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Probability Theory 1 Math a 9343 Chapter 1 Introductory Measure Theory (Lecture 01)

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Probability Theory 1 Math a 9343 Chapter 1 Introductory Measure Theory (Lecture 01) Probability Theory Basics from Measure Theory Probability Theory 1 Math A 9343 Chapter 1 Introductory Measure Theory (Lecture 01) Mohamed I. Riffi Department of Mathematics Islamic University of Gaza September 16, 2015 Mohamed I. Riffi Probability Theory 1 Math A 9343 2 Basics from Measure Theory Sets Collections of Sets Generators Probability Theory Basics from Measure Theory Outline 1 Probability Theory: An Introduction Mohamed I. Riffi Probability Theory 1 Math A 9343 Sets Collections of Sets Generators Probability Theory Basics from Measure Theory Outline 1 Probability Theory: An Introduction 2 Basics from Measure Theory Mohamed I. Riffi Probability Theory 1 Math A 9343 Collections of Sets Generators Probability Theory Basics from Measure Theory Outline 1 Probability Theory: An Introduction 2 Basics from Measure Theory Sets Mohamed I. Riffi Probability Theory 1 Math A 9343 Generators Probability Theory Basics from Measure Theory Outline 1 Probability Theory: An Introduction 2 Basics from Measure Theory Sets Collections of Sets Mohamed I. Riffi Probability Theory 1 Math A 9343 Probability Theory Basics from Measure Theory Outline 1 Probability Theory: An Introduction 2 Basics from Measure Theory Sets Collections of Sets Generators Mohamed I. Riffi Probability Theory 1 Math A 9343 Probability Theory Basics from Measure Theory Introduction Probability Space Let us introduce the probability space or probability triple (Ω; F; P), where Ω is the sample space; F is the collection of events; P is a probability measure. Mohamed I. Riffi Probability Theory 1 Math A 9343 Sets Probability Theory Collections of Sets Basics from Measure Theory Generators Set Operations Set Operations Let A; A1; A2;::: and B; B1; B2;::: be sets. Union: A [ B = fx : x 2 A or x 2 Bg; Intersection: A \ B = fx : x 2 A and x 2 Bg; Complement: Ac = fx : x 62 Ag; c Difference: A r B = A \ B ; Symmetric difference: A 4 B = (AB) [ (B r A). n n Also we use [k=1Ak for unions and \k=1Ak intersections. Mohamed I. Riffi Probability Theory 1 Math A 9343 Sets Probability Theory Collections of Sets Basics from Measure Theory Generators Some Additional Terminology Some Additional Terminology the empty set: φ; subset: A is a subset of B, A ⊂ B, if x 2 A ) x 2 B; disjoint: A and B are disjoint if A \ B = φ; power set: P(Ω) = fA : A ⊂ Ωg; fAn; n ≥ 1g is non-decreasing, An %, if A1 ⊂ A2 ⊂ · · · ; fAn; n ≥ 1g is non-increasing, An &, if A1 ⊃ A2 ⊃ · · · . Mohamed I. Riffi Probability Theory 1 Math A 9343 Sets Probability Theory Collections of Sets Basics from Measure Theory Generators The De Morgan Formulas The De Morgan Formulas n !c n [ \ c Ak = Ak k=1 k=1 n !c n \ [ c Ak = Ak k=1 k=1 Mohamed I. Riffi Probability Theory 1 Math A 9343 Sets Probability Theory Collections of Sets Basics from Measure Theory Generators Limits of Sets Limits of Sets Definition (2.1) Let fAn; n ≥ 1g be a sequence of subsets of Ω. We define 1 1 [ \ A∗ = lim inf An = Am n!1 n=1 m=n 1 1 ∗ \ [ A = lim sup An = Am n!1 n=1 m=n Mohamed I. Riffi Probability Theory 1 Math A 9343 Sets Probability Theory Collections of Sets Basics from Measure Theory Generators Proposition Proposition (2.1) Let fAn; n ≥ 1g be a sequence of subsets of Ω. (i) If A1 ⊂ A2 ⊂ A3 ··· , then 1 [ lim An = An: n!1 n=1 (ii) If A1 ⊃ A2 ⊃ A3 ··· , then 1 \ lim An = An: n!1 n=1 Mohamed I. Riffi Probability Theory 1 Math A 9343 Definition ∗ ∗ If the sets A∗ and A agree, then A = A∗ = A = limn!1 An and we say A = lim An = lim sup An = lim inf An; in other words, we say that An converges to A and write An ! A. Sets Probability Theory Collections of Sets Basics from Measure Theory Generators Proof of Proposition 2.1. Exercise 2.2 Mohamed I. Riffi Probability Theory 1 Math A 9343 Sets Probability Theory Collections of Sets Basics from Measure Theory Generators Proof of Proposition 2.1. Exercise 2.2 Definition ∗ ∗ If the sets A∗ and A agree, then A = A∗ = A = limn!1 An and we say A = lim An = lim sup An = lim inf An; in other words, we say that An converges to A and write An ! A. Mohamed I. Riffi Probability Theory 1 Math A 9343 Sets Probability Theory Collections of Sets