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HWsolution3

HWsolution3 - Math 447 Solutions to Assignment 3 Spring...

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Math 447 Solutions to Assignment 3 Spring 2009 Exercise 16.58 on page 284 Suppose that m * ( E ) < . Prove that E is measurable if and only if, for every ε > 0, there is a finite union of bounded intervals A such that m * ( E 4 A ) < ε (where E 4 A is the symmetric difference of E and A ). Solution First suppose that E is measurable, and fix a positive ε . For each positive integer n , let E n denote the set E ([ n - 1 , n ) [ - n, - n + 1)), which is a measurable set by Lemma 16.14 on page 278. Theorem 16.18 on page 280 implies that X n =1 m * ( E n ) = m * ( E ) < . The convergence of this infinite series of nonnegative terms means that there is a positive integer N such that n = N m * ( E n ) < ε/ 3. Let E 0 denote the bounded measurable set E \ S n = N E n . By part (ii) of Theorem 16.21 on page 283, there is an open set G containing E 0 such that m * ( G \ E 0 ) < ε/ 3. There is no loss of generality in supposing that G is a bounded set [simply intersect G with the interval ( - N, N )]. The open set G is then a countable union of bounded open intervals of finite total length. Split this collection of intervals into two subcollections A and B such that A consists of finitely many intervals, and the intervals in B have total length less than ε/ 3. Points in the symmetric difference E 4 A lie either in S n = N E n or in B or in G \ E 0 . Since each of these three sets has outer measure less than ε/ 3, the subadditivity property of outer measure implies that m * ( E 4 A ) < ε . Conversely, fix a positive ε , and suppose A is a finite union of bounded intervals such that m * ( E 4 A ) < ε . Slightly enlarging each of the intervals produces a finite collection A 0 of bounded open intervals such that m * ( E 4 A 0 ) < 2 ε . In particular, m * ( E \ A 0 ) < 2 ε .

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