HWsolution2

HWsolution2 - Math 447 Solutions to Assignment 2 n=1 Spring...

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Math 447 Solutions to Assignment 2 Spring 2009 Exercise 16.22 on page 273 Let E = S n =1 E n . Show that m * ( E ) = 0 if and only if m * ( E n ) = 0 for every n . Solution Since E n E , the monotonicity property of outer measure [part (ii) of Propo- sition 16.2 on page 269] shows that m * ( E n ) m * ( E ). If the right-hand side m * ( E ) = 0, then m * ( E n ) = 0 for every n . In other words, a subset of a set of measure zero is again a set of measure zero. On the other hand, the countable subadditivity of outer measure (Proposition 16.5 on page 272) implies that m * ( E ) n =1 m * ( E n ). If m * ( E n ) = 0 for every n , then the right-hand side equals 0, so m * ( E ) = 0. (This part of the exercise is Corollary 16.6 on page 272.) Exercise 16.24 on page 273 Given a subset E of R , prove that there is a G δ -set G containing E such that m * ( G ) = m * ( E ). Solution Recall from page 130 that a G δ set means the intersection of a countable number of open sets. By the definition of outer measure, there exists for each positive integer n an open set G n containing E such that m * ( G n ) m * ( E ) + 2 - n . Set G equal to the intersection T n =1 G n . Then G is a set of type G δ . Moreover, E G n for every n , so E G . Since G G n for every n , it follows that m * ( G ) m * ( E ) + 2 - n for every n , so m * ( G ) m * ( E ). By monotonicity of outer measure, m * ( E ) m * ( G ). Combining the preceding two inequalities shows that m * ( G ) = m * ( E ). Thus G has all the required properties. Exercise 16.28 on page 273 Fix α with 0 < α < 1 and repeat our “middle thirds” construction for the Cantor set except that now, at the n th stage, each of the 2 n - 1 open intervals we discard from [0 , 1] is to have length (1 - α )3 - n . (We still want to remove each open interval from the “middle” of a closed interval in the current level — it is important that the closed intervals that remain turn out to be nested.) The limit of this process, a set that we will name Δ α , is called a generalized Cantor set and is very much like the ordinary Cantor set. Note that Δ α is uncountable, compact, nowhere dense, and so on, but has nonzero outer measure. Indeed, check that m *
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HWsolution2 - Math 447 Solutions to Assignment 2 n=1 Spring...

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