Unformatted text preview: C of [0 ; 1] such that (i) ² ( C ) = ± and (ii) C contains no nonempty 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 nonempty open set E we have ² ( E \ S ) > and ² ( E \ S c ) > . 1...
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 Fall '10
 GREGORYMARGULIS
 Math, Sets, Empty set, Topological space, measure, Lebesgue measure, outer measure, Lebesgue integration

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