that contains A. That is: C is a σ-algebra, C A; for any σ-algebra D A we have: D C. (3.17 It is clear at once that if such a σ-algebra C exists, it is unique; but how to prove its existence? The simplest proof seems to be this: Let us consider all σ-algebras D that contain the class A. At least one such σ-algebra exists: namely, the σ-algebra P(X of all subsets of X (clearly, it contains A, and clearly it is a σ-algebra: all σ-algebra requirements are of the form: (if this or that, then a certain subset of X belongs to P(X; but every subset belongs to it. Then we take C = D. (3.18 D: D is a σ-algebra, and D A This class of sets clearly contains A (because all intersecants contain it; let us prove that it is a σ-algebra. We have to prove: X C; (3.19 A C A c C; (3.20 A 1, A 2,..., A n,... C A i C. (3.21 The first one is obvious: X belongs to each of intersecants D, all of them being σ-algebras. Let us prove (3.20: Suppose a set A belongs to the class C; then A belongs to every class D mentioned in (3.18. Since these classes are σ-algebras, we have also for each of them A c D; and a thing (A c belonging to each of the intersecants belongs to the intersection. In the same way (3.21 is proved: A i belongs to C because it belongs to every of the intersecants D which is because D are σ-algebras. The smallest σ-algebra containing a class of sets A is denoted σ(a. And the following term is used for it: the σ-algebra generated by A. In absolutely the same way the following theorem is proved: Theorem 3.8. For any class A of subsets of a set X there exists the smallest algebra C in X that contains A. The notation for this smallest algebra will be α(a. While we can describe the algebra α(a generated by a class of sets A easily enough (e. g., for A being the class of all intervals, finite or infinite, in the real line R 1 the algebra α(a is the class of all finite unions of intervals this class was mentioned before, 4

5 not so with the σ-algebra σ(a. Say, for the same example of the class of all intervals, is σ(a the class of all countable unions of intervals? No, e..g., it is easy to check that the set Ir of all irrational numbers belongs to the σ-algebra generated by all intervals, but it is not a countable union of intervals. Is σ(a the class of all countable unions of intervals, and complements of such unions? Again no: we can take irrational numbers in one half of the real line, and rational numbers in the other half. Is it the class of all countable unions of intervals, and their complements, and countable intersections of such unions? Again no but it is a little harder to invent a counterexample. Is it the class of all countable unions, and countable intersections thereof, and all countable unions of countable intersections of countable unions, and all countable intersections of countable unions of countable intersections of countable unions, etc.? Still it is no even if we are able to represent much more sets belonging to σ(a in this form. We can go in this direction specifying in a more precise and more elaborate way what etc. should mean but we see it is pretty complicated. So how can we work with σ-algebras generated by classes of sets? We can do so in some indirect ways. 5