A Probability Path.pdf

# Show that a c is the smallest class containing c

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Show that a (C) is the smallest class containing C which is closed under the formation of countable unions and intersections. 30. Let 8; be a-fields of subsets of Q fori = 1, 2. Show that the a-field 8t v 82 defined to be the smallest a-field containing both 8t and 82 is generated by sets of the form Bt n B2 where B; e 8; fori= 1, 2.

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1.9 Exercises 25 31. Suppose n is uncountable and let g be the a-field consisting of sets A such that either A is countable or A c is countable. Show g is NOT countably generated. (Hint: If 9 were countably generated, it would be generated by a countable collection of one point sets. ) In fact, if g is the a-field of subsets of n consisting of the countable and co-countable sets, g is countably generated iff Q is countable. 32. Suppose B1, B2 are a -fields of subsets of n such that B1 c Bz and B2 is countably generated. Show by example that it is not necessarily true that B1 is countably generated. 3 3. The extended real line. Let i = lR U {- oo} U { oo} be the extended or closed real line with the points -oo and oo added. The Borel sets B(JR) is the a-field generated by the sets [-oo,x],x E JR, where [-oo,x] = { -oo}U( -oo, x] . Show B(JR) is also generated by the following collections of sets: (i) [ -00, X), X E JR, (ii) (x, oo], x E JR, (ii) all finite intervals and { -oo} and { oo} . Now think of i = [ -oo, oo] as homeomorphic in the topological sense to [ -1, 1] under the transformation X X t-+ -- 1-lxl from [ -1, 1] to [ -oo, oo]. (This transformation is designed to stretch the finite interval onto the infinite interval.) Consider the usual topology on [ -1, 1] and map it onto a topology on [ -oo, oo] . This defines a collection of open sets on [ -oo, oo] and these open sets can be used to generate a Borel a-field. How does this a-field compare with B(JR) described above? 34. Suppose B is a a-field of subsets of n and suppose A ¢ B. Show that a(B U {A}), the smallest a-field containing both Band A consists of sets of the form ABu ACB', B, B' E B. 35. A a-field cannot be countably infinite. Its cardinality is either finite or at least that of the continuum. 36. Let n = {f, a, n, g}, and C = {{f, a, n}, {a, n}}. Find a(C). 37. Suppose n = Z, the natural numbers. Define for integer k kZ = {kz : z E Z}. Find B(C) when C is
26 1. Sets and Events (i) {3Z}. (ii) {3Z, 4Z}. (iii) {3Z, 4Z, 5Z}. (iv) {3Z, 4Z, 5Z, 6Z} . 38. Let n = IR 00 , the space of all sequences of the fonn (**) where Xi e R Let a be a pennutation of 1, . .. , n; that is , a is a 1-1 and onto map of {1, ... , n} {1, . . . , n}. If w is the sequence defined in(**), define a w to be the new sequence ( ) I Xu(j)• aw j = Xj, if j n, if j > n. A finite permutation is of the fonn a for some n; that is, it juggles a finite initial segment of all positive integers. A set A c Q is permutable if A= a A:= {aw: wE A} for all finite pennutations a . ( i) Let Bn , n 2: 1 be a sequence of subsets of R Show that and n {w = (xi, xz , . . . ) : L Xi E Bn i.o. } i=l n {w = (XJ,X2, .. . ) : V Xi E Bn i.o.} i=l are pennutable.

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