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Math320-Hw2

Math320-Hw2 - C of[0 1 such that(i ² C = ± and(ii C...

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Math 320 Measure Theory and Integration Assignment 2: Measures, Outer Measures, and Weird Sets The due date for this assignment is Thursday 9/16/2010. 1. Let ( X; A ) be a measure space and f A i g a sequence of A -measurable sets such that 1 X i =1 ( A i ) < 1 . (A) Use [ and \ A i . (B) Show that the set of points that belong to A k k has measure zero under . 2. Let X be a set and be an outer measure on 2 X . If E 2 X measure and there exists a & -measurable A E such that ( E ) = ( A ) ; show that E is & -measurable. 3. Let X be a set and be an outer measure on 2 X . Suppose for each E X , there exists a & -measurable set A such that E A and ( E ) = ( A ) . Show that is continous from above. In other words, if C 1 C 2 ::: X , then ( [ C i ) = lim i !1 ( C i ) . (As shown in class, this is not true in general for outer measures.) 4. (1) For each ± 2 (0 ; 1) , show that there is a closed set
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Unformatted text preview: C of [0 ; 1] such that (i) ² ( C ) = ± and (ii) C contains no non-empty open sets. (2) For each ± 2 (0 ; 1) , show that there exists an open set O & R such that O is dense in R and ² ( O ) < ± . (3) Use (2) (or otherwise) show that there exists S & [0 ; 1] such that ² ( S ) = 1 and S = [ 1 i =1 S i such that & & S i ± o = ; for i = 1 ; 2 ; 3 ::: , where & S stands for the closure of S and () o stands for the interior. (You may use the fact that & S has empty interior i/ & & S ± c is dense. Recall also the Baire Category Theorem.) 5. Show (using Problem 4 or otherwise) that there exists a Lebesgue measurable set S such that for every non-empty open set E we have ² ( E \ S ) > and ² ( E \ S c ) > . 1...
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