Math320-Hw2-Solutions

Math320-Hw2-Solutions - Math 320 Measure Theory and...

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Unformatted text preview: 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 \ to write the set of points which belong to in&nitely many A i . (B) Show that the set of points that belong to A k for in&nitely many values of k has measure zero under & . 2. Let X be a set and & & be an outer measure on 2 X . If E & 2 X has &nite & & 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 C of [0 ; 1] such that (i) ( 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 ( O ) < . (3) Use (2) (or otherwise) show that there exists S & [0 ; 1] such that ( 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 set5....
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Math320-Hw2-Solutions - Math 320 Measure Theory and...

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