real-analysis-hw3-with-solution-2007-9-25

real-analysis-hw3-with-solution-2007-9-25 - Real Analysis...

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Real Analysis Homework 3, due 2007-10-3 in class Show Your Work to Each Problem 1. (20 points) Let f : R n R be a continuous function. De fi ne the collection of sets P as X = © B R : f 1 ( B ) is measurable ª . Does P form a σ -algebra? If B R is a Borel set, does it follow that f 1 ( B ) is a Borel set? Give your reasons. Solution: Clearly , R P . If E P , by f 1 ( E c ) = ¡ f 1 ( E ) ¢ c (0.1) we know that E c P also. Similarly if { E k } k =1 is a collection of subsets of P , then by f 1 Ã [ k E k ! = [ k f 1 ( E k ) and f 1 Ã \ k E α ! = \ k f 1 ( E α ) (0.2) we know that k E k P and k E α P . Hence P is a σ -algebra. Similarly we can show that the set Λ = © B R : f 1 ( B ) is Borel measurable ª forms a σ -algebra. For any open set O R , the set f 1 ( O ) is also open. Hence f 1 ( O ) Λ . The same for closed set. Hence all open sets and closed sets are contained in Λ . Since Λ is a σ -algebra, if B R is a Borel set, then f 1 ( B ) must also be a Borel set. ¤ 2. (20 points) Do Exercise 12 in p. 48. Hint: You can use Theorem 3.29. Solution: We fi rst show that E 1 × E 2 R 2 is measurable.
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