solutions-1 - Solution to Homework #1, 36-754 27 January...

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Unformatted text preview: Solution to Homework #1, 36-754 27 January 2006 Exercise 1.1 (The product -field answers count- able questions) Let D = S S X S , where the union ranges over all countable sub- sets S of the index set T . For any event D D , whether or not a sample path x D depends on the value of x t at only a countable number of indices t . (a) Show that D is a -field. (b) Show that if A X T , then A X S for some countable subset S of T . Cf. the proof of Theorem 29 in the notes. (a): We must show that (i) T D , (ii) A D T \ A D and (iii) A n D S n A n D for any countable collection of sets A n . (i): Pick S = { t } for any t T , and take the base set to be , i.e, the base set is x T : x t t . Clearly, this set is T . (ii): Fix S . Then for any A X S , T \ A = T \ A Y t T \ S t = ( S \ A ) Y t T \ S t which is in X S ....
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This note was uploaded on 12/20/2011 for the course STAT 36-754 taught by Professor Schalizi during the Spring '06 term at University of Michigan.

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solutions-1 - Solution to Homework #1, 36-754 27 January...

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