Basics from Measure Theory Generators Example Let's actually go from what we know about the real numbers R. What's an example where we have the lim sup and lim inf are different? How about the sequence: 1 (−1)n+1 a = + n 2 2 Then the lim sup an = 1 and the lim inf an = 0. Mohamed I. Riffi Probability Theory 1 Math A 9343 Sets Probability Theory Collections of Sets Basics from Measure Theory Generators How can we use this to make a simple example for the case of sets? Example Let A1 = A3 = A5 = ··· = A and A2 = A4 = A6 = ··· = B where B is a proper subset of A. Then lim sup An = A and lim inf An = B. Mohamed I. Riffi Probability Theory 1 Math A 9343 Sets Probability Theory Collections of Sets Basics from Measure Theory Generators Set Relations Let A be a non-empty collection of subsets of Ω, and consider the following set relations: (a) A 2 A ) Ac 2 A; (b) A; B 2 A ) A [ B 2 A; (c) A; B 2 A ) A \ B 2 A; (d) A; B 2 A; B ⊂ A =) A n B 2 A; S1 (e) A1; A2;::: 2 A =) n=1 An 2 A; S1 (f) An 2 A; n ≥ 1; Ai \ Aj = φ, i 6= j =) n=1 An 2 A; T1 (g) An 2 A; n ≥ 1; =) n=1 An 2 A; S1 (h) An 2 A; n ≥ 1; An % =) n=1 An 2 A; T1 (i) An 2 A; n ≥ 1; An & =) n=1 An 2 A. Mohamed I. Riffi Probability Theory 1 Math A 9343 Sets Probability Theory Collections of Sets Basics from Measure Theory Generators Definition (2.2) A collection F of events is called an algebra or a field if Ω 2 A and (a) It is closed under complementation: A 2 F ) Ac 2 F (b) It is closed under taking finite union: A; B 2 F ) A [ B 2 F: Mohamed I. Riffi Probability Theory 1 Math A 9343 Sets Probability Theory Collections of Sets Basics from Measure Theory Generators Corollary It follows from these two conditions that algebras are also closed under finite intersections: A; B 2 F ) A \ B 2 F: Proof. De Morgan's Law states (A \ B)c = Ac [ Bc . Thus, A \ B = (Ac [ Bc )c Mohamed I. Riffi Probability Theory 1 Math A 9343 Sets Probability Theory Collections of Sets Basics from Measure Theory Generators Example Let F = f;; Ωg. This is the trivial algebra. In some sense, it is the \poorest" algebra. At the opposite extreme let F = P(Ω) the power set of Ω. This is the \richest" algebra. The true life is between these extremes. Mohamed I. Riffi Probability Theory 1 Math A 9343 Sets Probability Theory Collections of Sets Basics from Measure Theory Generators Definition (2.2) A collection F of events is called a σ-algebra or a σ-field if Ω 2 A and (a) It is closed under complementation: A 2 F ) Ac 2 F (e) It is closed under taking countable unions: 1 [ A1; A2;::: 2 F =) An 2 F: n=1 Mohamed I. Riffi Probability Theory 1 Math A 9343 Sets Probability Theory Collections of Sets Basics from Measure Theory Generators Discrete Probabilities You should not worry too much about σ-algebras when working in discrete probability. Then, you can just take the power set for your algebra. Mohamed I. Riffi Probability Theory 1 Math A 9343 Sets Probability Theory Collections of Sets Basics from Measure Theory Generators Definition Suppose F is a σ-algebra on Ω. Let A1; A2;::: ⊆ R. Then 1 1 \ [ lim sup An := Ak n=1 k=n = f! 2 Ω: 8 n 9 k ≥ n such that ! 2 Ak g = f! 2 Ω: ! belongs to infinitely many An'sg = “Infinitely many events occur": Mohamed I. Riffi Probability Theory 1 Math A 9343 Sets Probability Theory Collections of Sets Basics from Measure Theory Generators Definition Suppose F is a σ-algebra on Ω. Let A1; A2;::: ⊆ R. Then 1 1 [ \ lim inf An := Ak n=1 k=n = f! 2 Ω: 9 n such that 8 k ≥ n, ! 2 Ak g = f! 2 Ω: ! belongs to all except finitely many An'sg = \All except finitely many events An occur": Mohamed I. Riffi Probability Theory 1 Math A 9343 Solution From now on, whenever I say that A is an event, I implicitly mean that A 2 F. Sets Probability Theory Collections of Sets Basics from Measure Theory Generators Question Is A in the σ-algebra? Mohamed I. Riffi Probability Theory 1 Math A 9343 Sets Probability Theory Collections of Sets Basics from Measure Theory Generators Question Is A in the σ-algebra? Solution From now on, whenever I say that A is an event, I implicitly mean that A 2 F. Mohamed I. Riffi Probability Theory 1 Math A 9343 Solution This turns out to always be true, as long as Ω 6= ;. Indeed, take fag \ fagc = ; 2 F. Sets Probability Theory Collections of Sets Basics from Measure Theory Generators Question Are we assuming Ω 2 F? Mohamed I.
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