Bruce K. Driver Math 280 (Probability Theory) Lecture Notes November 2, 2006 File:prob.tex Contents Part I Background Material 1 Limsups, Liminfs and Extended Limits .................................................................................................... 3 2 Basic Probabilistic Notions ................................................................................................................ 7 Part II Formal Development 3 Preliminaries .............................................................................................................................. 13 3.1 SetOperations.......................................................................................................................... 13 3.2 Exercises............................................................................................................................... 15 3.3 Algebraic sub-structures of sets . 15 4 Finitely Additive Measures ................................................................................................................ 19 4.1 Finitely Additive Measures . 19 4.2 Examples of Measures . 20 4.3 SimpleIntegration....................................................................................................................... 22 4.4 Simple Independence and the Weak Law of Large Numbers . 24 4.5 Constructing Finitely Additive Measures . 26 5 Countably Additive Measures ............................................................................................................. 29 5.1 Distribution Function for Probability Measures on (R, BR).................................................................................... 29 5.2 Construction of Premeasures . 29 5.3 Regularity and Uniqueness Results . 31 5.4 Construction of Measures . 32 5.5 Completions of Measure Spaces . 34 5.6 A Baby Version of Kolmogorov’s Extension Theorem . 35 4 Contents 6 Random Variables ......................................................................................................................... 37 6.1 Measurable Functions . 37 7 Independence .............................................................................................................................. 43 7.1 π – λ and Monotone Class Theorems . 43 7.1.1 The Monotone Class Theorem . 44 7.2 Basic Properties of Independence . 46 7.2.1 An Example of Ranks . 50 7.3 Borel-Cantelli Lemmas . 51 7.4 Kolmogorov and Hewitt-Savage Zero-One Laws. 55 8 Integration Theory ........................................................................................................................ 57 8.1 A Quick Introduction to Lebesgue Integration Theory . 57 8.2 Integrals of positive functions . 60 8.3 Integrals of Complex Valued Functions . 63 8.4 Densities and Change of Variables Theorems . 67 8.5 Measurability on Complete Measure Spaces . 70 8.6 Comparison of the Lebesgue and the Riemann Integral . 70 8.7 Exercises............................................................................................................................... 72 8.7.1 Laws of Large Numbers Exercises . 73 Page: 4 job: prob macro: svmonob.cls date/time: 2-Nov-2006/14:21 Part I Background Material 1 Limsups, Liminfs and Extended Limits ¯ −α Notation 1.1 The extended real numbers is the set R := R∪ {±∞} , i.e. it and bn = n with α > 0 shows the necessity for assuming right hand side of is R with two new points called ∞ and −∞. We use the following conventions, Eq. (1.2) is not of the form ∞ · 0. ±∞ · 0 = 0, ±∞ · a = ±∞ if a ∈ R with a > 0, ±∞ · a = ∓∞ if a ∈ R with Proof. The proofs of items 1. and 2. are left to the reader. a < 0, ±∞ + a = ±∞ for any a ∈ R, ∞ + ∞ = ∞ and −∞ − ∞ = −∞ while Proof of Eq. (1.1). Let a := limn→∞ an and b = limn→∞ bn. Case 1., suppose ¯ ∞ − ∞ is not defined. A sequence an ∈ R is said to converge to ∞ (−∞) if for b = ∞ in which case we must assume a > −∞. In this case, for every M > 0, all M ∈ R there exists m ∈ N such that an ≥ M (an ≤ M) for all n ≥ m. there exists N such that bn ≥ M and an ≥ a − 1 for all n ≥ N and this implies ∞ ∞ ¯ an + bn ≥ M + a − 1 for all n ≥ N. Lemma 1.2. Suppose {an}n=1 and {bn}n=1 are convergent sequences in R, then: Since M is arbitrary it follows that an + bn → ∞ as n → ∞. The cases where 1 b = −∞ or a = ±∞ are handled similarly. Case 2. If a, b ∈ , then for every 1. If an ≤ bn for a.a. n then limn→∞ an ≤ limn→∞ bn. R ε > 0 there exists N ∈ N such that 2. If c ∈ R, limn→∞ (can) = c limn→∞ an. ∞ 3. If {an + bn}n=1 is convergent and |a − an| ≤ ε and |b − bn| ≤ ε for all n ≥ N. lim (an + bn) = lim an + lim bn (1.1) Therefore, n→∞ n→∞ n→∞ |a + b − (an + bn)| = |a − an + b − bn| ≤ |a − an| + |b − bn| ≤ 2ε provided the right side is not of the form ∞ − ∞. ∞ for all n ≥ N. Since n is arbitrary, it follows that limn→∞ (an + bn) = a + b. 4. {anbn}n=1 is convergent and Proof of Eq. (1.2). It will be left to the reader to prove the case where lim an lim (anbn) = lim an · lim bn (1.2) and lim bn exist in . I will only consider the case where a = limn→∞ an 6= 0 n→∞ n→∞ n→∞ R and limn→∞ bn = ∞ here. Let us also suppose that a > 0 (the case a < 0 is a provided the right hand side is not of the for ±∞ · 0 of 0 · (±∞) . handled similarly) and let α := min 2 , 1 . Given any M < ∞, there exists N ∈ N such that an ≥ α and bn ≥ M for all n ≥ N and for this choice of N, Before going to the proof consider the simple example where an = n and anbn ≥ Mα for all n ≥ N. Since α > 0 is fixed and M is arbitrary it follows bn = −αn with α > 0. Then that limn→∞ (anbn) = ∞ as desired. ¯ For any subset Λ ⊂ R, let sup Λ and inf Λ denote the least upper bound and ∞ if α < 1 greatest lower bound of Λ respectively. The convention being that sup Λ = ∞ lim (an + bn) = 0 if α = 1 if ∞ ∈ Λ or Λ is not bounded from above and inf Λ = −∞ if −∞ ∈ Λ or Λ is −∞ if α > 1 not bounded from below. We will also use the conventions that sup ∅ = −∞ and inf ∅ = +∞. while ∞ ¯ lim an + lim bn“ = ”∞ − ∞. Notation 1.3 Suppose that {xn} ⊂ is a sequence of numbers. Then n→∞ n→∞ n=1 R lim inf xn = lim inf{xk : k ≥ n} and (1.3) This shows that the requirement that the right side of Eq. (1.1) is not of form n→∞ n→∞ ∞−∞ is necessary in Lemma 1.2. Similarly by considering the examples an = n lim sup xn = lim sup{xk : k ≥ n}. (1.4) n→∞ 1 n→∞ Here we use “a.a. n” as an abreviation for almost all n. So an ≤ bn a.a. n iff there exists N < ∞ such that an ≤ bn for all n ≥ N. We will also write lim for lim infn→∞ and lim for lim supn→∞ . 4 1 Limsups, Liminfs and Extended Limits Remark 1.4. Notice that if ak := inf{xk : k ≥ n} and bk :=.