e1_07 - ECE 514, Fall 2007 Exam 1: Due 12:30pm in class,...

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ECE 514 , Fall 2007 Exam 1: Due 12:30pm in class, September 27, 2007 Solutions (version: September 25, 2007, 10:55) 75 mins.; Total 50 pts. 1. (12 pts.) a. Consider a sample space Ω . Suppose C 1 and C 2 are collections of subsets of Ω such that C 1 ⊂ C 2 . Show that σ ( C 1 ) σ ( C 2 ) . b. Let C be the collection of all closed intervals on R of the form [ a, b ] where a < b , a, b R . Is σ ( C ) equal to the Borel σ -algebra? Justify your answer fully. Ans.: a. We first claim that C 1 σ ( C 2 ) . This follows from the fact that C 1 ⊂ C 2 (given) and that C 2 σ ( C 2 ) (by definition of σ ( C 2 ) ). With the claim above, we can now show the desired result. First, notice that σ ( C 2 ) is a σ -algebra (by definition), and it also contains C 1 (by argument above). Hence, because σ ( C 1 ) is the smallest such σ -algebra, σ ( C 1 ) must be smaller than σ ( C 2 ) . This means that σ ( C 1 ) σ ( C 2 ) . b. The answer is yes. Let C 2 be the collection of all intervals on R . Now, σ ( C 2 ) is the Borel σ -algebra, by definition. Because C ⊂ C 2 , by part a we have σ ( C ) σ ( C 2 ) . So it remains to show that σ ( C 2 ) σ ( C ) . To show this, it suffices to show that σ ( C ) is a σ -algebra that contains C 2 (the desired result then follows because σ ( C 2 ) is the smallest such σ -algebra). First, it is clear that σ ( C ) is a σ -algebra. To show that it contains C 2 , we let A ∈ C 2 and show that A σ ( C
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e1_07 - ECE 514, Fall 2007 Exam 1: Due 12:30pm in class,...

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