A Probability Path.pdf

# The finite union is 4 arbitrary intersection as in 1

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The finite union is (4) Arbitrary intersection: As in (1) the arbitrary intersection is (5) Countable intersection: As in (2), the countable intersection is 00 nAj. j=l (6) Finite intersection: As in (3), the finite intersection is n nAj. j=l (7) Complementation: If A e C, then A c is the set of points not in A. (8) Monotone limits: If {An} is a monotone sequence of sets inC, the monotone limit lim An n-+oo is A j in case {An} is non-decreasing and is A j if {An} is non- increasing.

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12 1. Sets and Events Definition 1.5.1 (Closure.) Let C be a collection of subsets of n. C is closed under one of the set operations 1-8 listed above if the set obtained by performing the set operation on sets inC yields a set in C. For example, C is closed under (3) if for any finite collection At, ... , An of setsinC,U}=tAi eC. Example 1.5.1 1. Suppose n = IR, and C = finite intervals = {(a,b], -oo b < oo}. C is not closed under finite unions since (1, 2] U (36, 37] is not a finite interval. Cis closed under finite intersections since (a, b] n (c, d] = (a v c, dAb]. Here we use the notationavb = max{a, b} and aAb =min{ a, b}. 2. Suppose n = lR and C consists of the open subsets of JR. Then C is not closed under complementation since the complement of an open set is not open. Why do we need the notion of closure? A probability model has an event space. This is the class of subsets of n to which we know how to assign probabilities. In general, we cannot assign probabilities to all subsets, so we need to distinguish a class of subsets that have assigned probabilities. The subsets of this class are called events. We combine and manipulate events to make more complex events via set operations. We need to be sure we can still assign probabilities to the re- sults of the set operations. We do this by postulating that allowable set operations applied to events yield events; that is, we require that certain set operations do not carry events outside the event space. This is the idea behind closure. Definition 1.5.2 Afield is a non-empty class of subsets of n closed under finite union, finite intersection and complements. A synonym for field is algebra. A minimal set of postulates for A to be a field is (i) n eA. (ii) A e A implies A c e A. (iii) A, Be A implies AU BE A. Note if At, Az, A3 e A, then from (iii) At U Az U A3 = (At U Az) U A3 e A
1.5 Set Operations and Closure 13 and similarly if A1, ... , An E A, then U7= 1 A; E A. Also if A; E A, i = 1, ... , n, then n7= 1 A; e A since A; E A implies Af E A (from (ii)) n eA I implies UAfeA (from (iii)) i=l n (uAfr E A UAf implies (from (ii)) i=l 1=1 and finally (VAfr =0A• by de Morgan's laws so A is closed under finite intersections. Definition 1.5.3 A a-field B is a non-empty class of subsets of Q closed under countable union, countable intersection and complements. A synonym for a-field is a -algebra. A mimimal set of postulates forB to be a a-field is (i) Q E B.

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