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I 0 assume it sends the empty interval to 0 if a r

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˜ ( I ) [0 , ). Assume it sends the empty interval to 0. If A R define μ * ( A ) = inf { X n N ˜ ( I n ) : I 1 , I 2 , . . . open intervals, A [ n =1 ˜ ( I n ) } . (a) (20 points) Prove that μ * is an outer measure; in other words prove that μ * ( ) = 0, A B implies μ * ( A ) μ * ( B ), μ * ( S n =1 A n ) n =1 μ * ( A n ). (b) ? Give an example to show that μ * ( I ) = ˜ ( I ) for open intervals I may fail to hold. (c) ? Let f : R R be increasing and continuous. If I = ( a, b ) is an open interval, define ˜ ( I ) = f ( b ) - f ( a ). Prove: Defining μ * as above, all Borel sets are measurable. Is it necessary for f to be continuous? 4. (10 points) Let A R . Prove that A is measurable if and only if the characteristic function χ A of A is measurable. Here χ A ( x ) = 1 if x A , 0 if x / A . 5. (20 points) Let f n : R R be measurable for n = 1 , 2 , . . . and let D = { x : lim n →∞ f n ( x ) exists. } . Prove: D is measurable. 6. (20 points) Let E be a null set; that is, E R and m ( E ) = 0, where m is Lebesgue measure. Prove that its complement is dense; that is, the closure of R \ E is R .
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