Hw3-Math320

Hw3-Math320 - Math 320 Measure Theory and Integration...

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Math 320 Measure Theory and Integration Assignment 3: Measurable Functions and Their Unexpected Properties The due date for this assignment is 9/30 1. Consider the Cantor function : [0 ; 1] ! [0 ; 1] x 2 C , the Cantor set, we can write x = P 1 i =1 x n 3 n with x 0 = 0 or 2 and set ( x ) = 1 X i =1 x n 3 n ! := 1 X i =1 x n 2 1 2 n . Extend ( x ) to [0 ; 1] by ( x ) = sup f ( y ) : y 2 C; y < x g for each x 2 [0 ; 1] n C . (A) Show that is increasing and continuous but 0 = 0 almost everywhere on [0 ; 1] . (B) Show that maps C onto [0 ; 1] . 2. (A) Use problem 1 (or otherwise) show that there exists a Lebesgue measurable function f and a Lebesgue measurable set E such that f ( E ) is not Lebesgue measurable. (Hint: Consider f ( x ) = x + ( x ) .) (B) Show that there exists a Lebesgue measurable function f and a Lebesgue measurable set E such that f 1 ( E ) is not Lebesgue measurable. (C) Show that there exists a Lebesgue measurable set that is not a Borel set. 3. (A) Show that if
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